Boundary Crossing Probabilities for Diffusion Processes
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2025
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Melbourne Authors
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School of Mathematics and Statistics
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PhD thesis
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© 2025 Vincent Wenjun Liang
Abstract
The main objective of this thesis was to develop methods for approximating the probabilities for one-dimensional diffusion processes to stay between time-dependent boundaries in cases when a closed-form expression is unavailable. We also investigated resulting approximation errors. The considered problem is of fundamental importance in the theory of random processes and has diverse applications.
The thesis contains an extensive literature review, outlining the history of different methods and approaches used to attack this problem. The review was compiled with the additional aim of making it a useful reference for researchers and specialists working in applied areas who are interested in boundary crossing problems for diffusion processes.
We propose a refined version of the discrete-time \person{Markov} chain approximation method for computing boundary crossing probabilities of general diffusion processes on a finite time interval. For a broad class of boundaries and diffusion processes, we prove the convergence of the constructed approximations to the desired limit. Numerical results indicate that the convergence rate is $O(n^{-2})$ in the case of $C^2$-boundaries and uniform time grid with $n$ steps. This approach was extended to compute expectations of the \person{Feynman--Kac} type.
We also analyse the sensitivity of boundary crossing probabilities to perturbations of the boundary in the one-sided case. Considering the probability that the diffusion process stays below a boundary $g$ during a finite time interval as a functional $F(g),$ we prove the existence of and obtain an explicit representation for the \person{G\^ateaux} derivative of $F(g)$ for $g \in C^2$ in directions $h \in H \cup C^2,$ where~$H$ is the \person{Cameron--Martin} space. We also obtain auxiliary results of independent interest in both probability theory and partial differential equations.
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Keywords
Boundary crossing probabilities, First passage time, Diffusion processes, Brownian motion, Stochastic processes, Markov chain approximation, Feynman-Kac, Dirichlet problem, Moving boundary problem