DH 3rd floor SR

 

Relatively little is known about the ability of numerical methods for stochastic differential equations (SDE

Self-interacting diffusions are solutions to SDEs with a drift term depending

on the process and its normalized occupation measure $\mu_t$ (via an interaction

potential and a confinement potential): $$\mathrm{d}X_t = \mathrm{d}B_t -\left(

\nabla V(X_t)+ \nabla W*{\mu_t}(X_t) \right) \mathrm{d}t ; \mathrm{d}\mu_t = (\delta_{X_t}

- \mu_t)\frac{\mathrm{d}t}{r+t}; X_0 = x,\,\ \mu_0=\mu$$ where $(\mu_t)$ is the

process defined by $$\mu_t := \frac{r\mu + \int_0^t \delta_{X_s}\mathrm{d}s}{r+t}.$$

We establish a relation between the asymptotic behaviour of $\mu_t$ and the

asymptotic behaviour of a deterministic dynamical flow (defined on the space of

the Borel probability measures). We will also give some sufficient conditions

for the convergence of $\mu_t$. Finally, we will illustrate our study with an

example in the case $d=2$.

 

 

A wide variety of problems arising in applications require the sampling of a

probability measure on the space of functions. Examples from econometrics,

signal processing, molecular dynamics and data assimilation will be given.

In this situation it is of interest to understand the computational

complexity of MCMC methods for sampling the desired probability measure. We

overview recent results of this type, highlighting the importance of measures

which are absolutely continuous with respect to a Guassian measure.

 

 

I will describe some applications of the main techniques of rough paths

theory to problems not related to SDE

  Schramm Loewner Evolutions (SLE) are random planar curves (if κ ≤ 4) or growing compact sets generated by a curve (if κ > 4). We consider more general L

  A dimer configuration of a graph is a subset of the edges, such that every vertex is contained in exactly one edge of the subset. We consider dimer configurations of the honeycomb lattice on the cylinder, which are known to be equivalent to configurations of interlaced particles. Assigning a measure to the set of all such configurations, we show that the probability that particles are located in any subset of points on the cylinder can be written as a determinant, i.e. that the process is determinantal. We also examine Markov chains of interlaced particles on the circle, with dynamics equivalent to RSK.  

 

Gradient bounds are a very powerful tool to study heat kernel measures and

regularisation properties for the heat kernel. In the elliptic case, it is easy

to derive them from bounds on the Ricci tensor of the generator. In recent

years, many efforts have been made to extend these bounds to some simple

examples in the hypoelliptic situation. The simplest case is the Heisenberg

group. In this talk, we shall discuss some recent developments (due to H.Q. Li)

on this question, and give some elementary proofs of these bounds.

 

  The talk is based on my recent joint work with Zhechun Hu and Wei Sun. We consider a nonlinear filtering problem for general right continuous Markov processes associated with semi-Dirichlet forms. We show that in our general setting the filtering processes satisfy also DMZ (Duncan-Mortensen-Zakai) equation. The uniqueness of the solutions of the filtering equations are verified through their Wiener chaos expansions. Our results on the Wiener chaos expansions for nonlinear filters with possibly unbounded observation functions are novel and have their own interests. We investigate further the absolute continuity of the filtering processes with respect to the reference measures and derive the density equations for the filtering processes.

  The "Cubature on Wiener space" algorithm can be regarded as a general approach to high order weak approximations. Based on this observation we will derive many well known weak discretisation schemes and optimise the computational effort required for a given accuracy of the approximation. We show that cubature can also help to overcome some stability difficulties. The cubature on Wiener space algorithm is frequently combined with partial sampling techniques and we outline an extension to these methods to reduce the variance of the samples. We apply the extended method to examples arising in mathematical finance. Joint work of G. Gyurko, C. Litterer and T. Lyons