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

Abstract

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 Gâteaux differentiable in directions $h\in H\cup C^2$ and Fréchet-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.

38 pages, 4 figures. In this version: added Theorem 4 proving that F(g) is H-differentiable; added Theorem 5 on the error asymptotics in case of piece-wise linear approximating boundary; restored the proof of Theorem 2; added Example 3 to illustrate Theorem 3 (hyperbolic diffusion); added Example 4 to illustrate Theorem 5 (Brownian motion process and the Daniels boundary)