Fokker-Planck equations with terminal condition and related McKean probabilistic representation

Lucas Izydorczyk (1) , Nadia Oudjane (2) , Francesco Russo (1) , Gianmario Tessitore (3)

1 OC - Optimisation et commande
2 EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie
3 Dipartimento di Matematica e Applicazioni [Milano]

Usually Fokker-Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for existence and uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process.

Mots clés: Inverse problem; McKean stochastic differential equation; probabilistic representation of PDEs; time-reversed diffusion; Fokker Planck equation; and phrases Inverse problem

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Lucas Izydorczyk, Nadia Oudjane, Francesco Russo, Gianmario Tessitore. Fokker-Planck equations with terminal condition and related McKean probabilistic representation. Nonlinear Differential Equations and Applications, 2022, volume 29 (10), ⟨10.1007/s00030-021-00736-1⟩. ⟨hal-02902615v3⟩