### Abstract

Original language | English |
---|---|

Pages (from-to) | 345-358 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 346 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Oct 2008 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Mathematical Analysis and Applications*,

*346*(2), 345-358. https://doi.org/10.1016/j.jmaa.2008.05.072

}

*Journal of Mathematical Analysis and Applications*, vol. 346, no. 2, pp. 345-358. https://doi.org/10.1016/j.jmaa.2008.05.072

**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).

Research output: Contribution to journal › Article

TY - JOUR

T1 - The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications

AU - Yin, Juliang

AU - Mao, Xuerong

AU - NSF of Guangdong Province (Funder)

PY - 2008/10/15

Y1 - 2008/10/15

N2 - 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.

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.

KW - backward stochastic differential equations

KW - poisson point process

KW - comparison theorem

KW - Feynman–Kac formula

KW - viscosity solution

KW - PDIEs

U2 - 10.1016/j.jmaa.2008.05.072

DO - 10.1016/j.jmaa.2008.05.072

M3 - Article

VL - 346

SP - 345

EP - 358

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -