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

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  • Liang, V.
  • Borovkov, K.

Abstract

The paper analyses the sensitivity of the finite time horizon boundary non-crossing probability F(g) of a general time-inhomogeneous, one-dimensional diffusion process to perturbations of the boundary g. We prove that, for time-dependent boundaries g∈C2, this probability is Gâteaux differentiable in directions h∈H∪C2 and Fréchet-differentiable in directions h∈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 ḡ that are close to g and for which the computation of F(ḡ) is feasible. We also obtain auxiliary results of independent interest in both probability theory and PDE theory.

Suggested Citation

  • Liang, V. & Borovkov, K., 2025. "On time-dependent boundary crossing probabilities of diffusion processes as differentiable functionals of the boundary," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:spapps:v:190:y:2025:i:c:s0304414925001851
    DOI: 10.1016/j.spa.2025.104742
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    References listed on IDEAS

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