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- Title
Continuity problem for singular BSDE with random terminal time.
- Authors
Samuel, Sharoy Augustine; Popier, Alexandre; Sezer, Ali Devin
- Abstract
We study a class of non-linear Backward stochastic differential equations (BSDE) 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 +∞, i.e., singular. 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 ∞ at terminal time S. Our goal is to show existence of solutions to the BSDE for a range of singular terminal values under the assumption that S is solvable. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., it is continuous at time S and attains the terminal value with probability 1. We consider three types of terminal values: 1) Markovian: i.e., ξ is of the form ξ = g(ΞS) where Ξ is a continuous Markovian diffusion process, S is a hitting time of Ξ and g is a deterministic function 2) terminal conditions of the form ξ1 = ∞· 1(τ≥) and 3) ξ2 = ∞· 1(τ≥) where τ is another stopping time. For general ξ we prove that minimal supersolution has a limit at time S provided that F is left continuous at time S. Finally, we discuss the implications of our results about Markovian terminal conditions to the solution of non-linear elliptic PDE with singular boundary conditions.
- Subjects
STOCHASTIC differential equations; MATHEMATICAL singularities; PROBABILITY theory; DETERMINISTIC algorithms; EXISTENCE theorems
- Publication
ALEA. Latin American Journal of Probability & Mathematical Statistics, 2022, Vol 19, Issue 2, p1185
- ISSN
1980-0436
- Publication type
Article
- DOI
10.30757/ALEA.v19-49