A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems^*

¹ Department of Epidemiology and Biostatistics, Xuzhou Medical University, Xuzhou, Jiangsu 221004, PR China.
² School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, PR China.

^** Corresponding author: xiaolishun@xzhmu.edu.cn

Received: 5 December 2016
Accepted: 6 November 2019

Abstract

In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.