# Financial and Actuarial Mathematics Seminars

## 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

**Abstract**

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

**Abstract**

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!*

Abstract

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!*

Abstract

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

Abstract

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

Abstract:

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

Abstract:

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

Abstract:

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

Abststract:

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

Abstract:

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

M. LINDHOLM, R. RICHMAN, A. TSANAKAS, M.V. WUETHRICH

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.

## 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.

## 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.

Data:

- 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

Speakers:

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)

**Abstract: **

## 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

1:00PM GUILD-SUTC

## 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

10:00AM GUILD-SUTC

**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.

Timetable

- 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.

### Programme

- 09:00 - 09:30 Welcome - Coffee at Victoria Gallery and Museum

### 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