2021/12/21 by Xinfu Chen, John Chadam, Chen, Xinfu +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Analysis of PDEs (math.AP) #Diffusion and Search Dynamics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2112.10918
openalex publication_date 2021/12/21 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28
Consider the inverse first-passage problem: Given a diffusion process \\frakXt\t\geqslant 0 on a probability space (Ω,F,ℙ) and a survival probability function p on [0,∞), find a boundary, x=b(t), such that p is the survival probability that \frakX does not fall below b, i.e., for each t\geqslant 0, p(t)= ℙ(\ω∈Ω | \frakXs(ω) \geqslant b(s), ∀ s∈(0,t)\). In earlier work, we analyzed viscosity solutions of a related variational inequality, and showed that they provided the only upper semi-continuous (usc) solutions of the inverse problem. We furthermore proved weak regularity (continuity) of the boundary b under additional assumptions on p. The purpose of this paper is to study higher-order regularity properties of the solution of the inverse first-passage problem. In particular, we show that when p is smooth and has negative slope, the viscosity solution, and therefore also the unique usc solution of the inverse problem, is smooth. Consequently, the viscosity solution furnishes a unique classical solution to the free boundary problem associated with the inverse first-passage problem.