Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients

Symmetry Coefficients of Semilinear Partial Differential Equations

We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m > 1 and the equation is also linear in its derivatives of order m− 1 of the dependent variable, then the infinitesimal of the dependent variable is at most linear on the dependent variable. Many examples of important partial...

Backward Stochastic Differential Equations with Stochastic Monotone Coefficients

We prove an existence and uniqueness result for backward stochastic differential equations whose coefficients satisfy a stochastic monotonicity condition. In this setting, we deal with both constant and random terminal times. In the random case, the terminal time is allowed to take infinite values. But in aMarkovian framework, that is coupled with a forward SDE, our result provides a probabilis...

Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

Backward Doubly Stochastic Differential Equations With Monotone and Discontinuous Coefficients

In this paper, we use the Yoshida approximation to prove the existence and uniqueness of a solution for the backward doubly stochastic differential equation when the generator is monotone and continuous. Before that we present the results for existence and uniqueness of an adapted solution of the backward doubly stochastic differential equation under some generals conditions.

Strong solutions to stochastic differential equations with rough coefficients

We study strong existence and pathwise uniqueness for stochastic differential equations in R 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 bounds for the solution of the corresponding Fokker-Planck PDE,...