# Sample PhD Projects in Financial and Actuarial Mathematics

## General Interests

Markets; insurance; pricing; risk; random processes; stochastic systems

### Dr Ehsan Azmoodeh

#### Project 1. Fractional Brownian Motion and Rough Stochastic Volatility

In the seminal paperVolatility is roughappeared in 2018 in Quantitative Finance (18(6):933-949)Gatheral, Jaisson, and Rosenbaum showed that the instantaneous volatility is driven by a (rough) fractional Brownian motion, and hence contradicting decades of econometric analyses and practice.Fractional Brownian motion is a natural extension (through the so-called Hurst parameter H) of the classical Brownian motion employing in the Black–Scholes market model. The project is devoted todeep understanding of the emerge of rough volatility in financial modeling. The problems of pricing and hedging options will also be considered.

#### Project 2. Pathwise Methods in Finance and Stochastic Analysis

In early eighties, Hans F ̈ollmer in his article (Calcul d’Ito sans probabiliti ́es, S ́eminaire de probabilit ́es) derived a purely pathwise interpretation of the Itˆo integral. Since then, and in particular in the last decade, due to the development of utilizing rough paths (e. g., fractional Brownian motion) in mathematical finance, pathwise calculus got a massive attraction. The project aims to study to what extent pathwise calculus can be pushed. Applications in mathematical finance and stochastic analysis will be investigated.

**Project 3. Stein Method, Heavy-Tailed Distributions and Application in Ruin Theory**

Unarguably, the classical Cram ́er–Lundberg model introduced early twentieth century constitutes one of the backbones of the modern risk theory. Since then, a massive research have been devoted to extensions and generalization of the model in several vital directions. Motivated by the recent study (Corina D. Constantinescu, Jorge M. Ramirez & Wei R. Zhu (2019) An application of fractional differential equations to risktheory,Finance and Stochastics, vol. 23, pages 1001-1024), the primary goal of the project is to study the novel thought of application of Stein method and heavy-tailed distributions within the theory of ruin. The Mittag–Leffler distribution (a natural generalization of the exponential function), used to model waiting times, will play a central role. This distribution possesses various interesting features for applications, and in particular it is heavy-tailed. One of our devices will be Stein method. Stein method is a probabilistic technique to measure distance between probability measures using differential operators. Capturing characteristic differential operator of the Mittag–Leffler distribution falls within our main targets.

**Project 4. The Malliavin–Stein Method**

In 2005, Nualart and Peccati (Ann. Probab. Vol. 33, No 1, 177-193) discovered a striking result known nowadays as the fourth moment theorem in the context of weak convergence toward Gaussian distribution. A few years later, Nourdin and Peccati combined two probabilistic devices to quantify the fourth moment theorem. Since the publication of these two pathbreaking papers, many improvements and developments on this topic have been considered, and led to a fascinating range of applications including stochastic geometry, free probability theory and time series analysis. In this project, we study generalization of the method to non-Gaussian Wiener distributions considering potential application.Tools to be invited during the project are Malliavin calculus, Stein method, as well as a solid knowledge on real/functional analysis and ode theory is appreciated.

**Project 5. Algebraic Structures of Stein Operators**

Let μ denote a continuous probability measure on the real line. A differential operator L with polynomial coefficients satisfying Eμ[Lf] = 0 (for a large class of functionsf) is called a polynomial Stein operator for μ. Denote by PSO(μ) the class of polynomials Stein operators associated to μ. This project aims to study the algebraic structures of PSO(μ) to perceive significant statistical properties of μ. In a wider scope to initiate yet another fascinating bridge between probability theory and algebra.The project meets several mathematical disciplines ranging from complex analysis to non-commutative algebra

### Dr Carmen Boado Penas

**Project 1. How has the financial and economic crisis affected retirement behaviour?**

Although the net effect of the financial and economic crisis on retirement is far from clear it is possible to identify the prospective channels through which the effects will be transmitted and highlight interaction between household decisions concerning stock market investments, housing wealth and pension provision. In particular, while the current decline in home values and stock prices would be expected to lead to a decrease in the number of retirements, the effect of the increase of the unemployment is just the opposite. The aim of this project is to measure how the economic crisis may affect the retirement decisions of the individuals. The dissertation will develop an extension of Coile and Levine (2011) and will address questions about how the stock market wealth, housing wealth and unemployment ratio could influence on the retirees’ decisions,

**Project 2. Valuing Public Obligations in the Face of Economic and Longevity Risk **

The traditional methods of valuing long-term public obligations rely on specific assumptions: (i) the value of future claims as well as the duration of future payments are assumed known at the time when the programme is introduced, (ii) the discount factor, which translates into present terms the value of future claims, is determined in an ad-hoc fashion. From this starting point, the literature adds realism to the analysis by allowing claims and duration to be random, the so called, the “stochastic valuation methods”. The question of determining the correct discount factor for time and risk is left unresolved.

In contributing to this important debate about the value of public obligations, this project proposes an integrated approach correctly predicting uncertainty and risk. With this in mind, the research combines advances from several areas: modern finance theory for discounting and valuing cash-flows, econometrics for modeling of a population of outcomes (e.g. earnings or labour market participation), actuarial science and demography for modeling longevity and health risks, and finally macroeconomics to account for economic risk in the future.

### Professor Corina Constantinescu

**Project 1. Some exact and asymptotic results for the ﬁrst passage-time distribution of renewal jump-diffusions**

The investigation of the connections between the analytic and probabilistic aspects of the theory of diffusions and Markovian processes has been initiated by Kolmogorov (forward and backward Kolmogorov equations). Recently, the interest was reawakened in the context of ﬁnancial modeling. The most famous case is the Merton-Black-Scholes diffusion, which due to its mathematical tractability led to a large variety of explicit formulas for the transition and ﬁrst passage probabilities necessary for the pricing of options. For insurance modeling, jump-diffusion or renewal jump-diffusion processes are considered (small continuous changes modeled by diffusions, and ”catastrophic” changes modeled by pure jump processes) and exact or asymptotic results for the ﬁrst passage-time distribution are thought after.

**Project 2. A comparative study of pricing and risk management of equity-linked life insurance products**

Given the development of the equity-linked insurance market, there is a need for an overview of methods employed and models used in practice versus academia. The academic literature for variable annuities and similar products is to be compared with the methods currently used in practice for managing financial and mortality/longevity risks. The results should appease or raise warning flags in both life insurance and pension fund industries.

**Project 3. Fractional differential equations in risk theory**

In the ruin theory context, (integro-)differential equations are widely used. For instance, when the distribution of the inter-claim times are gamma distributed with the shape parameter integer, the ruin probability satisfies high order (integro-)differential equations. If the shape parameter is not an integer anymore, one needs to appeal to fractional calculus: analyze fractional differential equations and their boundary conditions.

### Dr Bujar Gashi

**Project 1. Forward-backward stochastic control with applications**

Forward-backward stochastic differential equations (FBSDE’s) appear in various applications. In particular they appear naturally in mathematical finance when dealing with pricing and hedging, and optimal investment and consumption. However, the control theory for such equations is still in infancy. The aim of this project is to study control problems for FBSDE’s, and to apply such results to problems in mathematical finance. In particular, we will study the pricing of options in a market with large investors, and optimal investment with stochastic interest rate.

**Project 2. Optimal investment with stochastic interest rate**

Most of the known methods for optimal investment assume that the interest rate is either constant or it changes according to a known law. However, the finance practice has shown that such an assumption is not realistic. As a result, various different models for interest rates have become available in recent years. The aim of this project is to study the optimal investment problem under the assumption that the interest rate changes randomly.

### Dr Olivier Menoukeu Pamen

**Project 1. Optimal dividend policy of an insurance firm. **** **

This project deals with the problem of finding an optimal dividend policy for a class of diffusion processes under some constraints (for e.g., transaction cost). More precisely, properties of the value function of this control problem would be studied and convergence of numerical schemes for the optimal value function would be looked at.

**Project 2. Derivative pricing under Markov regime switching.**

Random structural changes in the movements of stock (or spot) indices are not generally caused by internal events of the market but are related to the global socioeconomic and political environment. Markov modulated models (or regime switching) have been widely used in financial mathematics and economics to capture this exogenous cycles. The aim of this project is to price Bermudan swaptions under Markov regime switching model and to check the impact of the regime in the derivative pricing.

**Project 3. Risk sensitive control for jump-diffusion model under Markov regime switching.**

The studies of risk-sensitive optimal control have attracted much attention of researchers in the past decades. The aim of this project is to investigate a risk-sensitive optimal control problem for a Markov modulated Lévy process. As an application, we shall analyse the optimal wealth, investment and consumption for a risk sensitive control problem of an investor. A Malliavin calculus approach to such problem would also be explored.

### Dr Apostolos Papaioannou

**Project 1. Markov Arrival Process risk models in the presence of dividends strategies**

In this project, we can investigate the probability of ruin for a Markov Arrival risk model under various dividend strategies (constant, threshold, etc). This kind of risk processes can be used to model insurance portfolios with dependent risks .

**Project 2. Absolute ruin probabilities in time – dependent risk processes**

When the surplus process of an insurance portfolio becomes negative or the insurer is in deficit, the insurer could borrow money at a constant debit interest rate to repay the claims.

In this project, we can investigate the absolute ruin probabilities for a general class of dependent risk models.

**Project 3. Defective renewal equations for a class of dependent risk models**

In this project, we can investigate the derivation of defective renewal equations for many risk measures in a class of non –renewal risk models. Then, these defective renewal equations can be used to derive bounds for the corresponding risk measures.