Continuity problem for singular BSDE with random terminal time

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Samuel S. A., Popier A., SEZER A. D.

Alea (Rio de Janeiro), vol.19, no.2, pp.1185-1220, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 19 Issue: 2
  • Publication Date: 2022
  • Doi Number: 10.30757/alea.v19-49
  • Journal Name: Alea (Rio de Janeiro)
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.1185-1220
  • Keywords: Backward stochastic differential equation, stopping time, singularity, continuity problem, POSITIVE SOLUTIONS, DIFFERENTIAL-EQUATIONS, TERMINAL CONDITION, BOUNDARY, TRACE, JUMPS
  • Middle East Technical University Affiliated: Yes


© 2022, Alea (Rio de Janeiro).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