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Continuity problem for singular BSDE with random terminal time
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Date
2022-1-01
Author
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
Collections
Graduate School of Applied Mathematics, Article
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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.