Calculation of Probability of the Exit of a Stochastic Process from a Band by Monte-Carlo Method: A Wiener-Hopf Factorization

This paper considers a method of the calculation of probability of the exit from a band of the solution of a stochastic differential equation. The method is based on the approximation of the solution of the considered equation by a process which is received as a concatenation of Gauss processes, random partition of the interval, Girsanov transform and Wiener-Hopf factorization, and the Monte-Carlo method. The errors of approximation are estimated. The proposed method is illustrated by numerical examples.

On the Distribution of First Exit Time for Brownian Motion with Double Linear Time-Dependent Barriers

This paper focuses on the first exit time for a Brownian motion with a double linear time-dependent barrier specified by y=a+bt, y=ct, (a>0, b<0, c>0). We are concerned in this paper with the distribution of the Brownian motion hitting the upper barrier before hitting the lower linear barrier. The main method we applied here is the Girsanov transform formula. As a result, we expressed the density of such exit time in terms of a finite series. This result principally provides us an analytical expression for the distribution of the aforementioned exit time and an easy way to compute the distribution of first exit time numerically.

ON GIRSANOV AND GENERALIZED FEYNMAN–KAC TRANSFORMATIONS FOR SYMMETRIC MARKOV PROCESSES

Let X be a Markov process, which is assumed to be associated with a symmetric Dirichlet form [Formula: see text]. For [Formula: see text], the extended Dirichlet space, we have the classical Fukushima's decomposition: [Formula: see text], where [Formula: see text] is a quasi-continuous version of u, [Formula: see text] the martingale part and [Formula: see text] the zero energy part. In this paper, we investigate two important transformations for X, the Girsanov transform induced by [Formula: see text] and the generalized Feynman–Kac transform induced by [Formula: see text]. For the Girsanov transform, we present necessary and sufficient conditions for which to induce a positive supermartingale and hence to determine another Markov process [Formula: see text]. Moreover, we characterize the symmetric Dirichlet form associated with the Girsanov transformed process [Formula: see text]. For the generalized Feynman–Kac transform, we give a necessary and sufficient condition for the generalized Feynman–Kac semigroup to be strongly continuous.