arXiv subjects

Vincent Liang

Publications and source records attributed to Vincent Liang.

3 recordsLinked to original sources

arXiv2024

On time-dependent boundary crossing probabilities of diffusion processes as differentiable functionals of the boundary

The paper analyses the sensitivity of the finite time horizon boundary non-crossing probability $F(g)$ of a general time-inhomogeneous diffusion process to perturbations of the boundary $g$. We prove that, for boundaries $g\in C^2,$ this probability is G\^ateaux differentiable in directions $h \in H \cup C^2$ and Fr\'echet-differentiable in directions $h \in H,$ where $H$ is the Cameron--Martin space, and derive a compact representation for the derivative of $F$. Our results allow one to approximate $F(g)$ using boundaries $\bar{g}$ that are close to $g$ and for which the computation of $F(\bar{g})$ is feasible. We also obtain auxiliary results of independent interest in both probability theory and PDE theory. These include: (i) an elegant probabilistic representation for the limit of the derivative with respect to $x$ of the boundary crossing probability when the process starts at point $(t,x)$ in the time-space domain and $x\uparrow g(t),$ and (ii) a Shiryaev--Yor type martingale representation for the indicator of the boundary non-crossing event for time-dependent boundaries.

math.PR↗

arXiv2023

On extension of the Markov chain approximation method for computing Feynman--Kac type expectations

An efficient discrete time and space Markov chain approximation employing a Brownian bridge correction for computing curvilinear boundary crossing probabilities for general diffusion processes was recently proposed in Liang and Borovkov (2021). One of the advantages of that method over alternative approaches is that it can be readily extended to computing expectations of path-dependent functionals over the event of the process trajectory staying between two curvilinear boundaries. In the present paper, we extend the scheme to compute expectations of the Feynman--Kac type that frequently appear in option pricing. To illustrate our approximation scheme, we apply it in three special cases. For sufficiently smooth integrands, numerical experiments suggest that the proposed approximation converges at the rate $O(n^{-2})$, where $n$ is the number of steps on the uniform time grid used

math.PR↗

arXiv2021

On Markov chain approximations for computing boundary crossing probabilities of diffusion processes

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.

math.PR↗