Abstract
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 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 Ft that is continuous in probability at T (equivalently, 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 note 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.
Highlights
Introduction and definitionsA stochastic differential equation with a prescribed terminal condition is called a backward stochastic differential equation (BSDE)
The main idea in establishing the continuity of the minimal supersolution is to use the solution of a linear BSDE with terminal condition Yτ∞ · 1{τ≤T } as an upper bound on the time interval [0, τ ∧ T ]
The rest of this introduction discusses the implications of our results to stochastic optimal control, lists the assumptions we adopt and the results we will be using from prior work and gives a summary of what is known in the prior literature about the continuity of the minimal supersolution of BSDE with singular terminal conditions
Summary
A stochastic differential equation with a prescribed terminal condition is called a backward stochastic differential equation (BSDE). The main idea in establishing the continuity of the minimal supersolution is to use the solution of a linear BSDE with terminal condition Yτ∞ · 1{τ≤T } as an upper bound on the time interval [0, τ ∧ T ] The rest of this introduction discusses the implications of our results to stochastic optimal control, lists the assumptions we adopt and the results we will be using from prior work and gives a summary of what is known in the prior literature about the continuity of the minimal supersolution of BSDE with singular terminal conditions.
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