Estimates of space derivatives for functions of non-autonomous and McKean-Vlasov processes. Application to uniform weak error bounds induced by the approximating subsampled particle system
This thesis is split in three parts, all of which obtain derivative estimates for the solution to the backward Kolmogorov equation associated to diverse stochastic process and study their application to uniform weak error bounds. More specifically, throughout the document we consider the function of time and space obtained after evaluating a fixed test function at a process with deterministic initial condition (space component) and governed by a predetermined SDE (time component). This function satisfies the so-called backward Kolmogorov equation and we refer to it as the backward Kolmogorov function. The first part of the thesis studies space derivatives estimates for the backward Kolmogorov function associated to non-autonomous SDEs. It presents sufficient conditions for derivatives' decay in the time component and shows how these can be obtained from customary assumptions used in finite time PDE analysis when combined with monotonicity. The second part applies these results to the scenario where the underlying stochastic process is a McKean-Vlasov process and presents complementary results which lead to uniform in time gradient estimates of the backward Kolmogorov function. The obtained bounds do not rely on results available in the literature and are based on lax regularity assumptions of the coefficients and again monotonicity conditions. The third part develops a more intuitive, although more restricted, way of obtaining decaying in time derivative estimates of the backward Kolmogorov function associated to a McKean-Vlasov process. Moreover, it uses them to derive uniform weak error estimates for the time discretization of the subsampled particle system approximating a McKean-Vlasov process.