Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

BSDEs with diffusion constraint and viscous Hamilton–Jacobi equations with unbounded data

Andrea Cosso, Huyên Pham, and Hao Xing

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Abstract

We provide a stochastic representation for a general class of viscous Hamilton–Jacobi (HJ) equations, which has convex and superlinear nonlinearity in its gradient term, via a type of backward stochastic differential equation (BSDE) with constraint in the martingale part. We compare our result with the classical representation in terms of (super)quadratic BSDEs, and show in particular that existence of a viscosity solution to the viscous HJ equation can be obtained under more general growth assumptions on the coefficients, including both unbounded diffusion coefficient and terminal data.

Résumé

Nous donnons une représentation stochastique pour une classe générale d’équations d’Hamilton–Jacobi (HJ) visqueuses, convexes et super-nonlinéaires, au moyen d’équations différentielles stochastiques rétrogrades (EDSR) avec contraintes sur la partie martingale. Nous comparons nos résultats avec la représentation classique en termes d’EDSR (super)quadratiques, et montrons notamment que l’existence d’une solution de viscosité à l’équation visqueuse de HJ peut être obtenue sous des conditions de croissance plus générales, incluant des coefficients et une donnée terminale non bornées.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1528-1547.

Dates
Received: 27 May 2015
Revised: 26 January 2016
Accepted: 6 May 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773717

Digital Object Identifier
doi:10.1214/16-AIHP762

Mathematical Reviews number (MathSciNet)
MR3729627

Zentralblatt MATH identifier
06847054

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 35K58: Semilinear parabolic equations

Keywords
Backward stochastic differential equation (BSDE) Randomization Viscous Hamilton–Jacobi equation Deterministic KPZ equation Nonlinear Feynman–Kac formula

Citation

Cosso, Andrea; Pham, Huyên; Xing, Hao. BSDEs with diffusion constraint and viscous Hamilton–Jacobi equations with unbounded data. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1528--1547. doi:10.1214/16-AIHP762. https://projecteuclid.org/euclid.aihp/1511773717


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