Seminars in Mathematical Sciences
Wednesday 17th April - Allen Hart (University of Exeter) - Reservoir computing and Dynamical systems

Speaker:  Allen Hart (University of Exeter)

Date: Wed. 17th April at 3PM

Title: Reservoir computing and Dynamical systems

Abstract: A reservoir computer is a type of a recurrent neural network where most of the weights are random, and only a single layer of outer weights is trained by linear regression. Remarkably, this simple training procedure is sufficiently general that reservoir computers are still capable of universal approximation. Consequently, reservoir computing has become an interesting theoretical approach to machine learning problems involving time series, including time series forecasting and classification, as well as value function approximation in reinforcement learning.

This talk will be focused on analyzing a reservoir computer where the input is obtained from a deterministic or stochastic dynamical system, and considering questions like whether it is possible to forecast the trajectory of the system, learn topological invariants, and with what level of uncertainty?

Monday, 18th March - Prof. Alfred Muller (University of Siegen) - Decisions under uncertainty: sufficient conditions for almost stochastic dominance

Prof. Alfred Muller (University of Siegen)

Monday, 18th March

Title: Decisions under uncertainty: sufficient conditions for almost stochastic dominance

Decision making under risk involves a ranking of distributions, which is typically based on a method for assigning a real number to a distribution using a risk measure, a premium principle or a context of expected utility. As it is typically difficult to assess a concrete risk measure or utility function it is a well established idea to use stochastic dominance rules in form of stochastic orders to compare distributions. However, it is often equally difficult to completely specify a distribution. Therefore, it is an interesting question whether one can derive unambiguous decisions under partial knowledge of the distributions. In this talk we in particular address this question under the condition that we only know the mean and variance of the involved distributions or that we know the marginal distributions but not the copulas in a multivariate context. Under such assumptions we derive sufficient conditions for concepts of almost stochastic dominance that are based on restrictions on marginal utilities. The talk is based on joint work with Marco Scarsini, Ilia Tsetlin and Robert L. Winkler.

Friday 16th February 1-2 PM - Julie Bjørner Søe (University of Copenhagen) - What is the value of the annuity market?

Julie Bjørner Søe (University of Copenhagen)

Friday 16th February 1-2 PM

Title: What is the value of the annuity market?

Abstract: In the decumulation phase of a pension plan, consumption depends on the level of annuitization. We measure the welfare loss of an individual with a demand for annuitization if he has no access to annuitization or, equivalently, does not use such access. Unlike earlier studies of the value of the annuity option, both individuals with and without access to annuitization, respectively, are offered complete flexibility in the consumption/payout profile. In that sense, we assume that the financial institutions (are allowed to) design the best possible products in the two regimes, with and without annuitization. We find for realistic parameters that a patient individual with time-additive preferences loses 22% of wealth upon retirement if not annuitizing. Sensitivity studies show that the relative loss decreases with a higher interest rate, a higher market price of financial risk, a higher market price of mortality risk, more certainty in the lifetime distribution, and a lower elasticity of intertemporal substitution. Further, we analyze a suboptimal bank product based on conditional expected residual lifetime. This is a joint work with Mogens Steffensen.

Wednesday, 7th February 2 PM - Dr. Gholamali Aminian (The Alan Turing Institute) - Mean-field Analysis of Generalization Errors

Speaker: Dr. Gholamali Aminian (The Alan Turing Institute)

Date: Wednesday, 7th February 2PM

Title: Mean-field Analysis of Generalization Errors

Abstract: We propose a novel framework for exploring weak generalization error of algorithms through the lens of differential calculus on the space of probability measures. Specifically, we consider the KL-regularized empirical risk minimization problem and establish generic conditions under which the generalization error convergence rate, when training on a sample of size n, is $O(1/n)$. In the context of supervised learning with a one-hidden layer neural network in the mean-field regime, these conditions are reflected in suitable integrability and regularity assumptions on the loss and activation functions.

13th December 2023 - Dr. Xiaoyang Zhuo (Beijing Institute of Technology) - The Equivalent Expectations Measures Theory: Computing Risk and Returns of Contingent Claim Portfolios

Dr. Xiaoyang Zhuo (Beijing Institute of Technology)

13th December 2023

The Equivalent Expectations Measures Theory: Computing Risk and Returns of Contingent Claim Portfolios



In this talk, we generalize the Equivalent Expectation Measures Theory (Nawalkha and Zhuo, 2022, JF) to obtain analytical solutions of risk measures of contingent claims over a finite horizon date. We show that obtaining these analytical solutions requires construction of parallel universes, which are identical until a given horizon date, and distributionally identical and independent copies of each other after the horizon date. Using this framework, we derive analytical solutions of risk measures, such as variance, covariance, and higher-order moments and co-moments of the returns on equity options and fixed income securities. We also present expected return and risk measures for an affine-jump-diffusion factor pricing model which integrates option pricing and asset pricing in a unified framework.

22nd November 2023 - Prof. Yuliya Mishura (Taras Shevchenko National University of Kyiv) - Fractional regularity and irregularity

Prof. Yuliya Mishura (Taras Shevchenko National University of Kyiv)

22nd November 2023

Fractional regularity and irregularity


Abstract: We consider the following problems related to fractional processes:

-Asymptotic behavior of maximal functionals

-Small ball estimates for Volterra type processes

-Representation results for Gaussian processes in the generalized

-Representation theorems for Holder continuous Gaussian processes
satisfying small ball estimates

15th November 2023 - Prof. Thomas Mikosch (University of Copenhagen) - Extreme value theory for heavy-tailed time series

Prof. Thomas Mikosch (University of Copenhagen)

15th November 2023

Extreme value theory for heavy-tailed time series


Abstract: We will consider regularly varying time series. The name comes from the marginal tails which are of power-law type. Davis and Hsing (1995) and Basrak and Segers (2009) started the analysis of such sequences. They found an accompanying sequence (spectral tail process) which contains the information about the influence of extreme values on the future behaviour of the time series, in particular on extremal clusters. Using the spectral tail process, it is possible to derive limit theory for maxima, sums, point processes... of regularly varying sequences, but also re ned results like precise large deviation probabilities for these structures. In this talk, we will give a short introduction to regularly varying sequences and explain how the aforementioned limit results can be derived.


1st November 2023 - Dr. Francesco Ungolo (UNSW, Sydney) - Dirichlet Process Mixtures for dependence modelling in actuarial applications

Dr. Francesco Ungolo (UNSW, Sydney)

1st November 2023

Dirichlet Process Mixtures for dependence modelling in actuarial applications



Dirichlet Process Mixtures (DPM) are a flexible statistical tool which entails a regularization to non-parametric modelling techniques. In this talk, we focus on the development of regression models for the distribution of dependent time to events, where DPMs are used to account for dependence among these. We consider two common applications in actuarial science: the analysis of competing risk events, and the analysis of dependent lifetimes. 

For the analysis of competing risk events, the joint distribution of the time to events is characterized by a random effect, whose distribution follows a Dirichlet Process, explaining their variability. This entails an additional layer of flexibility of this joint model, whose inference is robust with respect to the misspecification of the distribution of the random effects. The model is analysed in a fully Bayesian setting, yielding a flexible Dirichlet Process Mixture model. The modelling approach is applied to the empirical analysis of the surrending risk in a US life insurance portfolio previously analysed by Milhaud & Dutang (2018).

The analysis of dependent lifetimes, such as husband and wife couples, develops the framework further by considering the effect of couple-specific covariates within the dependence relationship. The Dirichlet Process Mixture-based regression framework is therefore enriched to account simultaneously for both individual as well as group-specific covariates. The approach allows to account for right censoring and left truncation as typical of survival analysis. The model is illustrated to jointly model the lifetime of male-female couples in a portfolio of joint and last survivor annuities of a Canadian life insurer as analysed by Frees et. Al. (1996). 

The models show an improved in-sample and out-of-sample performance compared to traditional approaches assuming independent time to events. Furthermore, these offer additional insights on determinants of the dependence between time to events.



25th October 2023 - Rodrigue Kazzi (Vrije Universiteit Brussel) - Model uncertainty assessment for unimodal right-skewed distributions

Rodrigue Kazzi (Vrije Universiteit Brussel)

Wed. 25th October

Model uncertainty assessment for unimodal right-skewed distributions



Decisions such as setting premiums or capital requirements (Basel IV, Solvency II) are driven by risk measures of the portfolio loss distributions. However, the inherent uncertainty in the adopted model can lead to significant changes in the value of a risk measure. A common way

to assess this uncertainty is to determine the upper and lower risk bounds, that is, the largest and smallest possible values the risk measure can reach over a set of models that satisfy certain distributional assumptions.

The literature has so far offered risk bounds that may be deemed impractical for many actuarial

applications. This impracticality arises because either a limited set of distributional assumptions are considered – leading to overly wide risk bounds – or some assumptions are difficult to trust, rendering the bounds unsuitable for many scenarios of interest.

In this talk, we aim to derive risk bounds encompassing a broader set of distributional assumptions pertinent to actuarial modelling. The assumptions considered regarding the shape of loss distribution include unimodality, right-skewness, and symmetry following a concave transformation (e.g., log transformation or some power transformations) to the loss distribution. We also allow for the inclusion of additional assumptions about the loss distribution, including the moments, the range of potential loss values, moments on the distribution following a concave transformation, quantile-based information (e.g., knowledge of a particular quantile or the interquartile range), and trimmed moments, among others.

While the primary focus of the talk is the risk bounds for the (Range) Value-at-Risk, we

also show how to calculate bounds for several other measures, including expected utilities

and probability inequalities. The new risk bounds allow for a wide range of distributional assumptions to be incorporated, leading to significantly tighter bounds than the ones in the literature. The findings are illustrated using real-world datasets.





18th October 2023 - Deep Learning based algorithm for nonlinear PDEs in finance and gradient descent type algorithm for non-convex stochastic optimization problems with ReLU neural networks

Deep Learning based algorithm for nonlinear PDEs in finance and gradient descent type algorithm for non-convex stochastic optimization problems with ReLU neural networks

Dr. Ariel Neufeld, Nanyang Technological University (NTU)

Wednesday, 18th October 2023


In this talk, we first present a deep-learning based algorithm which can solve nonlinear parabolic PDEs in up to 10’000 dimensions with short run times, and apply it to price high-dimensional financial derivatives under default risk.
Then, we discuss a general problem when training neural networks, namely that it typically involves non-convex stochastic optimization.
To that end, we present TUSLA,  a gradient descent type algorithm (or more precisely : stochastic gradient Langevin dynamics algorithm) for which we can prove that it can solve non-convex stochastic optimization problems involving ReLU neural networks.
This talk is based on joint works with C. Beck, S. Becker, P. Cheridito, A. Jentzen, and  D.-Y. Lim, S. Sabanis, Y. Zhang, respectively.



4th October 2023 - Policy mirror descent for continuous state and action space MDPs: Convergence of the Fisher--Rao gradient flow

Policy mirror descent for continuous state and action space MDPs: Convergence of the Fisher--Rao gradient flow

Dr. David Siska (University of Edinburgh)

Wednesday, 4th October 2023


Mirror descent methods form the basis of some of the most efficient Reinforcement Learning (RL) algorithms. We study policy mirror descent for entropy-regularised, infinite-time-horizon, discounted Markov decision problems (MDPs) formulated on general Polish state and action spaces. The continuous-time version of the mirror descent flow is a Fisher--Rao gradient flow on the space of conditional probability measures.

In this talk, we show that the Fisher--Rao gradient flow converges exponentially (i.e., linearly) towards the optimal solution of the MDP. This is achieved even though the optimization objective is not convex by utilising the structure of the problem revealed by the performance difference lemma. Mathematically, the main difficulty stems from proving the well-posedness of the Fisher--Rao flow and the differentiability of the value function and Bregman divergence along the flow. The dual flow plays a key role in overcoming these difficulties.

We also prove the stability of the flow under perturbations of the gradient, arising, for example, from sampling or approximation of the Q-function. Specifically, we show that the value function converges exponentially fast to the optimal value function up to a bounded accumulated error associated with the perturbation. The stability result can be applied to deduce the convergence of natural policy gradient flows for more practical policy parameterizations. We study one possible application to log-linear policy classes and attain exponential convergence plus an additional error arising from a compatible function approximation.

This is joint work with James-Michael Leahy, Bekzhan Kerimkulov, Lukasz Spruch and Yufei Zhang.


7th June 2023 - Measures of Risk under Uncertainty

Measures of Risk under Uncertainty

Seminar with Dr. Fadina Tolulope (University of Essex)

Wednesday, 7th June 2023


A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure and present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as Expected Shortfall over Value-at-Risk, a relevant issue for risk management practice.


31st May 2023 - Signature-based models: theory and calibration

Signature-based models: theory and calibration

Seminar with Dr. Sara Svaluto-Ferro (University of Verona)

Wednesday, 31st May 2023


Universal classes of dynamic processes based on neural networks and signature methods have recently entered the area of stochastic modeling and Mathematical Finance. This has opened the door to robust and more data-driven model selection mechanisms, while first principles like no arbitrage still apply. 

In the first part of the talk we focus on signature SDEs whose characteristics are linear functions of a primary underlying process, which can range from a (market-inferred) Brownian motion to a general multidimensional tractable stochastic process. The framework is universal in the sense that any classical model can be approximated arbitrarily well and that the model characteristics can be learned from all sources of available data by simple methods. Indeed, we derive formulas for the expected signature in terms of the expected signature of the primary underlying process.

In the second part we focus on  a stochastic volatility model where the dynamics of the volatility are described by linear functions of the (time extended) signature of a primary underlying process. Under the additional assumption that this primary process is of polynomial type, we obtain closed form expressions for the squared VIX by exploiting the fact that the truncated signature of a polynomial process is again a polynomial process. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log-price and the squared VIX as linear functions of the signature of the corresponding augmented process. For both SPX and VIX options we obtain highly accurate calibration results.

The talk is based on joint works with Christa Cuchiero, Guido Gazzani, and Janka Möller.


17th May 2023 - Optimal voluntary disclosure with reputational benefits of silence

Optimal voluntary disclosure with reputational benefits of silence

Seminar with Adam Ostaszewsi

Wednesday, 17th May 2023


In a continuous-time setting we investigate how the management of a firm controls a dynamic choice between two generic voluntary disclosure decision rules (strategies): one a full and transparent disclosure referred to as candid, the other, referred to as sparing, under which items only above a dynamic threshold value are disclosed. We show how management are rewarded with a reputational premium for being candid.The candid strategy is, however, costly because the alternative of sparing behaviour shields from a downgrade in disclosed low values. We show how parameters of the model such as news intensity, pay-for-performance and time-to-mandatory-disclosure determine the optimal choice of candid versus sparing strategies and optimal times for management to switch between the two.

The private news updates received by management are modelled following a Poisson arrival process, occurring between the fixed (known) mandatory disclosure dates, such as fiscal years or quarters, with the news received by management generated by a background Black-Scholes model of economic activity and of its partial observation. The model presented develops a number of insights, based on a very simple ordinary differential equation (ODE) characterizing equilibrium in a piecewise-deterministic model, derivable from the background Black-Scholes model. When an at-most-single switching policy is assumed admissible, it is shown that a firm either employs a candid disclosure strategy throughout, or switches (alternates) between being candid and being sparing with the truth, in whatever order characterizes optimality, a feature confirmed in empirical findings.

3rd May 2023 - Mathematical foundations of dynamic learning based on reservoir computing

Mathematical foundations of dynamic learning based on reservoir computing

Seminar with Lukas Gonon

Wednesday, 3rd May 2023


In this talk we provide an introduction to reservoir computing and present our recent results on its mathematical foundations. Motivated by their performance in applications  -- ranging from realized volatility forecasting to chaotic dynamical systems -- we study approximation and learning based on random recurrent neural networks and more general reservoir computing systems.

We provide approximation and generalization error bounds for a novel class of infinite-dimensional reservoir systems. These are closely related to the random features method, which we will also discuss and illustrate by an application on learning exponential Lévy models.

The talk is based on joint works with Lyudmila Grigoryeva and Juan-Pablo Ortega.

19th April 2023 - Asymptotic Analysis of Deep Residual Networks and Global Convergence of Gradient Descent Methods

Asymptotic Analysis of Deep Residual Networks and Global Convergence of Gradient Descent Methods

Seminar with Renyuan Xu University of Southern California

Wednesday, 19th April 2023

Abstract: Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, we prove the existence of an alternative ODE limit, a stochastic differential equation, or neither of these. For each case, we also derive the limit of the backpropagation dynamics and address its adaptiveness issue. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

When the gradient descent method is applied to the training of ResNets, we prove that it converges linearly to a global minimum if the network is sufficiently deep and the initialization is sufficiently small. In addition, the global minimum found by the gradient descent method has finite quadratic variation without using any regularization in the training. This confirms existing empirical results that the gradient descent method enjoys an implicit regularization property and is capable of generalizing to unseen data.

This is based on a few papers with Rama Cont (Oxford), Alain Rossier (Oxford), and Alain-Sam Cohen (InstaDeep).

22nd March 2023 - Positively weighted kernel quadrature and a refined analysis of Nyström approximation

Positively weighted kernel quadrature and a refined analysis of Nyström approximation

Seminar with Satoshi Hayakawa University of OxfordI, AIMS Ghana

Wednesday, 22nd March

Abstract: We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d. samples from the underlying measure and evaluation of the kernel and that result in a small worst-case error. In addition to our theoretical results and the benefits resulting from convex weights, our experiments indicate that this construction can compete with the optimal bounds in well-known examples.

Furthermore, we will talk about a refined error analysis of Nyström approximation and a consequent theoretical guarantee of the proposed kernel quadrature, which improves the results already published in NeurIPS 2022. This is joint work with Harald Oberhauser and Terry Lyons.

8th March 2023 - Space-grid approximations of hybrid stochastic differential equations and their ruin probabilities

Space-grid approximations of hybrid stochastic differential equations and their ruin probabilities

Seminar with Oscar Peralta (Cornell University) Wednesday 8th March at 3PM via Zoom

Abstract: Hybrid stochastic differential equations are a useful tool for modeling continuously varying stochastic systems which are modulated by a random environment that may depend on the system state itself. In this talk, we establish the pathwise convergence of the solutions to hybrid stochastic differential equations through space-grid discretizations. While time-grid discretizations are a classical approach for simulation purposes, our space-grid discretization provides a link with multi-regime Markov modulated Brownian motions, leading to computational tractability. We exploit our convergence result to obtain efficient approximations for the ruin times and expected occupation times of the solutions of hybrid stochastic differential equations used to model risk reserve processes, results that are the first of their kind for such a robust framework.

8th February 2023 - Rough volatility: fact or artefact?

Rough volatility: fact or artefact?

Seminar with Purba Das Wednesday 8th February at 3pm.

Abstract: We investigate the statistical evidence for the use of `rough' fractional processes with Hurst exponent $H< 0.5$ for the modelling of volatility of financial assets, using a model-free approach.

Rough volatility fact or artefact (slides)

We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized $p$-th variation along a sequence of partitions. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional processes.   We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than $0.5$.

Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the roughness of the spot volatility process, realized volatility always exhibits `rough' behaviour with an apparent Hurst index $\widehat{H}<0.5$.

These results suggest that the origin of the roughness observed in realized volatility time-series lies in the estimation error rather than the volatility process itself.

This is joint work with Rama Cont.

7th December 2022 - Weak error rates of numerical schemes for rough volatility

Speaker: Paul Gassiat

Weak error rates of numerical schemes for rough volatility

Simulation of rough volatility models involves discretization of stochastic integrals where the integrand is a function of a (correlated) fractional Brownian motion of Hurst index H∈(0,1/2). We obtain results on the rate of convergence for the weak error of such approximations, in the special cases when either the integrand is the fBm itself, or the test function is cubic. Our result states that the convergence is of order (3H+1/2)∧1 for the Cholesky scheme, and of order H+12 for the hybrid scheme with well-chosen weights.

23rd November 2022 - A short journey from the mean value theorem to the Stein-type covariance identities

Speaker: Georgios Psarrakos

Abstract: In this talk, a method for the construction of Stein-type covariance identities for a nonnegative continuous random variable is proposed, by using a probabilistic analogue of the mean value theorem and weighted distributions. A generalized covariance identity is obtained, and applications focused on actuarial and financial science are provided. Some characterization results for gamma and Pareto distributions are also given. Identities for risk measures which have a covariance representation are obtained; these measures are connected with the Bonferroni, De Vergottini, Gini and Wang indices. Moreover, under some assumptions, an identity for the variance of a function of a random variable is derived, and its performance is discussed with respect to well known upper and lower bounds.


[1] Cacoullos, T. (1982). On upper and lower bounds for the variance of a function of a random variable. Annals of Probability 10, 799-809.

[2] Di Crescenzo, A. (1999). A probabilistic analogue of the mean value theorem and its applications to reliability. Journal of Applied Probability   36, 706-719.

[3] Furman, E. and Zitikis, R. (2008). Weighted premium calculation principles.

Insurance: Mathematics and Economics 42, 459-465.

[4] Landsman, Z. and Valdez, E.A. (2016). The tail Stein's identity with applications to risk measures. North American Actuarial Journal 20, 313-326.

[5] Psarrakos, G. (2022). How a probabilistic analogue of the mean value theorem yields stein-type covariance identities. Journal of Applied Probability 59, 350-365.

26th October 2022 - The stochastic local time-space integration introduced by Eisenbaum in to the case of Brownian sheet

Speaker: Antonie-Marie Bosgo, University of Yaounde I, AIMS Ghana

In this talk, we present the stochastic local time space integration introduced by Eisenbaum in [2] to the case of Brownian sheet. This allows us to prove a generalised Itô formula for Brownian sheet and derive Davie type inequalities (see [1]) for the Brownian sheet. Such estimates are useful to obtain regularity bounds for some averaging operators along Brownian sheet paths. These operators play a key role in the regularisation by noise theory of ordinary differential equation by random functions.

This is talk is based on a recent joint work with Moustapha Dieye and Olivier Menoukeu Pamen.

[1] A. M. Davie. Uniqueness of solutions of stochastic differential equations. International Mathematics Research Notices, Vol. 2007, 2007. [2] N. Eisenbaum. Integration with respect to local time. Potential analysis, 13(4):303–328, 2000.


12th October 2022 - Signature methods in stochastic portfolio theory

Speaker: Dr Christa Cuchiero, University of Vienna

In the context of stochastic portfolio theory we introduce a novel class of portfolios which we call linear path-functional portfolios. These are portfolios which are determined by certain transformations of linear functions of a collections of feature maps that are non-anticipative path functionals of an underlying semimartingale. As main example for such feature maps we consider (random) signature of the (ranked) market weights. We prove that these portfolios are universal in the sense that every continuous  (possibly path-dependent) portfolio function of the market weights can be uniformly approximated by signature portfolios. We also show that signature portfolios can approximate the log-optimal portfolio in several classes of non-Markovian models arbitrarily well and  illustrate numerically that the trained signature portfolios are remarkably close to the theoretical log-optimal portfolios. This applicability to non-Markovian markets makes these portfolios much more general than classical functionally generated portfolios usually considered in stochastic portfolio theory. 

Besides these universality features, the main numerical advantage lies in the fact that several optimization tasks like maximizing expected logarithmic utility or mean-variance optimization within the class of linear path-functional portfolios reduces to a convex quadratic optimization problem, thus making it computationally highly tractable.

We apply our method to real market data and show generic out-performance on out-of-sample data even under transaction costs.

The talk is based on joint work with Janka Möller.


28th June 2022 - On the geometry of filtering finite dimensional forward rate models

Speaker: Paul Eisenberg (Vienna University of Economics and Business)

14:00 Room MATH-106

Abstract: The Heath Jarrow Morton approach for money markets models the dynamics of forward rates directly. These rates describe the expected interest rate for a later point of time. We are interested in finite dimensional models for all these rates. We enhance this approach by adding an unobservable Markov chain to the model which influences the forward rates' dynamics. This leaves the question what is understood as observable. If continuous observations of risk neutral dynamics of the whole forward curve are assumed, then the 'hidden' Markov chain is fully observable as well. However, we like to understand the Markov chain is unobservable. To evade this problem one could (a) consider discrete observations only, (b) observations under the real measure or (c) assume that only parts of the forward curve can be observed. In this talk, we give a discussion of our findings regarding to approach (b) and (c).

28th June 2022 - Some behavioural impacts on optimal dividend strategies

Speaker: Julia Eisenberg (Vienna University of Technology)

14:00 Room MATH-106

Abstract: In this talk, we look at a dividend maximisation problem under a Brownian surplus and a Markov-switching preference rate model. The preference rate can attain two values - a positive and a negative. First, we discuss the optimal dividend payout strategy for the setting with a classical ruin concept - the ruin is declared when the surplus becomes negative. In the second part, the setting will be modified by a Parisian ruin with an exponential delay - the ruin is declared if the process stays negative during an exponentially distributed time interval.

In the first case, the optimal strategy turns out to be of a barrier type, being a finite barrier during the positive rate phases and infinite barrier (no dividends are paid) during the negative phases. We show that the finite barrier is a monotone function of the regime switching intensities. In the case of the Parisian delay, the optimal strategy depends on the relation between the expected income rate and the parameter of the exponential delay. The cases of long, medium and short expected delays have to be considered separately in order to find explicit expressions for the value function and the optimal strategy (remaining of a barrier type for short and medium delays).
If the expected delay is too long, the optimal strategy in the negative state can change from not paying dividends to a band strategy. 

27th April 2022 - Stochastic valuation of pension buy-out prices: the impact of dependence between mortality rates and financial markets

Speaker: Dr. Ayse Arik (Heriot-Watt University)

14:00 Room MATH-106

Abstract: Pension buy-outs are playing a significant role as a pension de-risking strategy. In this study, we investigate pension buy-out prices considering unexpected strong changes in mortality due to, for instance, the outbreak of a deadly pandemic. In particular, we investigate how buy-out premiums react if mortality shocks are accompanied by significant changes in financial markets. We apply jump diffusion models to define different dynamics under a risk-neutral pricing framework. We consider several models under different settings where we assume a strong dependence between mortality rates, short rates and/or asset returns to obtain buy-out prices for a hypothetical fully funded pension scheme. Our numerical  findings show that a downward shock on short rates or an unprecedented fall in asset prices combined with a mortality shock would have a much stronger impact on buy-out prices than upward shocks on mortality rates on their own.

23rd March 2022 - Volatility by jumps

Speaker: Prof.  Laura Ballotta (Bayse Business School)

15:00 via Zoom

Meeting ID: 957 4386 6632
Passcode: Wv#.@4QM

We offer a general framework based on time changed Lévy process for modelling the stochastic joint evolution of stock log-returns and their volatility, which includes risk factors of both diffusive and jump nature, and leverage effects originated by both factors. The proposed setting encompasses a large number of the most commonly used stochastic volatility models, allows for the construction of new potential alternative models, and enables a comparative study of their features in terms of volatility, volatility of volatility and correlation processes. We analyse the performance of these models in terms of calibration and fit of the volatility surface; attention is paid to the role of risk factors and distribution features for the purpose of a robust calibration performance.

9th March 2022 - E-backtesting risk measures

Speaker: Ruodo Wang (University of Waterloo)

15:00 via Zoom

Expected Shortfall (ES) is the most important risk measure in finance and insurance. One of the most challenging tasks in risk modeling practice is to backtest ES forecasts provided by financial institutions, based only on daily realized portfolio losses without imposing specific models. Recently, the notion of e-values has gained attention as potential alternatives to p-values as measures of uncertainty, significance and evidence. We use e-values and e-processes to construct a model-free backtest of ES, which can be naturally generalized to many other risk measures and statistical quantities. This talk is based on on-going joint work with Qiuqi Wang (Waterloo) and Johanna Ziegel (Bern).

23rd February 2022: Classical solutions of the Backward PIDE for Markov Modulated Marked Point Processes and Applications to CAT Bonds

Speaker: Katia Colaneri

15:00 via Zoom

Meeting ID: 942 3943 1818
Passcode: 2sU@Tz!*


The objective of this paper is to give conditions ensuring that the backward partial integro differential equation associated with a multidimensional jump-diffusion with a pure jump component has a unique classical solution; that is the solution is continuous, twice differentiable  in the diffusion component and  differentiable in time. Our proof uses a probabilistic argument  and extends the results of [Pham 1998] to processes with a pure jump component  where the jump intensity is modulated by a diffusion process. This result is particularly useful in some applications to pricing and hedging of financial and actuarial instruments, and we provide an example to pricing of CAT bonds.

9th February 2022: The time to ruin in a perturbed Cramer-Lundberg model

Speaker: Anita Behme

15:00 via Zoom

Meeting ID: 942 3943 1818
Passcode: 2sU@Tz!*


The existence of moments of ruin times in a perturbed Cramer-Lundberg model is governed by the general dynamics of the risk process, i.e. whether it is profitable or non-profitable. Whenever the risk process is profitable, we prove that the k-th moment of the ruin time (conditioned to be finite) exists if and only if the k+1-th moment of the claim sizes exists, thus generalizing a result shown earlier by Delbaen.

Moreover we provide general formulae for integer moments of the ruin time (whenever they exist) in terms of the scale function of the risk process and its derivatives and antiderivatives. From these we derive the asymptotic behaviour of the moments of the ruin time, as the initial capital tends to infinity.

Finally, we provide explicit formulae for the first two moments of the ruin time in some special cases.

6th October 2021: Exact simulation of Lévy subordinators

Speaker: Jia Wei Lim (Brunel University London)

15:00 via Zoom


In this talk, I will present on exact simulation methods of two particular classes of Lévy subordinators. Firstly, I will discuss on a simulation framework for truncated Lévy subordinators, which makes use of the property of the truncated jumps to set up a marked renewal process. Truncated subordinators have applications in the valuation of stochastic perpetuities and as Brownian subordinators. Secondly, I will present a simulation method for Generalised Inverse Gaussian processes, and in particular, the hitting time of Bessel process, which has applications in the pricing of barrier type options.

19th May 2021 : Linear optimal contracts in a Gaussian world

Speaker : Stéphane Villeneuve

15.00pm : via Zoom


Since the famous paper by Holmstrom and Milgrom, it is well known that linear contracts are optimal in a world where the output process is a Brownian motion and the preferences are CARA. In this talk, I will review probabilistic techniques for solving a principal-agent problem to show how true it is. I will extend the explicit linear solution of the principal-agent model in the general framework of a Gaussian process, including for instance fractional Brownian motion. The talk is based on a current project with Eduardo Abi Jaber.

5th May 2021 - Characterization of normal product and tetilla law in the second Wiener and Wigner chaoses.

Speaker : Dario Gasbarra

15.00pm via Zoom

Title : Characterization of normal product and tetilla law in the second Wiener and Wigner chaoses.

Abstract :

In the framework of classical probability, we consider the normal product distribution $F \sim N_1 \times N_2$ where $N_1, N_2$ are two independent standard normal random variables, and in the non-commutative setting of free probability, $F \sim ( S_1 S_2 + S_2 S_1 )/\sqrt{2}$, named as {\it tetilla law} by Deya and Nourdin, where $S_1, S_2$ are freely independent normalized semicircular random variables. We provide novel characterization of $F$ within the second Wiener (Wigner) chaos. More precisely, we show that for any generic element $F$ in the second Wiener (Wigner) chaos with variance one the laws of $F$ and $F_\infty$ match if and only if their moments satisfy $\mu_4 (F)= 9 \, (\mbox{resp. }\varphi(F^4)=2.5)$, and $\mu_{2r}(F)= ((2r-1)!!)^2 \, (\mbox{resp. }\varphi(F^{2r})=\varphi(F^{2r}_\infty ))$ for some $r \ge 3$. The presented work is in collaboration with Ehsan Azmoodeh from the University of Liverpool.

21st April 2021 - Statistical consistent term structures are flat

Speaker : Shijie Xu

15:00pm via Zoom

Title : Statistical consistent term structures are flat


We are interested in practical models for energy futures markets. The full term structure is in principle an infinite dimensional object, but from a practical view finite dimensionality is desired. Finite dimensional models for interest rate markets have essentially been shown to be flat by Filipovic and Teichmann (2004). This, however, does not hold for energy markets. From a practical perspective, it is desirable that a model allows for any volatility coming from an estimate. In this presentation, we show that such term structure models are flat.

17th March 2021 - Optimal trade execution with transient relative price impact and directional views: A 2nd-order variationally approach to a 3-dimensional non-convex free boundary problem

Speaker: Dirk Becherer

15:00pm via Zoom

Title: Optimal trade execution with transient relative price impact and directional views: A 2nd-order variational approach to a 3-dimensional non-convex free boundary problem


We solve the optimal execution problem to trade a large financial asset position within finite time in an illiquid market, where price impact is transient and possibly non-linear (in log-prices), like in models by J.P.Bouchaud, F.Lillo or J.Gatheral. We derive a complete solution for the optimal control problem of finite-fuel type, where mechanical price impact is 1.) multiplicative instead of additive (as in the of seminal articles by Obizhaeva/Wang 2013, Predoiu/Shaikhet/Shreve 2011 or Schied et al. 2010), and 2.) we allow for non-vanishing drift of the fundamental price process, i.e. for directional views about short-term price trends (as in Almgren/Chriss 2000, ch.4). Multiplicative impact in relative percentage terms (see Bertsimas/Lo 1998, ch.3) avoids the possibility of asset prices becoming negative. We prove a complete characterization of the regular three-dimensional free boundary surface which separates the no-/action regions for the non-convex three dimensional singular control problem by a family of characteristic curves, being described explicitly up to the solution of ODEs. While the free boundary description is almost as direct as in Obizhaeva/Wang, our analysis is more demanding by the lack of an apparent convexity structure to exploit. Yet, a key argument to prove global optimality turns out to show at first a local optimality for a candidate free boundary under smooth perturbations by 2nd-order variational arguments.

For the optimal trading application , the results can shed some light on phenomena, like e.g. a) how to profit from directional views (signals) about price trends by optimal (non-trivial) round-trips; b) down- or upward directional views lead to respective front- or back-loading in the optimal execution schedule of (sell) trading strategies; c) optimal trading strategies are qualitatively different and their profitability can depend non-monotonically on the resilience (transience) parameters for the price impact.

(This is joint work with Peter Frentrup.)

10th March 2021 -Doubly Reflected Backward Stochastic Differential Equations in the Predictable Setting

Speaker: Youssef Ouknine

15.00pm via Zoom

Title: Doubly Reflected Backward Stochastic Differential Equations in the Predictable Setting

Abstract: We introduce a specific kind of doubly reflected backward stochastic differential equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki’s condition, we show the existence of the solution (in consideration of the driver’s nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of Itô’s formula due to Gal’chouk (Math USSR Sbornik 40(4): 435–468, 1981) and Lenglart (Lect Notes Math 784:500–546, 1980), we provide a suitable a priori estimates which immediately implies the uniqueness of the solution.

(This is joint work with Siham Bouhadou and Ihsan Arharas)

24th February 2021 - Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime

Speaker: Bo Li

15:00pm via Zoom

Title: Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime


We study marked Hawkes processes with an intensity process which has a non-stationary baseline intensity, a general self-exciting function of event “ages” at each time and marks. The marks are assumed to be conditionally independent given the even times, while the distribution of each mark depends on the event time, that is, time-varying. We first observe an immigration-birth (branching) representation of such a non-stationary marked Hawkes process, and then derive an equivalent representation of the process using the associated conditional inhomogeneous Poisson processes with stochastic intensities. We consider such a Hawkes process in the high intensity regime, where the baseline intensity gets large, while the self-exciting function and distributions of the marks are unscaled, and there is no time-scaling in the scaled Hawkes process. We prove functional law of large numbers and functional central limit theorems (FCLT) for the scaled Hawkes processes in this asymptotic regime. The limits in the FCLT are characterized by continuous Gaussian processes with covariance structures expressed with convolution functionals resulting from the branching representation. We also consider the special cases of multiplicative self-exciting functions, and an indicator type of non-decomposable self-exciting functions (including the cases of “ceasing” and “delayed” reproductions as well as their extensions with varying reproduction rates), and study the properties of the limiting Gaussian processes in these special cases.

10th February 2021 - Peer-to-Peer Risk Sharing with an Application of Flood Risk Pooling

Speaker: Runhuan Feng

15:00 via Zoom


In contrast with classic centralized risk sharing, a novel peer-to-peer risk sharing framework is proposed. The presented framework aims to devise a risk allocation mechanism that is structurally decentralized, Pareto optimal, and mathematically fair. An explicit form for the pool allocation ratio matrix is derived, and convex programming techniques are applied to determine the optimal pooling mechanism in a constrained variance reduction setting. A tiered hierarchical generalization is also constructed to improve computational efficiency. As an illustration, these techniques are applied to a flood risk pooling example. Flood risk is known to be difficult to cover in practice, which contributes to the stagnant development for a private insurance market. It is shown that peer-to-peer risk sharing techniques provide an economically viable alternative to traditional flood policies.

16th December 2020 - Pricing equity-linked life insurance contracts with multiple risk factors by neural networks

Speaker: Karim Barigou

14: 00 via Zoom 

Abstract: This paper considers the pricing of equity-linked life insurance contracts with death and survival benefits in a general model with multiple stochastic risk factors: interest rate, equity, volatility, unsystematic and systematic mortality. We price the equity-linked contracts by assuming that the insurer hedges the risks to reduce the local variance of the net asset value process and requires a compensation for the non-hedgeable part of the liability in the form of an instantaneous standard deviation risk margin. The price can then be expressed as the solution of a system of non-linear partial differential equations. We reformulate the problem as a backward stochastic differential equation with jumps and solve it numerically by the use of efficient neural networks. Sensitivity analysis is performed with respect to initial parameters and an analysis of the accuracy of the approximation of the true price with our neural networks is provided

Click here to view presentation slides

2nd December Discrimination - Free Insurance Pricing

Speaker: Andreas Tsanakas

14:00 via Zoom


Abstract: We consider the following question: given information on individual policyholder characteristics, how can we ensure that insurance prices do not discriminate with respect to protected characteristics, such as gender? We address the issues of direct and indirect discrimination, the latter meaning that we can learn protected characteristics from non-protected ones. We provide rigorous mathematical definitions for direct and indirect discrimination, and we introduce a simple formula for discrimination-free pricing, that avoids both direct and indirect discrimination. Our formula works in any statistical model. We demonstrate its application on a health insurance example, using a state-of-the-art generalized linear model and a neural network regression model. An important conclusion is that discrimination-free pricing in general requires collection of policyholders' discriminatory characteristics, posing potential challenges in relation to policyholder's privacy concerns.

Click here to view presentation slides

Wednesday 25th November : Uncertainty and risk in a geophysical complex system: the invariant distribution of river discharge under stochastic rainfall.

Speaker : Jorge Mario Ramirez  

14:00 via Zoom

Abstract: We consider a paradigmatic example of a complex geophysical system responding to a stochastic input: the rainfall - discharge system on an arbitrary watershed. The goal is to derive interesting results regarding emerging patterns in the flow of uncertainty. This is a problem of great interest given the current levels of climatic uncertainty and is treated with mathematical techniques that are extendable to many diverse contexts in complex systems. Our goal is to study the relationship between the occurrence of large events in the rainfall (input) and those of the river discharge (output). Mathematically, the focus is on the characterization of high-dimensional invariant distributions of a piecewise deterministic Markov process and its relationship with the underlying dynamical model and the probabilistic properties of the input. We derive relationships between moments and tail weights of the input and output distributions, as well as emergent scaling relationships for the invariant distribution in terms of the particular connectivity properties of the system. Finally, in the context of our dynamical model, we outline a new mathematical interpretation of the "return period", which is the basic tool for risk estimation in hydrological systems.

This is joint work with: @Constantinescu, Corina (U. Liverpool) and @Sara Maria Vallejo Bernal (Universidad Nacional de Colombia).

Click here to view presentation slides

Wednesday 4th November - Matching marginals and sums

Speaker : Kais Hamza

Time: 14.00pm via Zoom

Abstract: For a given set of random variables $X_1,\ldots,X_d$ we seek as large a family as possible of random variables $Y_1,\ldots,Y_d$ such that the marginal laws and the laws of the sums match: $Y_i\eqd X_i$ and $\sum_iY_i\eqd\sum_iX_i$. We do so in two distinct settings. 1. Under the assumption that $X_1,\ldots,X_d$ are independent and belong to any of the Meixner classes, we give a full characterisation of the random variables $Y_1,\ldots,Y_d$ and propose a practical construction by means of a finite mean square expansion. 2. When $X_1,\ldots,X_d$ are identically distributed but not necessarily independent, using a symmetry-balancing approach we provide a universal construction with sufficient symmetry to satisfy the more stringent requirement that, for any symmetric function $g$, $g(Y)\eqd g(X)$. The same ideas are shown to extend to the non-identically but ``similarly'' distributed case.

This is joint work with Robert Griffiths.

Wednesday 21st October 2020 - Optimal Dividend Ratcheting Strategy

Speaker: Nora E Muler

Time : 14:00pm via Zoom

Abstract We introduce the problem of finding the optimal strategy to pay out dividend, if the dividends are paid according to a dividend rate that is not allowed to decrease (ratcheting strategies). The optimality criterion here is to maximize the expected value of the aggregate discounted dividend payments up to the time of ruin. In this talk we mainly focus on the diffusion setting. We present the case in which there are a finite number of possible dividend rates and the one in which the possible dividend rates belong to an interval. In both cases we derive the corresponding Hamilton-Jacobi-Bellman equation and characterize the optimal value function as the unique solution of this equation. Regarding the optimal dividend strategy: for each possible dividend rate there is a threshold such that if the current surplus is larger than this threshold, the dividend rate should be increased (change region) and if it is smaller the dividend rate should remain the same (non-change region). If the dividend rate belongs to an interval, we look for the curve that gives the boundary between the optimal change and non-change regions and obtain a quite explicit formulation. We also show some examples. This is a joint work with Hansjoerg Albrecher and Pablo Azcue.

Wednesday 7th October 2020 - Forecasting crude oil futures prices using the Kalman filter and Macroenomic news sentiment

Speaker: Paresh Date

14:00pm via Zoom 

Abstract: This talk outlines a methodology of combining traditional time series models with exogenous inputs derived from news analytics, in order to improve short term prediction of financial variables. The proposed methodology allows us to incorporate information from exogenous sources (such as news) into time series models in a transparent and interpretable fashion. The focus of the talk is a news-enhanced Kalman filter model and its use in forecasting of crude oil futures prices, although I would also outline numerical results in forecasting stock price volatilities using a similar, news-enhanced GARCH type model.

Click here to view the presenation slides 

Wednesday 30th September 2020 - Distribution and asymptotic results for Japanese double - debt risk model

Speaker: Hai Ha Pham

14:00pm via Zoom

Abstract: Inspired by double debt problem in Japan where the morgagor has to pay the remain loan even their morgage was destroyed by castastrophic disaster, we model morgarate loan to estimate risk of the lender by a renewal - reward process. We analyse the asymptotic behavior distribution of first hitting time which represents the probability of full repayment. We show that finite -time probability of full repayment converges exponentially fast to the infinite - time one. In a few concrete scenarios, we calculate the exact form of the infinite-time probability and the corresponding premiums.

Click here to view the presentation slides

Wednesday 16th September 2020 - Thermodynamic Approach to Whole-Life Insurance: An Evaluation Method of Surrender Risk

Speaker: Jiro Akahori

14:00pm via Zoom

Abstract: I will propose a new model for evaluating the risks, particularly the surrender risk, in life insurance.

In the model, the health condition of each insured is modeled by a diffusion process, and the mortality rate and the surrender rate are both modeled by its killing rate. By a continuous time approximation, finding the insurance premium is done by calculating Laplace transforms.

I will present two specific models where closed form formulas are obtained.

This is a joint work with Y. Ida, M. NIshida, and S. Tamada.

Wednesday 9th September 2020 - Option-Pricing without Probability: Good News and Bad News

Speaker: Tommi Sottinen

14:00pm: via Zoom

Abstract: We show, by using the Föllmer interpretation of the Itô integral, that the hedges of financial derivatives are independent of the probabilistic properties of the underlying asset. Also, by assuming a so-called conditional full support and rough enough paths on the underlying, we show that there are no reasonable arbitrage opportunities.

The good news are:

1. Stylized facts are irrelevant in option-pricing. 2. Black-Scholes formula is correct even when the underlying returns are not Gaussian. 3. Quadratic variation is all that matters, and it has nothing to do with probability.

The bad news are:

1. The use of martingale techniques like equivalent martingale measures are dubious. 2. Classical statistical estimation is wrong. 3. Errors in discrete hedging may converge arbitrarily slowly.

This talk is based on the articles [1] Bender, C., Sottinen, T. and Valkeila, E. (2008) Pricing by hedging and no-arbitrage beyond semimartingales. Finance and Stochastics 12, 441-468. [2] Bender, C., Sottinen, T. and Valkeila, E. (2011) Fractional processes as models in stochastic finance. Advanced Mathematical Methods for Finance. Series in Mathematical Finance, Springer, pp.75-103

and also to a lesser extend on the articles

[3] Gasbarra, D., Sottinen, T., and van Zanten, H. (2011) Conditional full support of Gaussian processes with stationary increments. Journal of Applied Probability 48, No. 2., 561-568. [4] Sottinen, T. and Viitasaari, L. (2016) Pathwise integrals and Ito-Tanaka Formula for Gaussian processes. Journal of Theoretical Probability 29, Issue 2, 590-616. [5] Pakkanen, M.S., Sottinen, T. and Yazigi, A. (2017) On the conditional small ball property of multivariate Levy-driven moving average processes, Stochastic Processes and their Applications, 127, Issue 3, 749-782. [6] Sottinen, T. and Viitasaari, L. (2018) Conditional-Mean Hedging Under Transaction Costs in Gaussian Models. International Journal of Theoretical and Applied Finance 21, no. 2.

Click here to view presentation slides

Wednesday 2nd September 2020 - Variance-gamma approximation by Stein's method

Speaker : Robert Gaunt

14:00pm : via Zoom

Abstract: The variance-gamma (VG) distributions form a four parameter family that includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and approximation on Wiener space. In this talk, we demonstrate how the probabilistic technique Stein's method can be used to derive Kolmogorov and Wasserstein distance error bounds for VG approximation. We illustrate this theory with applications which include VG approximation of functionals of isonormal Gaussian processes and Laplace approximation of random sums of independent mean zero random variables.

Wednesday 27th May 2020 - From the Lokka-Zervos alternative to Riemann Surfaces

Speaker : Professor Florin Avram (University of Pau)

14:00 Via Zoom

Abstract : We discuss the following question: For risk models with dividends and possible capital injections with proportional costs, when should we use capital injections rather than declaring bankruptcy? The talk is based on several joint papers (mostly future) with Dan Goreac, Jean-Francois Renaud, Bo Li, and Pingping Jiang.

Wednesday 22nd July 2020 - On the sum of independent random variables of the exponential family

Speaker : Edmond Levy

14:00pm via Zoom 

Abstract: We begin with a brief introduction and proceed with the hypo-exponential density. We then present a common framework to develop the formula for the Erlang density and that for sums of distinct independent Erlang distributed random variables. This new framework also leads to a novel expression for the density for sums of independent gamma distributed random variables. If time permits, we will explore an easy route to the state probabilities and waiting time distributions of some familiar stochastic processes.

Click here to view the presentation slides

Wednesday 24th June 2020 - Optimal mix between pay-as-you-go and funding for DC pension schemes in an overlapping generations model

Speaker: Professor Jennifer Alonso-Garcia

14:00pm via Zoom

Abstract: Public pension systems are usually pay-as-you-go financed, that is, current contributions cover the pension expenditures. However, some countries combine funding and pay-as-you-go within the first pillar. This article studies a mixed system where a part of the individual’s contribution accrues funded rights whereas the other part accrues pay-as-you-go rights. Diversification conditions between these two financing techniques are derived in a mean–variance framework for two distinct contexts: for a cohort entering the system (named ex-ante case) and for multiple cohorts coexisting at the same period of time (named ex-post case). The diversification benefits in presence of a liquidity constraint which ensures that the income from contributions is sufficient to cover the pension expenditures are also studied. We show that, on the one hand, diversification benefits individuals when the economy is dynamically efficient for the ex-ante case. On the other hand, diversification is unattractive when pay-as-ou-go and funding are positively correlated for the ex-post case.

Click here to view the presentation slides

Wednesday 10th June 2020 - Functional limit theorems for financial markets with long-range dependence

Speaker: Professor Yuliya Mishura (Taras Shevchenko National University of Kyiv, Ukraine)

14:00 Via Zoom

We start with additive stochastic sequence that is based on the sequence of iid random variables and has the coefficients that allow for this stochastic sequence to be dependent on the past. For such a sequence, we formulate the conditions of the weak convergence to some limit process in terms of coefficients and characteristic function of any basic random variable. We adapt the general conditions to the case where the limit process is Gaussian. Then we go to the multiplicative scheme in order to get the a.s. positive limit process that can model the asset price on the financial market. So, we assume that all multipliers in the pre-limit multiplicative scheme are positive, and this imposes additional restrictions on the coefficients, and in addition, we consider only Bernoulli basic random variables. We apply these general results to the case where the limit processes in the additive scheme are fractional Brownian motion (fBm) and Riemann-Liouville fBm.

This is a common talk with Sergij Shklyar and Kostiantyn Ralchenko

Click here to view presentation slides

Wednesday 20th May 2020 - Finite-time Ruin Probability for Markovian Skip-free Risk Processes

Speaker : Professor Pierre Patie (University of Liverpool)

14:00 Via Zoom

Abstract : In this talk, we study the ruin problem for a risk process whose dynamics is given as an upward skip-free continuous-time Markov process. It means that the surplus process has upward jumps of unit size, yet it can have downward transition of arbitrary magnitude. By resorting to the theory of Martin boundary, we start by providing a representation of the green function of this Markov process in terms of some fundamental excessive functions, extending the seminal work of Feller for diffusions. We proceed by deriving an expression of the Laplace transform of the time of ruin that may occur by a jump.  We end the talk by detailing some examples that illustrate our methodology.

Click here to view the presentation slides


Wednesday 11th March 2020 - Minimizing the expected time in Drawdown

Speaker: Leonie Brinker

13:00 Math 104

Abstract:  We consider a diffusion approximation to an insurance risk model. In this context the drawdown of the process is defined as the absolute distance from the running maximum. The insurer is allowed to buy proportional reinsurance to minimize the expected discounted time the drawdown process exceeds some critical value d.Both insurer and reinsurer charge premiums which are calculated via the expected value principle. We obtain explicit results for the value function, the optimal strategy and their dependence on the safety loading of the reinsurance premium.

The optimal strategy resulting from the minimization of drawdowns is solely influenced by negative deviation from the running maximum. As an extension to the model we introduce an incentive to grow. In particular, we assume that the  insurer pays out dividends following a barrier strategy. We consider the post-dividend process under proportional reinsurance and maximize the value of the expected discounted dividends minus a penalization for time spent in drawdown.

Wednesday 19th February 2020 - Matrix distributions, Mittag-Leffler functions and the modeling of heavy-tailed risks

Speaker: Professor Hansjoerg Albrecher, University of Lausanne

13:00 Mount Pleasant 126 Room 116

Abstract:  In this talk we discuss the extension of the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable to carry over fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also discuss related randomized versions involving Mittag-Leffler distributions and illustrate the versatility and parsimony of the proposed approach for the modelling of real-world insurance data.

Wednesday 12th February 2020 - Stochastic inversions and Kelvin Transform

Speaker: Larbi Alili, University of Warwick

14:00 MATH-106

Abstract: I will show that a space time inversion  of a strong Markov process X implies the existence of a Kelvin transform of harmonic functions.  We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions which are often not the spherical ones and excessive functions are given explicitly. We treat in details some examples.

Wednesday 5th February 2020 - Optimal Reinsurance and Investment in a Diffusion Model

Speaker: Hanspeter Schmidli, University of Cologne, Germany.

14:00 MATH-106

Abstract: We consider a diffusion approximation to an insurance risk model where an external driver models a stochastic environment. The insurer can buy reinsurance. Moreover, it is possible to invest in a financial market that depends on the insurance market. The financial market is also driven by the environmental process. Our goal is to maximise terminal expected utility. In particular, we consider the case of SAHARA utility functions. In the case of proportional and excess-of-loss reinsurance, we obtain explicit results. (Joint work with Matteo Brachetta, Pescara).

Tuesday 28th January 2020 - A ruin model with a resampled environment

Speaker: Guusje Delsing, University of Amsterdam

14:00: 126 Mount Pleasant Room 116

Abstract: We consider a Cramér-Lundberg risk setting, where the components of the underlying model change over time. These components could be thought of as the claim arrival rate, the claim-size distribution, and the premium rate, but we allow the more general setting of the cumulative claim process being modelled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs we resample the model components from a finite number of $d$ settings. It results in a setup that is particularly suited to describe situations in which the risk reserve dynamics are affected by external processes (such as the state of the economy, political developments, weather or climate conditions, and policy regulations).

We extend the classical Cramér-Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound  on the ruin probability, which can be viewed as an extension of Lundberg's inequality; importantly, here it is not required that the Lévy processes be spectrally one-sided. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk.

Wednesday 15th January 2020 - Statistical tools to manage longevity risk

Speaker: Ana Debron, Universitat Politècnica de València (Spain)

13:00 Mount Pleasant 126, Room 209

Abstract: The statistical methodology that this article proposes was applied with the aim of establishing an operating procedure which permits to detect associations between countries with similar mortality, in particular the European Union countries. Once confirmed the significant associations, we implemented a spatial model for panel data.

Wednesday 11th December 2019 - Dependence uncertainty bounds for the energy score

Speaker: Alfred Müller (University of Siegen)

14:00 MATH-106

Abstract: Authors: Alfred Müller, Carole Bernard

There is an increasing interest in recent years in methods for assessing the quality of probabilistic forecasts by so called scoring rules. For forecasting general multivariate distributions, however, there are only a very few scoring rules that are considered in the literature. In their fundamental paper, Gneiting and Raftery (2007) considered the so called energy score as an example of a scoring rule that is strictly proper for arbitrary multivariate distributions. Pinson and Tastu (2013) started a debate on the discrimination ability of this scoring rule with respect to the dependence structure.
In this paper we want to contribute to this discussion by deriving dependence uncertainty bounds for the energy score and the related multivariate Gini mean difference. This means that we derive bounds for the score under the assumption that we only know the marginals of the distributions, but do not know anything about the dependence structure, i.e. the copula. Using methods from stochastic orderings we will derive some analytical bounds that are sharp in some cases. In other cases we will derive interesting numerical bounds by using a variant of a swapping algorithm. It turns out that some of these bounds are attained for some non-standard copulas that are of interest in their own right.

Keywords: multivariate probabilistic forecasts, energy score, dependence uncertainty, stochastic orderings

Wednesday 4th December 2019 - Pricing and hedging with rough volatility

Speaker: Dr. David R. Banos (University of Oslo)

14:00 MATH-106

Abstract: Stochastic volatility models are not complete, as one is considering more than one source of noise. Nevertheless, in some cases, one can trade futures on the so-called forward volaility/variance. This is e.g. the case for the S&P 500 index and the VIX index where the latter is a measure of the stock market's expectation of the volatility implied by S&P 500. In such case, one can replicate any square-integrable derivative using classical technices, e.g. PDE-techniques.

There has been some research providing empirical evidence that volaility actually is rough in the sense of fractional Brownian motion. In this talk we consider a general framework where the stochastic volatility is of Volterra type (e.g. it can be driven fractional Brownian motion), in such case, the forward variance is no longer Markov and the PDE-technique fails. We use the so-called benchmark approach to price derivatives under the real-worl measure P, and not the classical risk-neutral measure Q and use Malliavin calculus to find hedging portfolios.

Wednesday 13th November 2019 - Epiphanies in Pension Design and Valuation

Speaker: Professor Mogens Steffensen, University of Copenhagen

14:00 MATH-106

Abstract: We discuss, on a principle rather than a technical ground, three instances where things are perhaps not the way you thought they were - with potential impact, theoretically or practically. a) In life-cycle portfolio choice, does one really need to take realized capital gains into account, or are age-based investment rules doing the job? b) In time-consistent mean-variance portfolio optimization, is normalization of the variance by current wealth really the 'right' thing to do, or is there a 'better' normalization? c) In multi-state models frequently used in life insurance and credit risk, does there exist such a thing as a set of forward transition rates?

Wednesday 30th October 2019 - Valuation of Insurance Liabilities: Merging Market - and Model-Consistency

Speaker: Prof. Jan Dhaene, KU Leuven, Belgium

14:00 MATH-106

Abstract: We investigate the valuation of liabilities related to an insurance policy or portfolio in a single period framework. We define a fair valuation as a valuation which is both market-consistent (mark-to-market for any hedgeable part of a claim) and model-consistent (mark-to-model for any claim that is independent of financial market evolutions). We introduce the class of hedge-based valuations, where in a first step of the valuation process, a ‘best hedge’ for the liability is set up, based on the traded assets in the market, while in a second step, the remaining part of the claim is valuated via an actuarial model. We also introduce the class of two-step valuations, the elements of which are very closely related to the two-step valuations which were introduced in Pelsser and Stadje (2014). We show that the classes of fair, hedge-based and two-step valuations are identical. 

Wednesday 23rd October 2019 - Peano curves, trees, and spheres

Speaker: Dr Daniel Meyer (University of Liverpool)

14:00 MATH-106

Abstract: In this talk I will explore connections between three different fields, namely random geometry, complex dynamics, and hyperbolic geometry. The first object is the "Brownian map'', introduced by Le Gall. This is a random metric space which is a topological 2-sphere. It can be thought of as a 2-dimensional analog of Brownian motion. The second construction, from complex dynamics, is the "mating of polynomials'' introduced by Douady and Hubbard. Here the Julia sets of two polynomials are glued together, which often results in a rational map (i.e., a holomorphic map on the Riemann sphere). The third construction, from hyperbolic geometry, are "manifolds that fiber over the circle'' and closely related "group invariant Peano curves'' by Cannon and Thurston. In each of these constructions certain trees (random, or coming from a dynamical system) are glued together, resulting in a sphere with a Peano (i.e., space filling) curve. This is an instance of "Sullivan's dictionary''. I will be giving an overview of these 3 objects/constructions, without going into details/technicalities.

Wednesday 20th March 2019 - Insurance risk pooling, loss coverage and social welfare: When is adverse selection not adverse?

Speaker: Pradip Tapadar (University of Kent)

14:00 MATH-105

Abstract: Restrictions on insurance risk classification may induce adverse selection, which is usually perceived as a bad outcome, both for insurers and for society. We suggest a counter-argument to this perception in circumstances where modest levels of adverse selection lead to an increase in `loss coverage’, defined as expected losses compensated by insurance for the whole population. This happens if the shift in coverage towards higher risks under adverse selection more than offsets the fall in number of individuals insured. We also reconcile the concept of loss coverage to a utilitarian concept of social welfare commonly found in economic literature. For iso-elastic insurance demand, ranking risk classification schemes by (observable) loss coverage always gives the same ordering as ranking by (unobservable) social welfare.

Wednesday 1st May 2019 - On a Missing Characteristic in the Theory of Semimartingales

Speaker: Alexander Schnurr (Head of Math Department in the University of Siegen)

Venue: MATH-105 14:00

Abstract: We extend the class of semimartingales in a natural way. This allows us to incorporate processes having paths that leave the state space. By carefully distinguishing between two killing states, we are able to introduce a fourth semimartingale characteristic which generalizes the fourth part of the Levy quadruple. Since three characteristics have become canonical over the years, we motivate the fourth characteristic also by considering Feller processes. Analyzing their generator, we find a natural fourth component which does not have an analogue in the theory of semimartingales yet. Our fourth characteristic completes a classical picture and allows to incorporate ane process (with killing) and non-conservative solutions to martingale problems in the semimartingale framework. Using the probabilistic symbol, we analyze the close relationship between the generators of certain Markov processes with killing and their (now four) semimartingale characteristics.

Wednesday 27th March 2019 - Medium Data and Socio-Economic Mortality

Speaker: Prof Andrew Cairns (Heriot Watt University)

14:00 MATH-105

Abstract: In this talk we will investigate what types of information about where you live affects mortality and life expectancy.


  • ONS population and deaths data at the level of small geographical areas (LSOA's)
  • socio-economic covariates for each LSOA
  • geographical covariates for each LSOA.

We will discuss first which combination of covariates have the strongest predictive power. Second we will investigate how much regional variation there is in our resulting models: is region a genuinely significant factor in the level of mortality or is observed regional variation simply reflecting differences in the socio-economic makeup of local populations.

The use of advanced statistical methods will allow us to investigate how much variation there is across England in mortality rates and life expectancy. We can then use that to inform mortality assumption setting in pension scheme valuations.

Wednesday 6th March 2019 - Yield Curves, Measure Transformation, and Applications in Chance-Risk Classification of German Pension Products

Speaker: Ralf Korn (TU Kaiserslautern)

14:00PM Room: MATH-105

Abstract: Yield curves display the equivalent fixed yield of zero bonds for different maturities when they are bought and hold until maturity. Thus, the yield curve describes the actual fixed income opportunities prefectly.

For reasons of e.g. product development or chance-risk classification of pension products simulation for up to 40 years is performed. We therefore examine the different forms of yield curves generated by classical affine models and the dynamic evolution of their distribution.

Finally, we present a measure transformation approach that can be used to conserve empirical distribution properties of yield curves with evolving time.

Wednesday 20th February 2019 - Cascade Sensitivity Measures

Speaker: Silvana Pesenti (Faculty of Actuarial Science and Insurance, Cass Business School, City, University of London)

14:00PM Room: MATH-105

Abstract: Sensitivity measures quantify the extent to which the distribution of a model output is affected by small changes (stresses) in an individual random input factor. For input factors that are dependent, a stress on one input should also precipitate stresses in other input factors. We introduce a novel sensitivity measure, termed cascade sensitivity, which captures the direct impact of the stressed input factor on the output, as well as indirect effects via other input factors that are dependent on the one being stressed. In this way, the dependence between inputs is explicitly taken into account. Representations of the cascade sensitivity measure, which can be calculated from one single Monte Carlo sample, are provided for two types of stress: a) a perturbation of the distribution of an input factor, such that the stressed input follows a mixture distribution, and b) an additive random shock applied to the input factor. These representations do not require simulations under different model specifications or the explicit study of the properties of the model's aggregation function, making the proposed method attractive for practical applications, as we illustrate through numerical examples.

Thursday 14th February 2019 - Leeds-Liverpool joint workshop on Optimization, Uncertainty, Actuarial and Financial Mathematics

Venue: 11:00-13:00 502-LT1 and 14:00-17:00 CHEM-GOS

11:30 - Jan Palczewski (University of Leeds) - Value of Stopping Games with Asymmetric Information

12:00 - Apostolos Papaioannou (University of Liverpool) and Lewis Ramsden (University of Hertfordshire) - On Risk Models with Dependent Delayed Capital Injections

14:00 - Katia Colaneri (University of Leeds) - Optimal Converge Trading with Unobservable Pricing Errors

14:30 - Carmen Boado Penas (University of Liverpool) - Automatic Balancing Mechanisms for Mixed Pension Systems under Different Investment Strategies

15:30 - Tiziano De Angelis (University of Leeds) - Optimal Dividends with Partial Information and Stopping of a Degenerate Reflecting Diffusion

16:00 - Paul Eisenberg (University of Liverpool) - Occupation Estimates

Click here for full details and abstracts

Wednesday 6th February 2019 - On the Wiener-Hopf Factorization

Speaker: Prof Takis Konstantopoulos (University of Liverpool)

13:00 MATH-211

Abstract: This is a talk on a classical topic seeing from a probabilist's viewpoint. Traditionally, the Wiener-Hopf method is a tool in applied mathematics used, for example, to solve PDEs on the plane with mixed boundary conditions.  Roughly speaking, it splits a complex function into a suitable product. For example, the design of a Wiener filter (linear estimation in stationary environment) is precisely a Wiener-Hopf factorization. Our interest is in the derivation of the law of the overall supremum of a random walk. The law can be characterized by its Laplace transform, a complex analytic function and the Wiener-Hopf factorization of it yields some magic that can be explained using Probability Theory. The ideas go back to Rogozin, Kolmogorov, et al, and are sometimes referred to as path decomposition methods. They boil down to a splitting of the path of a random walk into independent excursions, a simple application of the trivial "découpage de Lévy" identity, and a careful application of the two fundamental symmetries of a random walk:time-reversal and space-reflection. Our goal is to show but one instance where Probability and Analysis meet and where Probability can be used to explain purely analytical concepts and derive purely analytical identities.

Wednesday 12th December 2018 - Stochastic systems under parameter uncertainty

Speaker: Michel Mandjes (University of Amsterdam)

15:00PM Room: MATH-105

Abstract: Poisson processes are frequently used, e.g. to model the customer arrival process in service systems, or the claim arrival process in insurance models. In various situations, however, the fluctuations in the arrival rate are so severe that the Poisson assumption ceases to hold. In a commonly followed approach to remedy this, the deterministic parameter λ is replaced by a stochastic process Λ(t). In this way the arrival process becomes overdispersed.

The first part of this talk considers the case that the Poisson rate is sampled periodically, with a focus on an infinite-server queue fed by the resulting overdispersed arrival process. After having presented a functional central limit theorem, we concentrate on tail probabilities under a particular scaling of the arrival process and the sampling frequency. We derive logarithmic tail asymptotics, and in specific cases even exact tail asymptotics.

In the second part of the talk we embed our overdispersion setting in a more general framework. The probability of interest is expressed in terms of the composition of two Lévy processes, which can alternatively be seen as a Lévy process with random time change. For this two-timescale model we present exact tail asymptotics. The proof relies on an adaptation of classical techniques developed by Bahadur and Rao, in combination with delicate Edgeworth expansion arguments. The resulting asymptotics have a remarkable form, with finitely many sublinear terms in the exponent.

I finish my presentation by sketching a proof of convergence of the resampled M/M/1 queue (in heavy traffic) to reflected Brownian motion. (Joint work with Mariska Heemskerk, Julia Kuhn and Onno Boxma)

Tuesday 27th November 2018 - Minimum reversion in multivariate time series (with an application to human mortality data)

Speaker: Torsten Kleinov (Heriot-Watt University)

14:00PM Room: 126MP-110

Abstract: We propose a new multivariate time series model in which we assume that each individual component has a tendency to revert to the minimum of all components. Such a specification is useful to describe phenomena where the behaviour of the best performing member in a population which is subjected to random noise is mimicked by other members. We show that the proposed dynamics generate co-integrated processes, characterize the model’s asymptotic properties for the case of two populations and show the stabilizing effect on long term dynamics in simulation studies. An empirical study involving human survival data in different countries provides an example which confirms the occurrence of the phenomenon of reversion to the minimum in real data.

Wednesday 21st November 2018 - Distribution-constrained optimal stopping problems

Speaker: Christiane Elgert (TU Wien - Technical University of Vienna)

14:00PM Room: MATH-105

Abstract: We deal with financial and actuarial products whose payouts are driven by stochastic processes. The time point of the payouts is modelled by an stopping time or an adapted random probability measure. These stopping times (or adapted random probability measures) follow a given distribution and can depend on the payouts. Our target is to deduce the estimation of the worst-case situation, that means, the supremum of the expected payout over all stopping times satisfying the given marginals. We formulate our task as an optimal transport problem and prove the existence of an optimal strategy by using the methods and techniques from the optimal transport theory.

Wednesday 14th November 2018 - Cointegration in continuous time for factor models

Speaker: Prof. Fred Espen Benth (University of Oslo)

14:00PM Room: MATH-105

Abstract: We develop cointegration for multivariate continuous-time stochastic processes, both in finite and infinite dimension. Our definition and analysis are based on factor processes and operators mapping to the space of prices and cointegration. The focus is on commodity markets, where both spot and forward prices are analysed in the context of cointegration. We provide many examples which include the most used continuous-time pricing models, including forward curve models in the Heath-Jarrow-Morton paradigm in Hilbert space.

Wednesday 7th November 2018 - Integrated Unit Linked Collective Assets (ICA)

Speakers: Cordelia Rudolph and Axel Helmert (MSG Life Austria)

16:00PM Room: MATH-105

Abstract: The situation of life insurance companies and pension providers is challenging. One of the most difficult aspects is the low interest rate challenge. Together with the increasing requirements on solvency and other regulations this is developing pressure in search of new solutions. We have seen many attempts adopting long term guarantees in various new products, the results are not satisfying. In recent years, the opinion has prevailed that we need a more general approach. This leads us to the following questions: Is it possible to provide a high level of security combined with acceptable yields without traditional long term guarantees? Could we bring together the advantages of individual unit linked life insurance products with the traditional collective approach in a cost efficient automated environment?

Wednesday 24th October 2018 - Trends in the extreme value index

Speaker: Prof Laurens de Haan, (Erasmus University Rotterdam, The Netherlands)

14:00PM Room: MATH-105

Abstract: The first part of the talk will be an introduction to extreme value theory. I shall discuss the theoretical background, the tools for application and several specific applications.

A discussion of recent research follows. We consider extreme value theory for independent but not identically distributed observations. In particular, the observations do not necessarily share the same extreme value index. Assuming a continuously changing index we provide a non-parametric estimate for the functional extreme value index. Besides estimating the extreme value index locally, we also provide a global estimator for the trend and its joint asymptotic distribution. The asymptotic theory for the global estimator can be used for testing a pre-defined parametric trend in the index. In particular it can be applied to test whether the index remains constant across all observations. (Joint work with Chen Zhou).

Wednesday 17th October 2018 - Optimal control of piece-wise deterministic processes (PDP's)

Speaker: Dr Alexey Puinovskiy, (University of Liverpool)

14:00PM Room: MATH-105

Abstract: PDPs are widely used in Insurance, Reliability, Mathem. Epidemiology, Queueing Theory and so on. For the standard discounted model with a single objective, Dynamic Programming approach proved to be successful. In the case of several objectives (constrained model), Linear Programming is more appropriate. In this talk I will explain some theoretical results in these areas obtained jointly with Prof. F. Dufour (INRIA, France) and Prof. O. Costa (Uni. Of Sao Paolo, Brazil): SIAM J. Control Optim, 2016, V.54, N.3, p.1444-1474. If time permits, I will also mention the so called impulsive control theory. Where possible, the connection to insurance will be demonstrated.

Wednesday 10th October 2018 - Existence and Properties of Optimal Strategies for Distribution-Constrained Discrete-Time Optimization Problems

Speaker: Uwe Schmock, (TU Wien - Technical University of Vienna)

14:00PM Room: MATH-105

Abstract: We consider stochastic optimization problems in discrete time under distributional constraints. These problems are motivated by actuarial science, in particular to treat unit-linked insurance products with guarantees. They can also serve as benchmark models to account for customer behaviour, when the treatment as American option is not appropriate.

The basic mathematical set-up is an adapted stochastic process (interpreted as pay-outs) and (possibly randomized) stopping times to optimize the expected pay-out. The difference to classical optimal stopping problems is our constraint concerning the distribution of the stopping times or, more generally, the adapted random probability measures.

For these distribution-constrained optimization problems we prove the existence of an optimal strategy, the basic assumptions are suitable moment conditions. In special cases, optimal strategies are identified explicitly. (The talk is based on joint work with Christiane Elgert and Karin Hirhager.)

Thursday 13th September 2018 - Numerical methods for McKean-Vlasov equations: taming, Importance Sampling & LDP's

Speaker: Goncalo dos Reis (University of Edinburgh)

15:00PM Room: 126 Mount Pleasant, Teaching Room 209

Abstract: We present several recent results on numerical methods for McKean-Vlasov stochastic differential equations (MV-SDEs). Firstly on how to simulate MV-SDEs with drifts of superlinear growth, then how to employ the well-known Importance Sampling variance reduction technique for the simulation of certain quantities of interest involving expectations of the solution to MV-SDEs.

The presentation combines several joint works with:
S. Engelhardt and G. Smith, Simulation of McKean Vlasov SDEs with super linear growth, (arXiv:1808.05530)
G. Smith and P. Tankov, Importance Sampling for McKean-Vlasov SDEs, (arXiv:1803.09320)
W. Salkeld and J. Tugaut, Freidlin-Wentzell LDPs in path space for McKean-Vlasov equations and the Functional Iterated Logarithm Law, (arXiv:1708.04961)

Wednesday 9th May 2018 - Indifference pricing of life insurance contracts via BSDEs under partial information

Speaker: Katia Colaneri (University of Leeds)

14:00PM Room: MATH-103

Abstract: In this paper we investigate the pricing problem of a pure endowment contract when the insurer has a limited information on the mortality intensity of the policyholder. The payoff of this kind of policies depends on the residual life time of the insured as well as the trend of a portfolio traded in the financial market, where investments in a riskless asset, a risky asset and a longevity bond are allowed. We propose a modeling framework that takes into account mutual dependence between the financial and the insurance markets via an observable stochastic process, which affects the risky asset and the mortality index dynamics. Since the market is incomplete due to the presence of basis risk, in alternative to arbitrage pricing we use expected utility maximization under exponential preferences as evaluation approach, which leads to the so-called indifference price. Under partial information this methodology requires filtering techniques that can reduce the original control problem to an equivalent problem in complete information. Using stochastic dynamics techniques, we characterize the value function as well as the indifference price in terms of the solution to a quadratic-exponential backward stochastic differential equation. This is a joint work with Claudia Ceci and Alessandra Creatarola.

Thursday 19th July 2018 - Kendall Random Walks

Speaker: Prof. Barbara Jasiulis-Goldyn (University of Wroclaw, Poland)

14:00PM Room: MATH-029

Abstract: In this talk we introduce the Kendall random walks and the corresponding renewal processes. We prove asymptotic properties for them using regularly varying functions techniques. Applications to insurance will be discussed.

Wednesday 30th May 2018 - Ruin problem for correlated Brownian motions

Speaker: Lanpeng Ji (University of Leeds)

14:00PM Room: MATH-103

Abstract: Nowadays, insurance companies run different lines of businesses or collaborate with other companies, it is thus of interest to model these businesses using different risk processes, leading to the study of vector-valued risk models. In this talk, we will focus on the most recent findings for vector-valued Gaussian risk models. In particular, infinite-time simultaneous ruin probability of the (correlated) Brownian motions risk model will be discussed in detail, where we shall show the exact asymptotics of the ruin probabilities, by using the celebrated double-sum method combined with the theory of a quadratic programming problem. Additionally, the sojourn time of such risk model will be discussed, which has close relation with the Cumulative Parisian ruin, recently introduced in actuarial science.

This talk is based on joint works with Krzysztof Debicki (University of Wroclaw), Enkelejd Hashorva (University of Lausanne) and Tomasz Rolski (University of Wroclaw).

Wednesday 23rd May 2018 - Portfolio Optimisation with Semivariance

Speaker: Kwok Chuen Wong (Dublin City University)

14:00PM Room: MATH-103

Abstract: In this talk, I shall investigate dynamic portfolio management using semivariance of portfolio payoff as a portfolio risk measure. Comparing with variance which is widely used in the literature, semivariance is considered to be more plausible risk measure because semivariance penalizes adverse situations only. However, in the literature, it was shown that mean-semivariance optimisation under the Black-Scholes model has no optimal solution. Inspired by this non-existence result, we replace the mean term with the expected value of utility satisfying the Inada conditions, then utility-semivariance is solvable. By adding a downside risk management term such as semivariance, we numerically show that more than 90% of the respective deviation risk incured in the case of solely utility maximization can be reduced subject to less than 10% loss in utility as a tradeoff. Besides, I shall establish necessary and sufficient conditions under which the mean-semivariance optimisation possesses an optimal solution; which generalises the negative result in the literature. Moreover, I shall suggest the models under which such sufficient conditions are satisfied, thus, under these models, the explicit optimal solution to mean-semivariance optimisation can be obtained; such models can be applied into the themes of insurance and credit risk management. This talk is a joint work with Paolo Guasoni, Phillip Yam, and Harry Zheng.

Monday 14th - Thursday 17th May 2018 - Allotey workshop: connecting Liverpool to Africa through mathematics and data science

This event is dedicated to the memory of Professor Francis Allotey. Professor Francis is well known for the “Allotey Formalism” in X-ray spectroscopy. He obtained his master’s and doctorate degrees from Princeton University and Imperial College London, and he became the first Ghanaian full professor of mathematics at the Kwame Nkrumah University in 1974. He contributed to many international institutions including the Council of the prestigious Abdus Salam Centre since 1996. Among his many achievements for promoting science in Africa is his role in establishing AIMS Ghana in 2012.

See the workshop website for full details

Wednesday 31st January 2018 - The Wilkie Model – Past, Present and Future

Speaker: Professor David Wilkie (Heriot-Watt University)

14:00PM Room: MATH-103

Abstract: David Wilkie, originator of what actuaries have come to call “The Wilkie Model”, will describe, briefly, how it originated, how it has developed, and what future developments are being considering for it.

Wednesday 2nd May 2018 - Some challenges in portfolio optimization

Speaker: Bogdan Grechuk (University of Leicester)

14:00PM Room: MATH-103

Abstract: Portfolio optimization is the process of finding the proportions of various financial instruments to form a portfolio, which is better than any other feasible portfolio according to some criterion/criteria. The main challenges in portfolio optimization are (i) it is unclear which criteria to use; (ii) not only future rates of returns of financial instruments are unknown, but their distributions are unknown as well; (iii) for a group of individuals with different investment preferences, optimal group portfolio is not the sum of individual portfolios. In this talk, some new approaches for addressing these challenges will be discussed.

Wednesday 21st February 2018 - Determinants of tail risk in emerging and developed markets

Speaker: Devraj Basu and Bertrand Groslambert (Strathclyde University)

14:00PM Room: MATH-103

Abstract: We study the distribution of extreme events risk across emerging and developed stock markets and empirically identify the determinants of tail risk across countries. A recent literature has shown that rare disasters can explain some of the most important puzzles in finance and that tail risk is priced in the cross section of asset returns. We find a strong empirical relationship between tail risk and the quality of institutions even after economic and financial variables have been accounted for. Better governance substantially reduces the probability of extreme events. In addition, we find that what differentiates developed and developing countries concerning extreme stock market risk is the quality of their institutions, not the depth of their financial markets, nor the degree of financial and trade openness.

Wednesday 7th February 2018 - Spectral Backtests of Forecast Distributions with Application to Risk Management

Speaker: Alex McNeil (University of York)

13:00PM Room: Risk Institute Seminar Room

Abstract: In this talk we study a class of backtests for forecast distributions in which the test statistic  is a spectral transformation that weights exceedance events by a function of the modelled probability level. The choice of the kernel function makes explicit the user's priorities for model performance. The class of spectral backtests includes tests of unconditional coverage and tests of conditional coverage. We show how the class embeds a wide variety of backtests in the existing literature, and propose novel variants as well. We assess the size and power of the backtests in realistic sample sizes, and in particular demonstrate the tradeoff between power and specificity in validating quantile forecasts.

Wednesday 15th November 2017 - A test for the rank of the volatility process

Speaker: Prof Mark Podolskij (Aarhus University, Denmark)

14:00PM Room: MATH-103

Abstract: In this talk we present a test for the maximal rank of the matrix-valued volatility process in the continuous Ito semimartingale framework. Our idea is based upon a random perturbation of the original high frequency observations of an Ito semimartingale, which opens the way for rank testing. We develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses. This is joint work with Jean Jacod.

Friday 10th November 2017 - RELAX Actuarial Workshop

Venue: 128 Mount Pleasant

Speaker: Veronique Maume-Deschamps (Universite Lyon)

11:00 - On some non-parametric methods for extensions of spatial max-stable processes

Speaker: Manuel Morales (University of Montreal)

11:30 - On an Agent-based Simulator Model for the Limit-Order-Book and its Applications to Measuring Price Impact

Speaker: Andrei Badescu (University of Toronto)

12:00 - An IBNR-RBNS insurance risk model with marked Poisson arrivals

12:30 - Lunch

Speaker: Alfredo Egidio Dos Reis (University of Lisbon)

13:30 - Estimation of foreseeable and unforeseeable risks

Friday 26th May 2017 - A probabilistic approach to spectral analysis of growth-fragmentation equations

Speaker: Alexander Watson (University of Manchester)

2:00PM Room: MATH-104

Abstract: The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this talk, I discuss a probabilistic approach to the study of this asymptotic behaviour. The method is based on the Feynman-Kac formula and the identification of a driving Markov process. This is joint work with Jean Bertoin.

Friday 26th May 2017 - Beyond the Variance Risk Premium: Stock Market Index Return Predictability and Option-Implied Information

Speaker: Gabriel J. Power (Laval University)

10:30AM Room: Chadwick Building Seminar Room

Abstract: We investigate international stock market index return predictability using option-implied information (SP500, DAX, FTSE, CAC, and SMI). We document the predictive power of several forward-looking variables such as the variance risk premium and the Foster-Hart (FH) risk measure for horizons ranging from one to 250 days. Our results from predictive regressions as well as out-of-sample forecast tests suggest that the variance risk premium is a significant predictor of returns at horizons below one month, in addition to the previously documented quarterly predictability. Foster-Hart riskiness also has significant forecasting power for longer horizons. Overall, our results show that the VRP and FH risk are complementary. Risk-neutral skewness and kurtosis are often significant in predictive regressions, but in out-of-sample lose significance once VRP and FH are introduced in the model.

Friday 12th May 2017 - Eddie Stobart: what we do, our challenges, what data we collect

Speaker: Dr. Damon Daniels, Strategic Network Manager, Eddie Stobart

12:00PM MATH-104

Abstract: Damon (Liverpool alumus) will talk about how he came to at Eddie Stobart, and the beginnings of him trying to bring in some new approaches to the road network (some first steps analysis he did, and the tolls/methods he used that he picked up from his PhD). He will summarise the partnerships first project with Suhang Dai; the problem they had, and how working with Suhang has led to measureable business improvements for Eddie Stobart. Finally, outlining the future partnership plans; with big emphasis on potential projects they are hoping to develop across their Commercial, Finance, Operations and Network teams.

Friday 5th May 2017 - Forecasting algorithms for recurrent patterns in consumer demand

Speaker: Oleg Karpenkov, University of Liverpool

3:00PM MATH-106

Abstract: In this talk we discuss a new forecasting algorithm for recurrent patterns in consumer demand. We study this problem in two different settings: refill and supply models. We will consider several features of the algorithm concerning sampling, periodic approximation, denoising and forecasting. This is a joint research together with T.Boiko and B.Rakhimberdiev.

Friday 24th March 2017 - Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution

Speaker: Dr. Daniel Alai, University of Kent

2:00PM REN-SR1 (Rendall Building Seminar Room 1)


An important driver of longevity risk is uncertainty in old-age mortality, especially surrounding potential dependence structures.  We investigate a multivariate Pareto distribution that allows for the exploration of a variety of applications, from portfolios of standard annuities to joint-life annuity products for couples.  In past work, it has been shown that even a little dependence between lives can lead to much higher uncertainty.  Therefore, the ability to assess and incorporate the appropriate dependence structure, whilst allowing for extreme observations, significantly improves the pricing and risk management of life-benefit products.  Finally, we explore a generalization of the multivariate Pareto distribution via established links with Archimedean survival copulas.
Based on joint work with Profs Zinoviy Landsman and Michael Sherris.
Thursday 16th March 2017 - Asymptotic hedging of barrier option via parametrix and Fourier-Malliavin estimators based on discrete measures

Speakers: Dr Yuri Imamura, Tokyo University of Science and Dr. Nienlin Liu, Ritsumeikan University, Japan


Friday 10th March 2017 - On subexponential tails for negatively driven compound renewal processes with application to two-dimensional ruin problem

Speaker: Prof Dima Korshunov, Lancaster University


Abstract: We discuss subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and L'evy processes being motivated by Cram'er-Lundberg renewal risk process. These results allow to analyse the asymptotics of ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity and generic claim size is subexponential. (Particularly based on a joint work with Sergey Foss (Edinburgh) and Zbignew Palmowski (Wroclaw)).

Wednesday 25th January 2017 - Time fractional diffusion systems and their applications in control theory

Speaker: Prof. Chunhai Kou (College of Science, Donghua University,  Shanghai)

2:00PM Seminar Room 4 (JSM-SR4), Muspratt Building

Abstract: Recently anomalous diffusion systems have attracted increasing research interest since the introduction of continuous time random walks (CTRWS) and a large number of contributions have been given to them. The time fractional diffusion system, which replaces the first order time derivative by a fractional derivative, has been seen as the macroscopic presentation of a CTRW model.  From a physical view-point, this generalized diffusion equation can be used to describe transport processes with long memory in the spatially inhomogeneous environment. Here we attempt to explore the boundary feedback stabilization for a class of time fractional diffusion systems. By designing an invertible coordinate transformation, the Mittag-Leffler stability of the system studied is obtained.  Simulation result is also given at last to test the effectiveness of our results.  We hope that the results here could provide some insight into the control theory analysis for the time fractional diffusion system.

Friday 10th February 2017 - Exploiting Asymptotic Structure for Efficient Rare-event Estimation for Sums of Random Variables

Speaker: Thomas Taimre (The University of Queensland, Australia)

2:00PM MATH-106

Abstract: We consider the problem of estimating the right-tail probability of a sum of random variables when the density of the sum is not known explicitly, but whose asymptotic behaviour is known. We embed this asymptotic structure into a simple importance sampling estimator, in which we consider the radial and angular components of the distribution separately. By design, this estimator has a bounded relative error when the marginal tails decay exponentially. Moreover, we present a procedure to obtain a `good' approximation to the angular component as a mixture of Dirichlet distributions by using Bernstein polynomial approximation (cf. the Weierstrass approximation theorem). The estimator and procedure are applicable in both the heavy- and light-tailed settings, as well as for dependent and independent summands. We illustrate the approach with a series of examples. This is joint work with Patrick Laub.

Wednesday 8th February 2017 - An Endogenous Regime-Switching Continuous-Time Diffusion Model for S&P 500 Volatility Index

Speaker: Jie Cheng

2:00PM MATH-029

Abstract: We propose a new regime-switching continuous-time diffusion model for S&P 500 Volatility Index (VIX). Our model may be regarded as an extension of Choi et al. (2015) who developed a noval endogenous regime-switching mechanism for discrete-time Gaussian processes where the switching between regimes is driven by a latent factor potentially correlated with the innovations of the observed time series. We allow our regime-dependent diffusions to be continuous-time and non-Gaussian by considering nonlinear transformations of underlying Ornstein-Uhlenbeck (OU) processes, whereby the switching of regimes is driven by a latent factor potentially correlated with the innovations of the underlying OU processes. We apply the proposed model to time series of VIX at monthly, weekly and daily frequencies. In addition to .finding strong evidence of regime switching effect and endogeneity in the switching of regimes, we are also able to extract the latent factor that drives the switching of the regimes for further analysis. Our model appears to perform better for VIX at monthly and weekly frequencies than at daily frequency.

Wednesday 1st February 2017 - Ruin probabilities: exact and asymptotic results

Speaker: Prof. Zbigniew Palmowski, Wroclaw Technical University, Poland

2PM MATH-106

Abstract: Ruin theory concerns the study of stochastic processes that represent the time evolution of the surplus of a stylized non-life insurance company. The initial goal of early researchers of the field, Lundberg (1903) and Cramér (1930), was to determine the probability for the surplus to become negative. In those pioneer works, the authors show that the ruin probability decreases exponentially fast to zero with initial reserve tending to infinity when the net profit condition is satisfied and clam sizes are light-tailed.

During the lecture we explain when and why we can observe this phenomenon and discuss also the heavy-tailed case. We demonstrate main techniques and results related with the asymptotics of the ruin probabilities: Pollaczek-Khinchin formula, Lundberg bounds, change of measure, Wiener-Hopf factorization, principle of one big jump and theory of scale functions of Lévy processes.

Thursday 1st December 2016 - Parisian ruin theory for Lévy insurance risk processes

Speaker: Jean-François Renaud (UQAM)

3:00PM Room G-16

Abstract: In the last few years, the idea of Parisian ruin has attracted a lot of attention. In Parisian-type ruin models, the insurance company is not immediately liquidated when it defaults: a grace period is granted before liquidation. Roughly speaking, Parisian ruin occurs if the time spent below a pre-determined critical level is too long. In this talk, I will present recent results related to different definitions of Parisian ruin for spectrally negative Lévy processes.

Thursday 24th November 2016 - Ruin probabilities in Gamma risk models

Speaker: Wei Zhu (University of Liverpool) - RARE Seminar

3:00PM Room 201 Electrical Engineering

Abstract: In risk theory, the classical risk model assumes that each claim arrives after an exponential time. This talk considers a generalization of the exponential inter-claim times assumption of the classical model. Specifically, when the waiting times are Gamma distributed, we show that the ruin probability satisfies a fractional integro-differential equation, which has explicit solutions under certain assumptions on the claim size distribution.

The RARE (Risk Analysis, Ruin and Extremes) project supported by the EU Marie Curie International Research Staff Exchange Scheme has ran for 4 years (Dec 1, 2012- Nov 30, 2016) and has covered over 400 months of travel exchanges between 12 partner universities. IFAM has coordinated this project and benefited greatly from this network exchanges. Over 200 publications have been produced as result of this project.

Thursday 20th October 2016 - A dual parameter long-memory model

Speaker: Professor Rajendra Bhansali (University of Liverpool)

2:00PM Room MATH-027

Abstract: A new model for long-memory time series is introduced. It involves two memory parameters, d and c , say, and characterizes the correlation decay as a mixture of polynomial and logarithmic rates . This model includes as its special case the standard long memory model with a single memory parameter, d , in which the correlations decay only at a polynomial rate. Examples illustrating some situations in which the standard model does not apply but the new model does do so are presented. A mathematical definition of the class of dual parameter long memory models is given and this class is extended to include also the class of dual parameter intermediate memory models. The class of parametric dual-parameter FARIMA models, called DFARIMA models, is also introduced and the notions of strong, weak and mixed long and intermediate memory are defined. Non-parametric and semi-parametric estimation of the parameters of the new model by the dual parameter extensions of the standard logperiodogram and local Whittle methods is considered together with the maximum likelihood estimation of the parameters of the DFARIMA model. Asymptotic properties of the estimates are investigated and it is shown that the standard single-parameter estimation methods can be badly biased when the dual parameter model holds. The usefulness of the asymptotic results for observed series of finite length is investigated by a simulation study. An application of the dual parameter model to internet packet traffic is also discussed.

Thursday 22nd September 2016 - On the Fatou property of convex functions on the duals of Orlicz spaces

Speaker: Dr. Keita Owari (Ritsumeikan University, Japan)

4:00PM Room 103

Wednesday 9th March 2016 - Herd-like Behaviour and the Psychology of Market Bubbles

Speaker: Colm Fitzgerald

15:00 - Lecture Theatre 2 Life Sciences

Lecture sponsored by the Worshipful Company of Actuaries

Colm Fitzgerald will discuss methodologies that can be used to assess and manage various forms of risks related to group psychology, e.g. herd-like behaviour, financial market bubbles, etc. He will use the concept of the narrative, draw the distinction between a narrative and an analysis and will use this approach to define what he refers to as 'narrative risk'. He will look back at historical bubbles to point out what we can learn and what we cannot learn from them. He will also discuss probable current bubbles.

Colm is a Fellow of the Institute & Faculty of Actuaries and the Society of Actuaries in Ireland. He lectures in actuarial science in University College Dublin and is a member of the Education Board and the Board of Examiners of the Institute & Faculty of Actuaries. Previously, he spent most of his career working as a trader, finishing up as Head of Quantitative Trading in Bank of Ireland Global Markets. His research interests include the psychology of risk, trading models, the application of actuarial techniques in wider fields, applying classical thought and forestry.


  • 15:00 - 15:15 Opening remarks
  • 15:15 - 16:00 Presentation by Colm Fitzgerald
  • 16:00 - 16:30 PhD and UG students’ presentations
  • 16:30 - 16:35 Prizes
  • 16:35 - 17:00 Coffee and posters
Monday 7th March 2016 - Tail process and extremes of heavy tailed sequences

Speaker: Prof. Bojan Basrak (University of Zagreb, Croatia)

3:00PM, Lecture Theatre 029

Abstract: We describe how one can characterise dependence structure in a stationary heavy tailed sequence using the notion of tail process. This theory is applied to study extreme values of dependent regularly varying sequences, such as GARCH processes for instance. We will also discuss the convergence of partial sums and corresponding point processes, covering some recent results in the literature.

Tuesday 1st March 2016 - RARE workshop on Stochastic Analysis and Applications

As part of the Risk Analysis, Ruin and Extremes (RARE EU-IRSES 318984) project, we are pleased to host a one day workshop at the University of Liverpool.


Morning Session (Seminar room 521, Cedar House)

  • 09:30 - 10:15 Andrea Macrina (University College, London)
  • 10:15 - 10:45 Camilo Garcia Trillos (University College, London)
  • 10:45 - 11:15 Coffee at Victoria Gallery and Museum
  • 11:15 - 12:00 Mihalis Zervos (London School of Economics)
  • 12:00 - 12:30 Yuri Imamura (Ritsumeikan University, Japan)
  • 12:30 - 14:00 Lunch at Victoria Gallery and Museum

Afternoon Session (Lecture room 203 (E3) Electrical Engineering)

  • 14:00 - 14:45 Kai Liu (University of Liverpool)
  • 14:45 - 15:15 Zhongyang Sun (Nankai University, China)
  • 15:15 - 15:45 Coffee at Victoria Gallery and Museum
  • 15:45 - 16:30 Toshihiro Yamada (Tokyo University, Japan)
  • 16:30 - 17:00 Alexey Piunovskiy (University of Liverpool)

This workshop has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 318984 –RARE