Mario Lefebvre†, Marouane Chafi‡
Abstract: First-passage problems are considered for a Wright-Fisher diffusion process {X(t), t ≥ 0}, which is important in population genetics and mathematical finance. Let τ(x) be the first time that, starting from X(0) = x, is equal to 0 or to 1. We compute the expected area covered by the process in the interval [0, τ(x)]. We also compute the probability that the process will take on a value smaller than or equal to 0 before a value greater than or equal to 1 in the case when there are uniform jumps according to a Poisson process. Finally, stochas- tic control problems called homing problems are solved explicitly in particular cases for a controlled version of the process {X(t), t ≥ 0}.
Keywords: Brownian motion, first-passage time, stochastic optimal control, dynamic programming, non-linear differential equation.
MSC: 60J70, 93E20.
DOI 10.56082/annalsarscimath.2025.3.39
† mario.lefebvre@polymtl.ca, Department of Mathematics and Industrial Engineering, Polytechnique Montreal, P.O. Box 6079, Station Centre-Ville, Montreal, Qc, Canada
‡ marouane.chafi@polymtl.ca, Department of Mathematics and Industrial Engineering, Polytechnique Montreal, P.O. Box 6079, Station Centre-Ville, Montreal, Qc, Canada
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 17 no 3, 2025
