We propose a discrete-time discrete-space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the proposed approximation with the Brownian bridge correction is $O(n^{-2})$ in the case of $C^2$ boundaries and a uniform time grid with n steps.

The two-sided nonlinear boundary crossing probabilities for one-dimensional Brownian motion and related processes have been studied in Fu and Wu (2010) based on the finite Markov chain imbedding technique. It provides an efficient numerical method to computing the boundary crossing probabilities. In this paper we extend the above results for high-dimensional Brownian motion. In particular, we obtain the rate of convergence for high-dimensional boundary crossing probabilities. Numerical results are also provided to illustrate our results.

Given an autoregressive process X of order p(i.e.Xn = a1Xn−1 + ··· + apXn−p + Ynwhere the random variables Y1, Y2,... are i.i.d.), we study the asymptotic behaviour of the probability that the processdoes not exceed a constant barrier up to time N (survival or persistenceprobability). Depending on the coefficients a1,...,ap and the distribution ofY1, we state conditions under which the survival probabilitydecays polynomially, faster than polynomially or converges to a positive constant. Specialemphasis is put on AR(2) processes.

Using martingale methods, we derive a set of theorems of boundary crossing probabilities for a Brownian motion with different kinds of stochastic boundaries, in particular compound Poisson process boundaries. We present both the numerical results and simulation experiments. The paper is motivated by limits on exposure of UK banks set by CHAPS. The central and participating banks are interested in the probability that the limits are exceeded. The problem can be reduced to the calculation of the boundary crossing probability from a Brownian motion with stochastic boundaries. Boundary crossing problems are also very popular in many fields of statistics.

Given two correlated Brownian motions (Xt)t≥ 0 and (Yt)t≥ 0 with constant correlation coefficient, we give the upper and lower estimations of the probability ℙ(max0 ≤s≤tXs≥ a, max0 ≤s≤tYs≥ b) for any a,b,t > 0 through explicit formulae. Our strategy is to establish a new reflection principle for two correlated Brownian motions, which can be viewed as an extension of the reflection principle for one-dimensional Brownian motion. Moreover, we also consider the nonexit probability for linear boundaries, i.e. ℙ (Xt ≤ at+c,Yt ≤ bt+d, 0≤ t≤T) for any constants a, b≥0 and c,d, T > 0.

In this paper we analyze different forms of fractional relaxation equations of order ν ∈ (0, 1), and we derive their solutions in both analytical and probabilistic forms. In particular, we show that these solutions can be expressed as random boundary crossing probabilities of various types of stochastic process, which are all related to the Brownian motion B. In the special case ν = ½, the fractional relaxation is shown to coincide with Pr{sup0≤s≤tB(s) < U} for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the gamma density, we obtain more and more complicated fractional equations.

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.

Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation of the boundary crossing probabilities is feasible. In particular, we partially generalize and improve results obtained by Pötzelberger and Wang in the case when the approximating boundaries are piecewise linear. Applications to barrier option pricing are also discussed.

For the original Moran dam with independent and identically distributed inputs a representation of the stationary distribution is given which readily provides a geometric rate of convergence to this distribution. For the integer-valued case the stationary distribution can be expressed in terms of simple boundary crossing probabilities for the underlying random walk.