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October 2015 Second-order BSDEs with jumps: Formulation and uniqueness
Nabil Kazi-Tani, Dylan Possamaï, Chao Zhou
Ann. Appl. Probab. 25(5): 2867-2908 (October 2015). DOI: 10.1214/14-AAP1063


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.


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Nabil Kazi-Tani. Dylan Possamaï. Chao Zhou. "Second-order BSDEs with jumps: Formulation and uniqueness." Ann. Appl. Probab. 25 (5) 2867 - 2908, October 2015.


Received: 1 May 2013; Revised: 1 January 2014; Published: October 2015
First available in Project Euclid: 30 July 2015

zbMATH: 1325.60091
MathSciNet: MR3375890
Digital Object Identifier: 10.1214/14-AAP1063

Primary: 60H10 , 60H30

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

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.25 • No. 5 • October 2015
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