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Title: Quasi-linear PDEs and forward-backward stochastic differential equations: weak solutions
Authors: Feng, Chunrong
Wang, Xince
Zhao, Huaizhong
Keywords: Forward-backward stochastic differential equations
Weak solutions
Quasi-linear partial differential equations
Probabilistic representation
Infinite horizon
Issue Date: 2017
Publisher: © Elsevier
Citation: FENG, C., WANG, X. and ZHAO, H., 2017. Quasi-linear PDEs and forward-backward stochastic differential equations: weak solutions. Journal of Differential Equations, 264 (2), pp. 959-1018.
Abstract: 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 H1ρ(Rd). 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 L2ρ(Rd; Rd) ⊗ L2ρ(Rd; Rk) ⊗ L2ρ(Rd; Rk×d). 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.
Description: This paper was published in the journal Journal of Differential Equations and the definitive published version is available at https://doi.org/10.1016/j.jde.2017.09.030.
Sponsor: We would like to acknowledge the financial support of Royal Society Newton Advanced Fellowship NA150344.
Version: Accepted for publication
DOI: 10.1016/j.jde.2017.09.030
URI: https://dspace.lboro.ac.uk/2134/26622
Publisher Link: https://doi.org/10.1016/j.jde.2017.09.030
ISSN: 0022-0396
Appears in Collections:Published Articles (Maths)

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