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November 2002 Representation theorems for backward stochastic differential equations
Jin Ma, Jianfeng Zhang
Ann. Appl. Probab. 12(4): 1390-1418 (November 2002). DOI: 10.1214/aoap/1037125868


In this paper we investigate a class of backward stochastic differential equations (BSDE) whose terminal values are allowed to depend on the history of a forward diffusion. We first establish a probabilistic representation for the spatial gradient of the viscosity solution to a quasilinear parabolic PDE in the spirit of the Feynman--Kac formula, without using the derivatives of the coefficients of the corresponding BSDE. Such a representation then leads to a closed-form representation of the martingale integrand of a BSDE, under only a standard Lipschitz condition on the coefficients. As a direct consequence we prove that the paths of the martingale integrand of such BSDEs are at least c\`{a}dl\`{a}g, which not only extends the existing path regularity results for solutions to BSDEs, but contains the cases where existing methods are not applicable. The BSDEs in this paper can be considered as the nonlinear wealth processes appearing in finance models; our results could lead to efficient Monte Carlo methods for computing both price and optimal hedging strategy for options with nonsmooth, path-dependent payoffs in the situation where the wealth is possibly nonlinear.


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Jin Ma. Jianfeng Zhang. "Representation theorems for backward stochastic differential equations." Ann. Appl. Probab. 12 (4) 1390 - 1418, November 2002.


Published: November 2002
First available in Project Euclid: 12 November 2002

zbMATH: 1017.60067
MathSciNet: MR1936598
Digital Object Identifier: 10.1214/aoap/1037125868

Primary: 60H10
Secondary: 34F05 , 90A12

Keywords: adapted solutions , anticipating stochastic calculus , Backward SDE's , path regularity , viscosity solutions

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.12 • No. 4 • November 2002
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