Stochastic partial differential equations with singular terminal condition

Stochastic Partial Differential Equations

Backward Stochastic Differential Equations with Random Stopping Time and Singular Final Condition

Introduction. Let (Ω,F ,P) be a probability space, B = (Bt)t≥0 a Brownian motion defined on this space, with values in Rd. (Ft)t≥0 is the standard filtration of the Brownian motion. Also given are τ a {Ft}-stopping time, ξ a real, Fτ -measurable random variable, called the final condition, and f :Ω× R+ × R×Rd → R the generator. We wish to find a progressively measurable solution (Y,Z), with val...

Nonlinear Stochastic Partial Differential Equations with Singular Diffusivity and Gradient Stratonovich Noise

We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by ...

Forward-Backward Doubly Stochastic Differential Equations with Random Jumps and Stochastic Partial Differential-Integral Equations

In this paper, we study forward-backward doubly stochastic differential equations driven by Brownian motions and Poisson process (FBDSDEP in short). Both the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) and stochastic Hamiltonian systems arising in stochastic optimal control problems with random jum...

A Primer on Stochastic Partial Differential Equations

These notes form a brief introductory tutorial to elements of Gaussian noise analysis and basic stochastic partial differential equations (SPDEs) in general, and the stochastic heat equation, in particular. The chief aim here is to get to the heart of the matter quickly. We achieve this by studying a few concrete equations only. This chapter provides sufficient preparation for learning more adv...