First Passage Time Moments of Jump-Diffusions with Markovian Switching

Using an integral equation associated with generalized backward Kolmogorov's equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of jump-diffusions with Markovian switching. The results are used to find the expectation of first passage time of some financial models. 1. Introduction Owing to the increasing demands on regime-switching diffusions in financial engineering and wireless communications, much attention has been drawn to switching jump diffusion processes. For example, one of the early efforts of using such hybrid models for financial applications can be traced back to [1, 2], in which both the appreciation rate and the volatility of a stock depend on a continuous Markov chain. The introduction of such models makes it possible to describe stochastic volatility in a relatively simpler manner. A stock market may be considered to have two “modes” or “regimes”, up and down, resulting from the state of the underlying economy, the general mood of investors in the market, and so on. The rationale is that in the different modes or regimes, the volatility and return rates are very different. For instance, in a stock market, the regimes can be roughly divided into two states, bull market and bear market. The market sentiment and reaction to the two states are in stark contrast. Normally, a bare market is more volatile than that of a bull market. Another need is to capture the features of insurance policies that are subject to the economic or political environment changes. It is thus sensible and necessary to take such regime changes into consideration. Also due to the needs from modeling points of view, formulation of dividend optimization problem often results in complex models in order to take into consideration various scenarios arising in finance and insurance practice. For example, in lieu of the diffusion models or jump-diffusion models, one may consider such models with regime switching modulated by a continuous time Markov chain, in which the Markov chain is used to delineate the random environment. Adopting a Markov regime-switching model is an easy way to capture all the cyclical features of the drift and volatility of asset return depending on market environment. For a long time, lots of empirical evidences have supported the existence of regime-switching property in financial markets. For example, Lewellen found that expected returns on equities change over time in [3]. Schwert concluded that the volatilities of stock returns also vary substantially over time in