# PhD projects

The stochastic analysis and mathematical finance research group consists of two overlapping research subgroups: (1) stochastic analysis and (2) mathematical finance. The research in this group concentrates its interests on infinite dimensional stochastic analysis, including stochastic differential equations on infinite dimensional manifolds, stochastic partial differential equations (especially stochastic Navier-Stokes and Euler equations in relation to turbulence phenomena), stochastic analysis on Riemannian and Finslerian manifolds, Feynman path integrals, and applications to mathematical physics, biology and mathematical finance, in discrete and continuous times.

Below we present a brief descriptions of some projects offered by members of this group. Students interested in PhD projects in this group are welcome to contact individual members of staff directly.

## Stochastic Partial Differential Equations (Zdzislaw Brzezniak)

Zdzislaw Brzezniak is interested in the theory of ferromagnetism which was initiated by Weiss, see Brown (1978) and references therein,  and further developed by Landau and Lifshitz (1935) and Gilbert (1955). According to this theory the orientation of the magnetic moment $M$ of a ferromagnetic material occupying  a 3-dimensional region $D$ at temperatures below the critical (so-called Curie) temperature is of constant length and satisfies a certain system of degenerate parabolic equations called now Landau-Lifshitz-Gilbert (LLG) equation. The stationary solutions of this equation correspond to the equilibrium states of the ferromagnet and are not unique in general. An important problem in the theory of ferromagnetism is to describe the phase transitions between different equilibrium states induced by thermal fluctuations. Therefore, the LLG equation needs to be modified in order to incorporate random fluctuations of the field $H$ into the dynamics of the magnetization $M$ and to describe noise-induced transitions between equilibrium states of the ferromagnet. The program to analyze noise induced transitions was initiated by Neel (1946) and further developed in  Brown (1963), Kamppeter (1999) and others. Recent mathematical publications include  Kohn,  Reznikoff and  Vanden-Eijnden, Magnetic elements at finite temperature and large deviation theory,  J. Nonlinear Sci. 15 (2005), Brzezniak, Goldys and Jegaraj, Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation,  Appl. Math. Res. Express. (2012)  and Large deviations for a stochastic Landau-Lifshitz equation, extended version, arXiv:1202.0370, Banas, Brzezniak and Prohl, Convergent finite element discretization of the stochastic Landau-Lifshitz-Gilbert equation, (preprint).

## Configuration Space Analysis and Multicomponent Stochastic Systems  (Alexei Daletskii)

Alexei Daletskii is interested in study of multicomponent interecting particle systems. This research is motivated by applications in statistical mechanics and biology. An infinite system of interacting particles living in a Euclidean space or a Riemannian manifold $X$ is modeled by a measure $\mu$ of the space $\Gamma_X$ of countable subsets of $X$ (the configuration space). There are many exciting problems which can be studied in this framework:

• study of the relationship between properties of Gibbs measures on configuration spaces of Riemannian manifolds and geometry and topology of underlying manifolds
• study of the (stochastic) dynamics in concrete statistical mechanical or biological models (in particular, described by Poisson and Gibbs cluster point processes)
• study of phase transitions in particular models of statistical mechanics
• applications of measures on $\Gamma_X$ to the representation theory of  diffeomorphism groups of $X$

## Option Pricing in Financial Models with Friction (Alet Roux)

Alet Roux is interested in the pricing and hedging of options in financial models with friction, for example transaction costs and borrowing and lending constraints. She is also interested in optimal stopping problems arising from the pricing of American and game options. There are many interesting problems that can be studied in this framework:

• Developing efficient algorithms for computing option prices, hedging strategies and optimal stopping times.
• Developing theoretical results on the representation of option prices in terms of dual variables.
• Developing the option pricing theory in terms of the modern pricing paradigms offered by utility maximization, indifference, expected shortfall, etc.

## PhD projects

Overview

Projects in individual research areas: