Abstract
First, we present a model for the instantaneous variance of an Hilbert space valued Ornstein-Uhlenbeck process (OU process), with dynamics given by a stochastic partial differential equation (SPDE). This model allows for a stochastic volatility with state dependent jumps. The existence of an appropriate variance process will be deduced from the existence of infinitely divisible processes on the cone of positive-definite Hilbert-Schmidt operators and by solving an associated class of ordinary differential equations, allowing for a reconstruction of the Lévy-Khintchine exponent of the variance process.
Second, we extend the Wishart processes to Hilbert-Schmidt operator framework.
This is based on a work in progress with Sonja Cox, Christa Cuchiero, and Sven Karbach.