Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators

Pierre Etoré (1) , Miguel Martinez (2)

1 IPS - Inférence Processus Stochastiques
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

In this paper we study time-inhomogeneous versions of one-dimensional Stochastic Differential Equations (SDE) involving the Local Time of the unknown process on curves. After proving existence and uniqueness for these SDE under mild assumptions, we explore their link with Parabolic Differential Equations (PDE) with transmission conditions. We study the regularity of solutions of such PDE and ensure the validity of a Feynman-Kac representation formula. These results are then used to characterize the solutions of these SDE as time-inhomogeneous Markov Feller processes.

Mots clés: Feynman-Kac representation formula; Divergence Form Operators; time inhomogeneous Skew Brownian Motion; Stochastic Differential Equations with Local Time; time inhomogeneous Markov processes

Domaines: Probabilités [math.PR]

Pierre Etoré, Miguel Martinez. Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators. Stochastic Processes and their Applications, 2018, 128 (8), pp.2642-2687. ⟨10.1016/j.spa.2017.09.018⟩. ⟨hal-01356270v2⟩