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

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

Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs

We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999), as well as Ma and Zhang (2000), using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others

Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation

This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao and Liu \cite{GL}, this extends the corresponding results collected in Freidlin and Wentzell \cite{FreidlinWentzell}. However, we use a different line of argument, adapting the PDE method of Fleming \cite{Fleming} and Evans and Ishii \cite{EvansIshii} to the path-dependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path-dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics