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PreprintarXiv licencearXiv

On the First Hitting Time Problems for Diffusion Processes: Local Time-Space Approach

  • Jerome Detemple
  • Yerkin Kitapbayev
  • Danila Shabalin
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Abstract

Using the local time-space calculus of Peskir (2005) and the method developed in Mijatovic (2010), we derive a new integral representation for the backward probability associated with the first-passage time (FPT) of a diffusion process through a time-dependent barrier. We present a complete three-step numerical algorithm: first, the problem is reduced to a Volterra-type integral equation; second, its kernel is approximated by a Markov chain; finally, the resulting equation is solved using a quadrature rule. The method is implemented for several representative examples, and its convergence properties are established. An extension to double barrier problems is carried out

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Cite

BibTeXRIS
BibTeX
@misc{detemple2026first,
  title = {{On the First Hitting Time Problems for Diffusion Processes: Local Time-Space Approach}},
  author = {Jerome Detemple and Yerkin Kitapbayev and Danila Shabalin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.13732},
  eprint = {2608.13732},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.13732},
}
RIS
TY  - UNPB
TI  - On the First Hitting Time Problems for Diffusion Processes: Local Time-Space Approach
AU  - Jerome Detemple
AU  - Yerkin Kitapbayev
AU  - Danila Shabalin
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2608.13732
UR  - https://arxiv.org/abs/2608.13732
AB  - Using the local time-space calculus of Peskir (2005) and the method developed in Mijatovic (2010), we derive a new integral representation for the backward probability associated with the first-passage time (FPT) of a diffusion process through a time-dependent barrier. We present a complete three-step numerical algorithm: first, the problem is reduced to a Volterra-type integral equation; second, its kernel is approximated by a Markov chain; finally, the resulting equation is solved using a quadrature rule. The method is implemented for several representative examples, and its convergence properties are established. An extension to double barrier problems is carried out
ER  -