Multidimensional Nonlinear Boundary Crossing Problems with Applications

Abstract

We study the first passage time of a multidimensional simple random walk crosses a certain type of nonlinear boundary, which occurs in various disciplines, including finance applications such as one-factor models. Under some regularity conditions, we derive the asymptotic expansion for the expected stopping time. The evaluation is possible due to a device that first rewrite the problem as an one dimensional Markov random walk crossing a linear boundary, and then approximate this Markov random walk by a sequence of uniformly ergodic Markov random walks. By such, we are able to study mutlidimesional nonlinear problems through generalizing existing one dimensional linear Markov renewal theories. Numerical simulations are given for illustration. Generalizations are also presented. Keywords：First passage probabilities, ladder height distribution, Markov renewal theory, rate of convergence.