On time-dependent boundary crossing probabilities of diffusion processes as differentiable functionals of the boundary
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913472878477312 |
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| author | Liang, Vincent Borovkov, Konstantin |
| author_facet | Liang, Vincent Borovkov, Konstantin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16787 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On time-dependent boundary crossing probabilities of diffusion processes as differentiable functionals of the boundary Liang, Vincent Borovkov, Konstantin Probability 60J60 (Primary) 60G40, 60G46 (Secondary) 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. |
| title | On time-dependent boundary crossing probabilities of diffusion processes as differentiable functionals of the boundary |
| topic | Probability 60J60 (Primary) 60G40, 60G46 (Secondary) |
| url | https://arxiv.org/abs/2401.16787 |