The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications

Juliang Yin, Xuerong Mao, NSF of Guangdong Province (Funder)

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem andthe linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type.
LanguageEnglish
Pages345-358
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume346
Issue number2
DOIs
Publication statusPublished - 15 Oct 2008

Fingerprint

Backward Stochastic Differential Equation
Comparison Theorem
Feynman-Kac Formula
Siméon Denis Poisson
Jump
Differential equations
Partial differential equations
Integral equations
Integral Equations
Coefficient
Girsanov Theorem
Minimal Solution
Linear partial differential equation
Existence and Uniqueness Results
Viscosity Solutions
Second order differential equation
Existence and Uniqueness
Partial differential equation
Viscosity
Class

Keywords

  • backward stochastic differential equations
  • poisson point process
  • comparison theorem
  • Feynman–Kac formula
  • viscosity solution
  • PDIEs

Cite this

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title = "The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications",
abstract = "This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem andthe linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type.",
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author = "Juliang Yin and Xuerong Mao and {NSF of Guangdong Province (Funder)}",
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The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications. / Yin, Juliang; Mao, Xuerong; NSF of Guangdong Province (Funder).

In: Journal of Mathematical Analysis and Applications, Vol. 346, No. 2, 15.10.2008, p. 345-358.

Research output: Contribution to journalArticle

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AB - This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem andthe linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type.

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