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

UNSTEADY CHARACTERISTICS OF THE SHOCK PROPAGATION IN A CONVERGENT SHOCK TUBE WITH SMALL ANGLE

The whole evolution of the incident shock propagation in a convergent shock tube with small angle is studied in detail by using the direct numerical simulation. Specifically, the shape of the curved shock and the unsteady flow patterns which differs from the K-H instability, have been evaluated. The results show that as a disturbance of the inclined wall on the shock, the bending position of the incident shock represents periodically changed and its non-dimensional wavelength is larger when the convergent angle becomes greater, indicating a faster response to the curvature variation. At the same time, two different flow instable patterns for the shock propagation in the area reduction channel are discovered, one of which is the asymmetric shock bifurcations when the reflected shock from the collision of the right wall interacts with the boundary layer. This instability is closely related to the unsteady vortex shedding behind the bifurcated feet, resulting in the dramatic pressure fluctuation. Another pattern occurs when the reflected shocks generated by the curved incident shock impinge on the upper and lower walls. The collision position moves at a modest speed, which causes the formation of small vortices near the reflection regions