On strong solutions for positive definite jump-diffusions

TL;DR

This paper proves the existence of unique strong solutions for a class of positive definite jump-diffusions, extending results for Wishart processes and introducing new models with power-law diffusion coefficients.

Contribution

It establishes existence conditions for complex matrix-valued SDEs, including affine, Wishart, and GARCH-like processes, broadening the scope of stochastic models on positive definite matrices.

Findings

Existence of unique strong solutions for affine diffusion processes.

Extension of Wishart process results to broader classes.

Conditions for SDEs with power-law diffusion coefficients.

Abstract

We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics. Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the alpha-th positive semidefinite power of the process itself with 0.5<alpha<1 and obtain existence conditions for them. In the case of a diffusion coefficient which is linear in the process we likewise get a positive definite analogue of the univariate GARCH diffusions.