The martingale problem for a class of nonlocal operators of diagonal type

TL;DR

This paper investigates the uniqueness of weak solutions for systems of stochastic differential equations driven by independent stable processes with diagonal coefficient matrices, using the martingale problem approach.

Contribution

It establishes the uniqueness of weak solutions for a class of nonlocal operators of diagonal type through martingale problem analysis.

Findings

Proves uniqueness of weak solutions for systems with diagonal coefficient matrices.

Extends martingale problem methods to nonlocal operators with stable processes.

Provides conditions for well-posedness of such stochastic systems.

Abstract

We consider systems of stochastic differential equations of the form \[ \d X_t^i = \sum_{j=1}^d A_{ij}(X_{t-}) \d Z_t^j\] for $i=1,\dots,d$ with continuous, bounded and non-degenerate coefficients. Here $Z_t^1,\dots,Z_t^d$ are independent one-dimensional stable processes with $\alpha_1,\dots,\alpha_d\in(0,2)$. In this article we research on uniqueness of weak solutions to such systems by studying the corresponding martingale problem. We prove the uniqueness of weak solutions in the case of diagonal coefficient matrices.