Quasi-linear PDEs and forward-backward stochastic differential equations: Weak solutions

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:264

Article

Feng, Chunrong¹  Wang, Xince¹  Zhao, Huaizhong¹ 

[1] Loughborough Univ Technol, Dept Math Sci, Loughborough LE11 3TU, Leics, England

关键词: Forward backward stochastic differential equations;    Weak solutions;    Quasi-linear partial differential equations;    Probabilistic representation;    Parabolic;    Elliptic;   

DOI  :  10.1016/j.jde.2017.09.030

来源: Elsevier

【 摘 要 】

In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space H-rho(1)(R-d). For this, we study first the solutions of forward-backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space L-rho(2)(R-d;R-d)circle times L-rho(2)(R-d;R-k)circle times L-rho(2)(R-d;R-kxd). This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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