Generalized Wishart processes under nonlinear expectation

Abstract

We develop a theory of generalized Wishart processes within the framework of $G$-expectation, with $G$-Dyson Brownian motion and the $G$-Wishart process as two typical examples. In contrast to the classical setting, we show that $G$-Dyson Brownian motion may exhibit collisions and provide a sufficient condition ensuring the non-collision of the eigenvalues of generalized Wishart processes. We then derive the equations governing the high-dimensional limits of the empirical spectral measures under $G$-expectation, leading to the corresponding semicircle law for $G$-Dyson Brownian motion and Marchenko--Pastur law for the $G$-Wishart process.