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 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.