On strong solutions for positive definite jump diffusions

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 α th positive semidefinite power of the process itself with 0.5 < α < 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.