First-passage times of regime switching models

Abstract

The probability of a stochastic process to first breach an upper and/or a lower level is an important quantity for optimal control and risk management. We present those probabilities for regime switching Brownian motion. In the 2- and 3-state model, the Laplace transform of the (single and double barrier) first-passage times is–up to the roots of a polynomial of degree 4 (respectively 6)–derived in closed-form by solving the matrix Wiener–Hopf factorization.11The matrix Wiener–Hopf factors of regime switching models are defined via a set of quadratic matrix equations (see, e.g., London et al., 1982; Barlow et al., 1990; Kennedy and Williams, 1990; Rogers and Shi, 1994; Asmussen, 1995). This concept was expanded to regime switching jump diffusions by Jiang and Pistorius (2008). This extends single barrier results in the 2-state model by Guo (2001b). If the quotient of drift and variance is constant over all states, we show that the Laplace transform can even be inverted analytically.