Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

Abstract

We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

Boundary crossing probabilities for high-dimensional Brownian motion

“…Hence, we expect that the above results could be extended to Y-type time tunnels illustrated in Figure 1. Further using a one-to-one transformation, we also expect that the above results are able to extend Y-type time tunnels for certain diffusion processes satisfying dX(t) = b(t, X(t)) dt + σ (t, X(t)) dW (t), for example the Ornstein-Uhlenbeck (OU) processes and the Brownian bridge; see [7]. Numerical examples of BCPs of Y-type time tunnels for Brownian motion and the OU process will be provided to illustrate the theoretical results in Section 5.…”

“…The following theorem is cited directly from [7] with the minor modification of condition (A) by allowing finite discontinuous points of the first kind. We do not repeat the proof here.…”

“…Recently, Fu and Wu [7] developed a computationally efficient method for BCPs based on the finite Markov chain imbedding (FMCI) technique. In this paper we extend the approach in [7] to high-dimensional Brownian motion. The reason for adopting the method are four fold: conceptually simple, flexible toward the boundaries, easy to program, and efficient in computation.…”

“…The paper is organized in the following way. In Section 2, notation and results in [7] are reviewed, including the approximate BCPs for one-dimensional Brownian motion and their convergence rates. In Section 3, two-dimensional and higher-dimensional Brownian 544 J. C. FU AND T.-L. WU motions are considered and their BCPs are studied.…”

Boundary non-crossing probabilities for fractional Brownian motion with trend

“…In the literature, most of contributions treat the case when the Gaussian process X(t), t ≥ 0 is a Brownian motion which allows to calculate the boundary non-crossing probability P (X(t) + f (t) < u, t ∈ [0, T ]), for some trend function f and two given constants T, u > 0 by various methods (see, e.g., [1,15]). For particular f including the case of a piecewise constant function, explicit calculations are possible, see, e.g., [17].…”