Iterative weak approximation and hard bounds for switching diffusion

TL;DR

This paper introduces a new iterative framework for approximating switching diffusion processes, simplifying complex PDE systems into independent equations, with proven convergence and bounds, supported by numerical experiments.

Contribution

The paper presents a novel convergent iteration method for weak approximation of switching diffusions, enabling efficient numerical solutions through decoupling PDE systems.

Findings

Convergence of the iterative approximation is rigorously proven.

Upper and lower bounds for solutions are effectively constructed.

Numerical results confirm the theoretical accuracy and efficiency.

Abstract

We establish a novel convergent iteration framework for a weak approximation of general switching diffusion. The key theoretical basis of the proposed approach is a restriction of the maximum number of switching so as to untangle and compensate a challenging system of weakly coupled partial differential equations to a collection of independent partial differential equations, for which a variety of accurate and efficient numerical methods are available. Upper and lower bounding functions for the solutions are constructed using the iterative approximate solutions. We provide a rigorous convergence analysis for the iterative approximate solutions, as well as for the upper and lower bounding functions. Numerical results are provided to examine our theoretical findings and the effectiveness of the proposed framework.