PreprintarXiv licencearXiv
Volterra Integral Reduction for Boundary Diffusion Problems
Abstract
This paper addresses a class of integral representations of the form f(t,x)=g(t,x)+∫0t k(t,s) p(t−s,x,y) ∂x f(s,y+) ds, 0 ≤ t ≤ T, where f is unknown, p is the transition density of a diffusion process, and $g, k$ are prescribed functions. For an arbitrary diffusion process with sufficiently regular coefficients, we prove that this problem is equivalent to a Volterra integral equation of the second kind. This reduction provides a unified framework for both theoretical analysis and numerical approximation. An example of the implementation in the context of financial mathematics is presented.
Related
- A 2+1-dimensional extended Dym equation: moving boundary problems solvable via Painlevé II symmetry reduction
- An Extended 2+1-Dimensional Gardner Equation: On Moving Boundary Problems Solvable via Ermakov-Painlevé II Symmetry Reduction
- A novel extension of the canonical 2+1-dimensional Dym equation. Moving boundary problems solvable via Painleve' II Reduction
- Boundary integral equation based solvers for elastic transmission problems in complex media
- Carleson measures, Volterra integral operators and multipliers for QK spaces
BibTeX
@misc{shabalin2026volterra,
title = {{Volterra Integral Reduction for Boundary Diffusion Problems}},
author = {Danila Shabalin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03669},
eprint = {2609.03669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03669},
}
RIS
TY - UNPB
TI - Volterra Integral Reduction for Boundary Diffusion Problems
AU - Danila Shabalin
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03669
UR - https://arxiv.org/abs/2609.03669
AB - This paper addresses a class of integral representations of the form f(t,x)=g(t,x)+∫_0^t k(t,s) p(t−s,x,y) ∂_x f(s,y^+) ds, 0 ≤ t ≤ T, where f is unknown, p is the transition density of a diffusion process, and $g, k$ are prescribed functions. For an arbitrary diffusion process with sufficiently regular coefficients, we prove that this problem is equivalent to a Volterra integral equation of the second kind. This reduction provides a unified framework for both theoretical analysis and numerical approximation. An example of the implementation in the context of financial mathematics is presented.
ER -