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

Abstract

We compute some functionals related to the generalised joint Laplace transforms of the first times at which two-dimensional jump processes exit half strips. It is assumed that the state space components are driven by Cox processes with both independent and common (positive) exponential jump components. The method of proof is based on the solutions of the equivalent partial integro-differential boundary-value problems for the associated value functions. The results are illustrated on several two-dimensional jump models of stochastic volatility which are based on non-affine analogues of certain mean-reverting or diverting diffusion processes representing closed-form solutions of the appropriate stochastic differential equations.

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

“…The Laplace transforms of the first passage times from intervals for mean-reverting and diverting one-dimensional jumpdiffusion processes were recently computed in Gapeev and Stoev [15]. Some functionals related to the generalised Laplace transforms of the first times at which some two-dimensional jump-diffusion processes exit half strips were recently computed in Gapeev and Stoev [16]. In the present paper, we derive closed-form expressions for the functionals related to the generalised joint Laplace transforms of the first passage times as stopping problems for two-dimensional diffusion-type Markov processes with switching coefficients.…”