## Mean-variance hedging with random volatility jumps.(English)Zbl 1035.91028

From authors’ abstract: The authors introduce a general framework for stochastic volatility models, with the risk 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 $$(\omega,\eta)\in (\Omega\times H, {\mathcal T}^\Omega\otimes{\mathcal T}^H, P^\Omega\otimes P^H)$$.
In particular, the authors allow for random discontinuities in the volatility $$\sigma$$ and the drift $$\mu$$. First it is characterized the set of equivalent martingale measures, then compute the mean-variance optimal measure $$\widetilde P$$, using some results of Schweizer on the existence of an adjustment process $$\beta$$. It is shown the examples where the risk premium $$\lambda= (\mu-r)/\sigma$$ follows a discontinuous process, and explicit calculations for $$\widetilde P$$.

### MSC:

 91B28 Finance etc. (MSC2000) 60H30 Applications of stochastic analysis (to PDEs, etc.)
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