Weak Error for Continuous Time Markov Chains Related to Fractional in Time P(I)DEs

TL;DR

This paper derives precise error bounds for the difference between transition densities of multidimensional CTMCs and solutions of fractional in time PDEs, linking stochastic processes with fractional differential equations.

Contribution

It provides sharp error estimates connecting continuous-time Markov chains with fractional in time PDEs involving Caputo derivatives and stochastic differential operators.

Findings

Established explicit error bounds for transition densities

Linked CTMCs with fractional PDE solutions accurately

Applicable to Brownian and stable driven SDEs

Abstract

We provide sharp error bounds for the difference between the transition densities of some multidimensional Continuous Time Markov Chains (CTMC) and the fundamental solutions of some fractional in time Partial (Integro) Differential Equations (P(I)DEs). Namely, we consider equations involving a time fractional derivative of Caputo type and a spatial operator corresponding to the generator of a non degenerate Brownian or stable driven Stochastic Differential Equation (SDE).