Seminar in Probability for Student.

University of Bath, Claverton Down, Bath, BA2 7AY, UK

xpjp20@bath.ac.uk

++44 (0)1225 38 56 37

**Skills**

Probability

99%

Analysis

60%

Statistics

35%

Algebra

1%

**Keywords**

Stochastic Calculus

Brownian Motion

Malliavin Calculus

Reinforced Branching Processes

This a new seminar for student at University of Bath. The goal is to share specific knowledge in the area of probability and develop more interaction between postgraduate students. Undergraduate are accepted too for intenting the seminar. Every two weeks at 10.15AM in the Cosy Meeting Room, a postgraduate student will present one topic related to probability.

Minicourse.

In this presentation I will introduce the reinforced branching process with fitness and explain how general branching process theory can be used to analyse its growth dynamics.

In a stochastic optimal control problem, we aim to maximise some reward in the presence of uncertainty. (Slightly) more precisely, given an SDE whose coefficients depend on some parameter, we take a function which depends on the path of a solution to the SDE. The problem is then to maximise the expected value of the function over all admissible parameters. In this talk, I will introduce the main tools used to study stochastic optimal controlproblems. In particular, I will outline the Dynamic Programming Principle and show a connection to Hamilton-Jacobi-Bellman equations, which we can consider as a non-linear analogue to the Feynman-Kac representation. I will also draw on my own research to present some illustrative examples.

In this talk I will illustrate a wide range of possible definitions of solutions to finite and infinite dimensional stochastic differential equations. As a matter of fact, different notions of solutions are developed in order to deal with the particular features of the specific equations being the topic of study. A pratical example will also be discussed. This talk will be kept as self-contained and accessible as possible.

Topic: From a stochastic differential equation, we are able to define some notions of quantitative analysis (on symmetric diffusion) by introducing semi-group, generator, carre du champ and functional inequalities. We will reverse this approach. More precisely from a diffusion Markov Triple (Set, invariant measure and carre du champ only) we will construct a generator and "produce" a SDE. Finally we illustrate this "Alchemy" by working on a example.

Topic: Let there be a single particle at the origin of the d-dimensional integer lattice, and let a particle come from infinity and be aggregated according to the harmonic measure of the boundary. Repeating this procedure we have a growing aggregate of particles, and this is the classic diffusion limited aggregation (DLA) model. Since it's creation in 1981 it has been almost impervious to a strong mathematical treatment, and variations to the model have been considered. This talk will introduce the multi-particle DLA, where there is a Poisson cloud of particles of intensity K on the integer lattice that perform continuous time random walks, and particles attach to the aggregate if they attempt to occupy it. We will show that if K<1 in 1-dimension there is sub-linear growth, and if K>1 then there is linear growth. Higher dimensional results will be discussed, including the remarkable conjecture that there is linear growth in 2 or more dimensions if K>0.

Stochastic Differential Equations: An Introduction with Applications, Authors: Oksendal, Bernt.

Applied Stochastic Control of Jump Diffusions, Authors: Oksendal, Bernt, Sulem-Bialobroda, Agnes.

Stochastic Integration and Differential Equations, Authors: Protter, Philip E.

Brownian Motion, Authors: Hida, T.

Diffusion Processes and their Sample Paths: Authors: Itô, Kiyosi, McKean, Henry P. Jr.