Theoretical analysis of numerical schemes for stochastic (partial) differential equations

The theory and numerics of stochastic differential equations (SDEs) are well understood in the context of equations with `regular’ coefficients. An important effort is currently being made in the field, in the attempt to go beyond the canonical setup. However, the challenge is huge, and results are often very specialised and not easily extendable to different equations.

This project aims at developing a solid theoretical analysis of numerical schemes for backward SDEs and stochastic partial differential equations (SPDEs) whose coefficients have very low regularity (typically elements in the class of Schwarz distributions).

As an example, the type of SPDEs mentioned above features naturally in physical problems modelled by transport equations in porous media, like water flowing through porous rocks: in this case the velocity of the flow is modified at the level of individual molecules, because the size of the water molecules is comparable to that of rock’s pores. This can be mathematically modelled by taking the velocity as a very `rough’ function of space (e.g., Schwarz distributions).

A starting point for this work would be as follows. We should consider sequences of regularised versions of the stochastic equation under study (i.e., involving a mollification of the `rough’ coefficients). This approach is natural but hardly trivial because the convergence (and convergence rate) of the scheme will be strongly depending on a clever choice of the regularising sequence and on the norms that one adopts on the relevant functional spaces.

Current theoretical work of Dr Issoglio on these kinds of backward SDEs and SPDEs as well as some early results on numerical methods for SDEs can guide the start of the project. The potential outcomes of such study are likely to be of interest to the wide community of researchers working in stochastic analysis and PDE theory. Keywords: numerical methods, stochastic differential equations, BSDEs, SPDEs, irregular coefficients.