Boundary crossing densities for Brownian motion:

His research focuses on a theoretical problem in stochastic analysis: the “boundary crossing problem.” This problem aims to determine the hitting time of a Brownian motion to a specific boundary, represented as a time-dependent function f(t). Stochastic calculus—essential for solving complex probability problems and differential equations—is a key tool in modeling this hitting time, with applications in finance and other applied fields due to its ability to predict the behavior of random processes. The work specifically focuses on finding the probability density of the hitting time when the boundary is a parabola rather than a constant, which introduces additional complexity. Using the heat equation with a moving boundary, along with tools from complex analysis and numerical methods, the study extends previous solutions to this problem and proposes new strategies for handling varying boundary conditions. This expands the applicability of stochastic analysis to moving boundary problems in fields such as mathematical finance.