First Passage Times of Regime Switching Models

The probability of a stochastic process to first breech upper and/or lower levels are important quantities 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 -- derived in closed-form by solving the matrix Wiener-Hopf factorization. This extends single barrier results in the 2-state model by Guo [2001]. If the quotient of drift and variance -- in Finance known as the Sharpe ratio -- is regime-independent, we show that the Laplace transform can be inverted analytically. Due to their analytical tractability and their ability to well explain many empirical phenomena, regime switching models have recently gained remarkable attention. Two examples modeling persistently changing long-term trends demonstrate this popularity. Due to their assumption of independent increments, Lévy processes are unable to capture such effects