Conference Agenda

In the past few years there has been a surge in interest in the estimation of solutions to stochastic differential equations on Riemannian manifolds. The need for such methods arose from a desire to be able to sample from probability measures on Riemannian manifolds. Taking inspiration from Euclidean space, the Euler discretisation of the over-damped Langevin diffusion has been used to tackle the problem of sampling. However rates for weak convergence of the algorithm had not yet been proved without a reliance on embedding the manifold into a high dimensional copy of Euclidean space.

Analysis of the Euler scheme in the weak sense also raises the question about convergence rates in the strong sense. By following closely to the approach laid out in the seminal works of Milstein, we show how to generate high order strong schemes on a Riemannian manifold with non-positive curvature. In particular, we present the Milstein correction to the Euler scheme which yields a scheme of order 1.

The talk will give an overview of recent results obtained in joint work with Karthik Bharath, Akash Sharma, and Michael Tretyakov.

Regularisation by noise in the context of stochastic differential equations (SDEs) with coefficients of low regularity, known as singular SDEs, refers to the beneficial effect produced by noise so that the singularity from the coefficients is smoothed out yielding well-behaved equations. Kinetic SDEs, also sometimes called second order SDEs, as one typical type of stochastic Hamiltonian systems, describe the motion of a particle perturbed by some random external force. The difference of comparing it with usual SDEs is that the noise of the space position vanishes and only appears in the direction of velocity, hence less noise gets involved in the system. In this talk we will discuss about the regularization effect by the degenerate noise for the singular kinetic SDEs from numerical approximation and also particles approximation view point.

In this talk the approximation of jump-diffusion stochastic differential equations with discontinuous drift, possibly degenerate diffusion coefficient, and Lipschitz continuous jump coefficient is studied. These stochastic differential equations can be approximated with a jump-adapted approximation scheme with a convergence rate $3/4$ in $L^p$. This rate is optimal for jump-adapted approximation schemes. We present an advanced adaptive approximation scheme to improve this convergence rate. Our scheme obtains a strong convergence rate of at least $1$ in $L^p$ in terms of the average number of evaluations of the driving noises.