论文标题
随机终端时间的单数BSDE的连续性问题
Continuity problem for singular BSDE with random terminal time
论文作者
论文摘要
我们研究了一类具有超级线性驱动器过程F的非线性BSDE，适用于过滤F和随机时间间隔[[0，s]]，其中S是F的停止时间。我们的目标是在这种情况下向BSDE展示解决方案的存在。我们将通过证明BSDE的最小化度是一种解决方案，即具有概率1。我们考虑三种类型的终端值：1）Markovian：I.e. $ξ$是$之一，$之一表格$ $ = $ \ infty $ $ \ times $ 1 {$τ$ \ $ \ le $ s}和3）$ 2 $ 2 = $ \ infty $ $ $ \ times $ 1 {$τ$> s}，其中$τ$是$τ$是另一个停止时间。对于一般$ξ$，我们证明最小的超溶液在时间时是连续的。只要F在时间S时保持连续。如果BSDE具有最低限度的超级级别，则可以解决对给定的BSDE和过滤的停止时间，并且最终值$ \ Inftty $在终端S.终端S.的概念S的概念中扮演了许多重要的角色。最后，我们讨论了我们的结果对马尔可夫终端条件的含义与具有单数边界条件的非线性椭圆PDE的溶液。
We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $ξ$ is allowed to take the value +$\infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $ξ$ is of the form $ξ$ = g($Ξ$ S) where $Ξ$ is a continuous Markovian diffusion process and S is a hitting time of $Ξ$ and g is a deterministic function 2) terminal conditions of the form $ξ$ = $\infty$ $\times$ 1 {$τ$ $\le$S} and 3) $ξ$ 2 = $\infty$ $\times$ 1 {$τ$ >S} where $τ$ is another stopping time. For general $ξ$ we prove the minimal supersolution is continuous at time S provided that F is left continuous at time S. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value $\infty$ at terminal time S. The concept of solvability plays a key role in many of the arguments. Finally, we discuss implications of our results on the Markovian terminal conditions to solution of nonlinear elliptic PDE with singular boundary conditions.
