Feynman–Kac formulas for regime-switching jump diffusions and their applications

Abstract

This work develops Feynman-Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated to a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal, and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.

Almost Sure and Moment Exponential Stability of Regime-Switching Jump Diffusions

“…Then for any ̺ > ε > 0, there exists a positive constant T such that E[|X(t)| p ] ≤ e −(̺−ε)t for all t ≥ T. This, together with Lemma 3.1 of Zhu et al (2015), implies that there exists some positive number M so that…”

“…For any t ≥ 0 and (x, i) ∈ R n × M, we consider the function f (t, x, i) := e c 3 t V (x, i). Condition (i) and Lemma 3.1 of Zhu et al (2015) imply that…”

“…Moreover, in view of Zhu et al (2015), for each initial condition (x 0 , α 0 ) ∈ R n ×M, the system represented by (2.3) and (2.5) (or equivalently, (2.3) and (2.7)) has a unique strong solution (X(·), α(·)) = (X x 0 ,α 0 (·), α x 0 ,α 0 (·)); the solution does not explode in finite time with probability one. In addition, the generalized Itô lemma reads…”

“…Consequently increasing attention has been drawn to the study of regime-switching jump diffusions in recent years. Some recent work in this vein can be found in Shao and Xi (2014), Xi (2009), Yin and Xi (2010), Zhu et al (2015), Zong et al (2014) and the references therein.…”

Exit Problems as the Generalized Solutions of Dirichlet Problems

“…If L is a non-local operator corresponding to a Lévy jump diffusions, which have become popular in the recent development in financial modeling, see [10,13,24], the discontinuity of the random path X ∼ L brings extra difficulty in studying the boundary behavior (see [5,17,29,34]). To the best of our knowledge, the verification for a Feynman-Kac functionals to be a solution, or even a generalized viscosity solution (as discussed in this paper) of the Dirichlet PDE, has not been thoroughly studied for jump diffusion in the extant literature.…”