Finite–dimensional Markovian realizations for stochastic volatility forward–rate models

Abstract

We consider forward-rate models of Heath-Jarrow-Morton type, as well as more general infinite-dimensional stochastic differential equations, where the volatility-diffusion term is stochastic in the sense of being driven by a separate hidden Markov process. Within this framework, we use the previously developed Hilbert-space-realization theory in order to provide general necessary and sufficient conditions for the existence of a finite-dimensional Markovian realization for the stochastic volatility models. We illustrate the theory by analysing a number of concrete examples.