Mean-variance hedging with random volatility jumps

We introduce a general framework for stochastic volatility models, with the risky asset dynamics given by: dX(t)(omega, eta) = mu(t)(eta)X-t(omega, eta)dt + sigma(t)(eta)X-t(omega,eta)dW(t)(omega) where (omega, eta) is an element of (Omega X H, F(Omega)circle timesF(H), p(Omega)circle timesP(H)). In particular, we allow for random discontinuities in the volatility sigma and the drift mu. First we characterize the set of equivalent martingale measures, then compute the meanvariance optimal measure (P) over tilde, using some results of Schweizer on the existence of an adjustment process beta. We show examples where the risk premium lambda = (mu - r)/sigma follows a discontinuous process, and make explicit calculations for (P) over tilde.