arXiv subjects

Danila Shabalin

Publications and source records attributed to Danila Shabalin.

2 recordsLinked to original sources

arXiv2026

Volterra Integral Reduction for Boundary Diffusion Problems

This paper addresses a class of integral representations of the form \begin{equation} f(t,x)=g(t,x)+\int_0^t k(t,s)\, p(t-s,x,y)\, \partial_x f(s,y^+)\,ds, \qquad 0 \le t \le T, \end{equation} 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.

math.PR↗

arXiv2026

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

Using the local time-space calculus of Peskir (2005) and the method developed in Mijatovic (2010), we derive a new integral representation for the distribution of 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 method. The method is implemented for several representative examples, and its convergence properties are established. An extension to double barrier problems is carried out.

math.PR↗