Integro-partial differential equations with singular terminal condition

Popier, Alexandre

doi:10.1016/j.na.2017.01.012

Cited by 5 publications

(2 citation statements)

References 38 publications

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“…In Ahmadi et al (2021); Kruse and Popier (2016b); Popier (2006); Sezer et al (2019), we study non-linear BSDE with singular terminal condition at a deterministic terminal time T . Such BSDE generalize parabolic diffusion-reaction PDE with singular final trace (Graewe et al (2018); Marcus and Véron (1999); Popier (2017)) and they can be used to represent value functions of a class of stochastic optimal control problems with terminal constraints (Ankirchner et al (2014); Graewe et al (2018); Kruse and Popier (2016b) and the references therein).…”

Section: Introduction and Definitionsmentioning

confidence: 99%

Continuity problem for singular BSDE with random terminal time

Popier¹,

Samuel²,

Sezer³

2022

ALEA

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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(Ξ S ) 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 ξ 1 = ∞ • 1 {τ ≤S} and 3) ξ 2 = ∞ • 1 {τ >S} 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.

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Section: Introduction and Definitionsmentioning

confidence: 99%

Continuity problem for singular BSDE with random terminal time

Popier¹,

Samuel²,

Sezer³

2022

ALEA

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“…We model real-life problems usually result in mathematical form as follows: functional equations, ordinary differential equations (ODEs) or partial differential equations (PDEs), integral or IDEs, and stochastic equations. Most mathematical modeling of physical phenomena contains IDEs [6][7][8][9][10][11]. The theory and application of integro-differential equations play an important role in many fields like engineering sciences, fluid dynamics, and nonlinear optics [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Solutions of Nonlinear Integro-Partial Differential Equations by the Method of $(G^{'})$

Gusu

Bulo

2022

Advances in Mathematical Physics

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In this article, a special expansion method is implemented in solving nonlinear integro-partial differential equations of 2 + 1 -dimensional using a special expansion method of G ′ / G , 1 / G . We obtained the solutions for 2 + 1 -dimensional nonlinear integro-differential equations in real physical phenomena. The method is applied on 2 + 1 -dimensional space time and solved in three different cases: hyperbolic, trigonometric, and rational functions. The obtained solutions for each result were illustrated by graphical plots using Wolfram Mathematica 9.0 software packages. Furthermore, the obtained results are exactly fit with exact solutions which solves the complicity of finding the solution for nonlinear integro-partial differential equations. Finally, the method is powerful and effective to solve partial differential equations of nonlinear integro form.

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Continuous Viscosity Solutions to Linear-Quadratic Stochastic Control Problems with Singular Terminal State Constraint

Horst

Xia

2020

Appl Math Optim

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This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub-and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument.

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Integro-partial differential equations with singular terminal condition

Cited by 5 publications

References 38 publications

Continuity problem for singular BSDE with random terminal time

Continuity problem for singular BSDE with random terminal time

Solutions of Nonlinear Integro-Partial Differential Equations by the Method of $(G^{'})$

Continuous Viscosity Solutions to Linear-Quadratic Stochastic Control Problems with Singular Terminal State Constraint

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Integro-partial differential equations with singular terminal condition

Cited by 5 publications

References 38 publications

Continuity problem for singular BSDE with random terminal time

Continuity problem for singular BSDE with random terminal time

Solutions of Nonlinear Integro-Partial Differential Equations by the Method of G ′

Continuous Viscosity Solutions to Linear-Quadratic Stochastic Control Problems with Singular Terminal State Constraint

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Solutions of Nonlinear Integro-Partial Differential Equations by the Method of $(G^{'})$