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 $F\left(g\right)$ of a general time-inhomogeneous, one-dimensional diffusion process to perturbations of the boundary $g$. We prove that, for time-dependent 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 $F\left(g\right)$ using boundaries $\bar{g}$ that are close to $g$ and for which the computation of $F\left(\bar{g}\right)$ is feasible. We also obtain auxiliary results of independent interest in both probability theory and PDE theory.