## Electronic Journal of Probability

### Second order BSDEs with jumps: existence and probabilistic representation for fully-nonlinear PIDEs

#### Abstract

In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in an accompanying paper. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of some fully non-linear partial integro-differential equations, which is the point of the second part of this work. We prove a non-linear Feynman-Kac formula and show that solutions to 2BSDEJs provide viscosity solutions of the associated PIDEs.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 65, 31 pp.

Dates
Accepted: 14 June 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067171

Digital Object Identifier
doi:10.1214/EJP.v20-3569

Mathematical Reviews number (MathSciNet)
MR3361253

Zentralblatt MATH identifier
1321.60125

Rights

#### Citation

Kazi-Tani, Nabil; Possamaï, Dylan; Zhou, Chao. Second order BSDEs with jumps: existence and probabilistic representation for fully-nonlinear PIDEs. Electron. J. Probab. 20 (2015), paper no. 65, 31 pp. doi:10.1214/EJP.v20-3569. https://projecteuclid.org/euclid.ejp/1465067171

#### References

• Applebaum, David. Lévy processes and stochastic calculus. Cambridge Studies in Advanced Mathematics, 93. Cambridge University Press, Cambridge, 2004. xxiv+384 pp. ISBN: 0-521-83263-2.
• Barles, Guy; Buckdahn, Rainer; Pardoux, Etienne. Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60 (1997), no. 1-2, 57–83.
• Becherer, Dirk. Bounded solutions to backward SDE's with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16 (2006), no. 4, 2027–2054.
• Bichteler, Klaus. Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 (1981), no. 1, 49–89.
• Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9.
• Bismut, Jean-Michel. Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 (1973), 384–404.
• Bouchard, Bruno; Touzi, Nizar. Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49 (2011), no. 3, 948–962.
• Crépey, Stéphane; Matoussi, Anis. Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison. Ann. Appl. Probab. 18 (2008), no. 5, 2041–2069.
• Dellacherie, Claude; Meyer, Paul-André. Probabilités et potentiel. (French) Chapitres I á IV. Edition entièrement refondue. Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XV. Actualités Scientifiques et Industrielles, No. 1372. Hermann, Paris, 1975. x+291 pp.
• El Karoui, N.; Peng, S.; Quenez, M. C. Backward stochastic differential equations in finance. Math. Finance 7 (1997), no. 1, 1–71.
• Essaky, E. H. Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bull. Sci. Math. 132 (2008), no. 8, 690–710.
• Hamadène, S. and Ouknine, Y. (2008). Reflected backward SDEs with general jumps, preprint, sl arXiv:0812.3965.
• Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1.
• Karandikar, Rajeeva L. On pathwise stochastic integration. Stochastic Process. Appl. 57 (1995), no. 1, 11–18.
• Kazi-Tani, N., Possamaï, D., Zhou, C. (2012). Second order BSDEs with jumps: formulation and uniqueness, preprint, arXiv:1208.0757.
• Morlais, Marie-Amelie. Utility maximization in a jump market model. Stochastics 81 (2009), no. 1, 1–27.
• Neufeld, Ariel; Nutz, Marcel. Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18 (2013), no. 48, 14 pp.
• Neufeld, A., Nutz, M. (2014). Nonlinear Lévy processes and their characteristics, preprint, sl arXiv:1401.7253.
• Nutz, Marcel. Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17 (2012), no. 24, 7 pp.
• Nutz, M., van Handel, R. (2012). Constructing sublinear expectations on path space, preprint, sl arXiv:1205.2415.
• Pardoux, E.; Peng, S. G. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55–61.
• Possamai, D., Zhou, C. (2010). Second order backward stochastic differential equations with quadratic growth, preprint sl arXiv:1201.1050.
• Royer, Manuela. Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process. Appl. 116 (2006), no. 10, 1358–1376.
• Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 (2012), no. 1-2, 149–190.
• Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Dual formulation of second order target problems. Ann. Appl. Probab. 23 (2013), no. 1, 308–347.
• Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 (2011), no. 67, 1844–1879.
• Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4.
• Tang, Shan Jian; Li, Xun Jing. Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 (1994), no. 5, 1447–1475.