Strong solutions to stochastic differential equations with rough coefficients

Nicolas Champagnat (1, 2) , Pierre-Emmanuel Jabin (3)

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 IECL - Institut Élie Cartan de Lorraine
3 Department of Mathematics [College Park]

We study strong existence and pathwise uniqueness for stochastic differential equations in $\RR^d$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and $L^p$ bounds for the solution of the corresponding Fokker-Planck PDE, which can be proved separately. This allows a great flexibility regarding the method employed to obtain these last bounds. Hence we are able to obtain general criteria in various cases, including the uniformly elliptic case in any dimension, the one-dimensional case and the Langevin (kinetic) case.

Mots clés: strong solutions; stochastic differential equations; Fokker-Planck equation; degenerate diffusion matrix; pathwise uniqueness; rough drift; rough diffusion matrix; maximal operator

Domaines: Probabilités [math.PR]

Nicolas Champagnat, Pierre-Emmanuel Jabin. Strong solutions to stochastic differential equations with rough coefficients. Annals of Probability, 2018, 46 (3), pp.1498-1541. ⟨10.1214/17-AOP1208⟩. ⟨hal-00799242v2⟩