1,220 research outputs found
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 FourniƩ 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
- ā¦