Continuity problem for singular BSDE with random terminal time

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2022-1-01

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Samuel, Sharoy Augustine
Popier, Alexandre
Sezer, Ali Devin

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All Rights Reserved.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 1 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 (Formula Presented) and 3) (Formula Presented) where Ƭ 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

Subject Keywords

Backward stochastic differential equation, stopping time, singularity, continuity problem, POSITIVE SOLUTIONS, DIFFERENTIAL-EQUATIONS, TERMINAL CONDITION, BOUNDARY, TRACE, JUMPS, Backward stochastic differential equation, Continuity problem, Singularity, Stopping time

URI

https://hdl.handle.net/11511/101830

Journal

Alea (Rio de Janeiro)

DOI

https://doi.org/10.30757/alea.v19-49

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Graduate School of Applied Mathematics, Article

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Citation Formats

S. A. Samuel, A. Popier, and A. D. Sezer, “Continuity problem for singular BSDE with random terminal time,” Alea (Rio de Janeiro), vol. 19, no. 2, pp. 1185–1220, 2022, Accessed: 00, 2023. [Online]. Available: https://hdl.handle.net/11511/101830.