2024/10/28 by S. E. Chorfi, Abderrahmane Habbal, Chorfi, S. E. +7 · 1 citation
Economics, Econometrics and Finance · Mathematics · #35K565 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Numerical methods in inverse problems #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2410.21541
openalex publication_date 2024/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate inverse backward-in-time problems for a class of second-order degenerate Mean-Field Game (MFG) systems. More precisely, given the final datum (u(⋅, T),m(⋅, T)) of a solution to the one-dimensional mean-field game system with a degenerate diffusion coefficient, we aim to determine the intermediate states (u(⋅,t0),m(⋅,t0)) for any t0 ∈ [0, T), i.e., the value function and the mean distribution at intermediate times, respectively. We prove conditional stability estimates under suitable assumptions on the diffusion coefficient and the initial state (u(⋅,0),m(⋅,0)). The proofs are based on Carleman's estimates with a simple weight function. We first prove a Carleman estimate for the Hamilton-Jacobi-Bellman (HJB) equation. A second Carleman estimate will be derived for the Fokker-Planck (FP) equation. Then, by combining the two estimates, we obtain a Carleman estimate for the mean-field game system, leading to the stability of the backward problems.