Abstract. In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial differential equations of parabolic type. Some emphasis is put on the fact that the who...

This paper aims to study a degenerate parabolic problem for solenoidal vector field in which the time derivative acts on moving body. We propose fully-discrete finite element scheme combined with backward Euler’s method saddle-point variational formulation. The convergence of this numerical is proved and error estimates some stable pairs are also established.

This paper is concerned with the Sobolev type weak solutions of one class second order quasilinear parabolic partial differential equations (PDEs, for short). First all, similar to Feng, Wang and Zhao [9] Wu Yu [29], we use a family coupled forward-backward stochastic (FBSDEs, short) which satisfy monotonous assumption represent classical PDEs. Then, based on PDEs approximating PDEs, prove exis...

Abstract Optimal control of diffusion processes is intimately connected to the problem solving certain Hamilton–Jacobi–Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate potential iterative optimisation techniques, in particular considering applications importance sampling and rare event simulation, focusing problems without ...

The boundary stabilization problem for a class of linear and nonlinear pseudo-parabolic differential equations is considered. The proposed control laws are used to achieve global exponential stability for the linear system and semi-global exponential stability for the nonlinear system in the H 1-sense. An H 2 bound of the solution for the nonlinear system is also derived. A numerical example is...

We study the deterministic counterpart of a backward–forward stochastic differential utility, which has recently been characterized as the solution to the Cauchy problem related to a PDE of degenerate parabolic type with a conservative first order term. We first establish a local existence result for strong solutions and a continuation principle, and we produce a counterexample showing that, in...

In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform backward Euler used temporal Furthermore, preconditioning approach is also to ensure uniform convergence. Numerical experiments show that method first-order accuracy in time almost space.

A new kind of backward stochastic differential equations (in short BSDE), where the solution is a pair of processes adapted to the past of the driving Brownian motion, has been introduced by the authors in [63. It was then shown in a series of papers by the second and both authors (see [8, 7, 9, 103), that this kind of backward SDEs gives a probabilistic representation for the solution of a lar...

the compact finite difference schemes have been found to give simple ways of reaching the objectives of high accuracy and low computational cost. during the past two decades, the compact schemes have been used extensively for numerical simulation of various fluid dynamics problems. these methods have also been applied for numerical solution of some prototype geophysical fluid dynamics problems ...