Weak solutions for forward--backward SDEs--a martingale problem approach

Details Weak solutions for forward--backward SDEs--a martingale problem approach Weak solutions for forward--backward SDEs--a martingale problem approach

2009-01-19

arxiv.org/abs/0901.2790v1

Annals of Probability 2008, Vol. 36, No. 6, 2092-2125

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP0383 the Annals of Probability (http://www.imstat.org/aop/) by the Institute o

Scientific paper

10.1214/08-AOP0383

In this paper, we propose a new notion of Forward--Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the forward--backward stochastic differential equations (FBSDEs). The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of an FBSDE. We first prove a general sufficient condition for the existence of the solution to the FBMP. In the Markovian case with uniformly continuous coefficients, we show that the weak solution to the FBSDE (or equivalently, the solution to the FBMP) does exist. Moreover, we prove that the uniqueness of the FBMP (whence the uniqueness of the weak solution) is determined by the uniqueness of the viscosity solution of the corresponding quasilinear PDE.

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