On some functionals of the first passage times in models with switching stochastic volatility

We compute some functionals related to the joint generalised Laplace transforms of the first times at which two-dimensional diffusion-type Markov processes exit half strips. It is assumed that the state space components are driven by constantly correlated Brow- nian motions and the dynamics of the coefficients are described by a continuous-time Markov chain. The method of proof is based on the solutions of the equivalent boundary- value problems for systems of elliptic-type partial differential equations for the associated value functions. The results are illustrated on several two-dimensional continuous mean- reverting or diverting models of switching stochastic volatility