Boundary Non-Crossing Probabilities as Functionals of the Deterministic Variance Clock

We study finite-horizon first-passage time and boundary non-crossing probabilities for Gaussian martingales, viewed as continuous local martingales obtained by running Brownian motion on a deterministic variance clock associated with deterministic volatility. Our aim is to quantify how the associated survival probability changes when the variance clock is perturbed. Using a deterministic time change representation, we reduce the problem to a Brownian boundary-crossing problem with a transformed horizon and a transformed boundary. This allows us to combine time change arguments with recent differentiability results for boundary-crossing probabilities. Under suitable regularity assumptions, we derive a first-order sensitivity formula with respect to the variance clock. The derivative splits naturally into two components: one produced by the deformation of the transformed boundary and one produced by the variation of the terminal transformed horizon. Several explicit examples are provided, including affine barriers and nonlinear deterministic clocks. These examples show in particular that, for nonconstant boundaries, redistributing variance over calendar time can change the finite-horizon survival probability even when the terminal variance is kept fixed.