The Annals of Applied Probability

Second-order BSDEs with jumps: Formulation and uniqueness

Nabil Kazi-Tani, Dylan Possamaï, and Chao Zhou

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Abstract

In this paper, we define a notion of second-order backward stochastic differential equations with jumps (2BSDEJs for short), which generalizes the continuous case considered by Soner, Touzi and Zhang [ Probab. Theory Related Fields 153 (2012) 149–190]. However, on the contrary to their formulation, where they can define pathwise the density of quadratic variation of the canonical process, in our setting, the compensator of the jump measure associated to the jumps of the canonical process, which is the counterpart of the density in the continuous case, depends on the underlying probability measures. Then in our formulation of 2BSDEJs, the generator of the 2BSDEJs depends also on the underlying probability measures through the compensator. But the solution to the 2BSDEJs can still be defined universally. Moreover, we obtain a representation of the $Y$ component of a solution of a 2BSDEJ as a supremum of solutions of standard backward SDEs with jumps, which ensures the uniqueness of the solution.

Article information

Source
Ann. Appl. Probab. Volume 25, Number 5 (2015), 2867-2908.

Dates
Received: May 2013
Revised: January 2014
First available in Project Euclid: 30 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1438261055

Digital Object Identifier
doi:10.1214/14-AAP1063

Mathematical Reviews number (MathSciNet)
MR3375890

Zentralblatt MATH identifier
1325.60091

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Second-order backward stochastic differential equation backward stochastic differential equation with jumps mutually singular measures quasi-sure analysis

Citation

Kazi-Tani, Nabil; Possamaï, Dylan; Zhou, Chao. Second-order BSDEs with jumps: Formulation and uniqueness. Ann. Appl. Probab. 25 (2015), no. 5, 2867--2908. doi:10.1214/14-AAP1063. https://projecteuclid.org/euclid.aoap/1438261055


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