Uniqueness in Law for a Class of Degenerate Diffusions with Continuous Covariance

TL;DR

This paper proves the well-posedness of the martingale problem for a class of degenerate diffusions with continuous covariance matrices, extending results to certain non-constant drift fields.

Contribution

It establishes uniqueness and existence of solutions for degenerate diffusions with continuous, positive-definite covariance and structured drift matrices, including non-constant vector fields.

Findings

Martingale problem is well-posed under continuous, positive-definite covariance.

Results extend to non-constant drift fields with nondegenerate Jacobian.

Well-posedness holds for structured lower-diagonal block matrices B.

Abstract

We study the martingale problem associated with the operator $L u = \partial_s u + 1/2 \sum_{i,j=1}^{d_0} a^{ij} \partial_{ij} u + \sum_{i,j=1}^d B^{ij} x^j \partial_i u$, where $d_0 \leq d$. We show that the martingale problem is well-posed when the function $a$ is continuous and strictly positive-definite on $\bb R^{d_0}$ and the matrix $B$ takes a particular lower-diagonal, block form. We then localize this result to show that the martingale problem remains well-posed when $B$ is replaced by a sufficiently smooth vector field whose Jacobian matrix satisfies a nondegeneracy condition.