On Some Functionals of the First Passage Times in Models With Switching Stochastic Volatility

Abstract

We compute some functionals related to the joint generalized 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 Brownian 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.

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

“…Aiming to obtain closed-form expressions for the functionals, we assume that the considered jump processes are driven by compound Poisson processes with (positive) exponentially distributed jumps. The same functionals related to the generalised Laplace transforms of the first exit times from half strips were computed in [18] in parallel for a two-dimensional continuous diffusion-type processes with switching coefficients. It is known that optimal stopping problems for multi-dimensional continuous-time Markov processes are analytically more difficult than the problems for the one-dimensional ones and the solutions of the former are rarely found explicitly.…”