2019/11/16 by Ahmadi, Mahdi, Popier, Alexandre, Sezer, Ali Devin
#35K57 #35K67 #60G40 #60H30 #60H99 #60J65 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1911.07016
We consider a class of Backward Stochastic Differential Equations with superlinear driver process f adapted to a filtration supporting at least a d dimensional Brownian motion and a Poisson random measure on \mathbb Rm- \0\. We consider the following class of terminal conditions ξ1 = ∞ ⋅ 1_\τ1 ≤ T\ where τ1 is any stopping time with a bounded density in a neighborhood of T and ξ2 = ∞ ⋅ 1AT where At, t ∈ [0,T] is a decreasing sequence of events adapted to the filtration \mathcal Ft that is continuous in probability at T. A special case for ξ2 is AT = \τ2 > T\ where τ2 is any stopping time such that P(τ2 =T) =0. In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We further show that the first exit time from a time varying domain of a d-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density; therefore such exit times can be used as τ1 and τ2 to define the terminal conditions ξ1 and ξ2. The proof of existence of the density is based on the classical Green's functions for the associated PDE.