Second-order bsdes with jumps: Formulation and uniqueness

Annals of Applied Probability

Volumn 25, Issue 5, 2015, Pages 2867-2908

  Kazi Tani, Nabil ^a   Possamaï, Dylan ^b   Zhou, Chao ^c  

^a UNIVERSITÉ LYON 1   (France)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

^c NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Backward stochastic differential equation with jumps; Mutually singular measures; Quasi sure analysis; Second order backward stochastic differential equation

Indexed keywords

EID: 84941565268     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/14-AAP1063     Document Type: Article

References (37)

1. AVELLANEDA, M., LÉ VY, A. and PARAS, A. ( 1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73-88.
2. BARLES, G., BUCKDAHN, R. and PARDOUX, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60 57-83. MR1436432
3. BECHERER, D. (2006). Bounded solutions to backward SDE's with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16 2027-2054. MR2288712
4. BICHTELER, K. (1981). Stochastic integration and Lp-theory of semimartingales. Ann. Probab. 9 49-89. MR0606798
5. BILLINGSLEY, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
6. BION-NADAL, J. and KERVAREC, M. (2012). Risk measuring under model uncertainty. Ann. Appl. Probab. 22 213-238. MR2932546
7. BISMUT, J.-M. (1973). Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 384-404. MR0329726
8. BOUCHARD, B. and TOUZI, N. ( 2011). Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49 948-962. MR2806570
9. COHEN, S. N. (2012). Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces. Electron. J. Probab. 17 no. 62, 15. MR2959068
10. CRÉPEY, S. and MATOUSS I, A. (2008). Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison. Ann. Appl. Probab. 18 2041-2069. MR2462558
11. DENIS, L., HU, M. and PENG, S. (2011). Function space s and capacity related to a sublinear expectation: Application to G-Brownian motion paths. Potential Anal. 34 139-161. MR2754968
12. DENIS, L. and MARTINI, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 827-852. MR2244434
13. EL KAROUI, N. (1981). Les aspects probalilistes du contrǒle stochastique. In Ecole D'Eté de Probabilité s de Saint-Flour IX-1979. Lecture Notes in Math. 876 73-238.
14. EL KAROUI, N., HUUNGUYEN, D. and JEANBLANC-PICQUÉ, M. (1987). Compactification methods in the control of degenerate diffusions : Existence of an optimal control. Stochastics 20 169-219. MR0878312
15. EL KAROUI, N., KAPOUDJIAN, C., PARDOUX, E., PENG, S. and QUENEZ, M. C. (1997). Reflected solutions of backward SDE's, and related obstacle problems for PDE's. Ann. Appl. Probab. 25 702-737. MR1434123
16. FUJIWARA, T. and KUNITA, H. (1985). Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25 71-106. MR0777247
17. HU, M. and PENG, S. (2009). G-Lé vy processes under sublinear expectations. Preprint. Available at arXiv:0911.3533.
18. JACOD, J. (1979). Calcul Stochastique et Problémes de Martingales. Springer, Berlin. MR0542115
19. JACOD, J. and SHIRYAEV, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin. MR0959133
20. KARANDIKAR, R. L. (1995). On pathwise stochastic integration. Stochastic Process. Appl. 57 11-18. MR1327950
21. KAZI-TANI, N., POSSAMAÏ , D. and ZHOU, C. (2012). Second-order BSDEs with jumps: Existence and probabilistic representation for fully-nonlinear PIDEs. Preprint. Available at arXiv:1208.0763.
22. LÉPINGLE, D. and MÉ MIN, J. (1978). Sur l'intégrabilité uniforme des martingales exponentielles. Z. Wahrsch. Verw. Gebiete 42 175-203. MR0489492
23. LÉ PINGLE, D. andMÉMIN, J. (1978). Intégrabilité uniforme et dans Lr des martingales exponentielles. I n Seminar on Probability, Rennes 1978 (French) Exp. No. 9, 14. Univ. Rennes, Rennes. MR0602524
24. MÉ MIN, J. (1978). Décompositions multiplicatives de semimartingales exponentielles et applications. In Séminaire de Probabilité s, XII (Univ. Strasbourg, Strasbourg, 1976/1977). 35-46. Springer, Berlin. MR0519991
25. NEUFELD, A. and NUTZ, M. (2014). Measurability of semimartingale characteristics with respect to the probability law. Stochastic Process. Appl. 124 3819-3845. MR3249357
26. NEUFELD, A. and NUTZ, M. (2014). Nonlinear Lé vy processes and their characteristics. Preprint. Available at arXiv:1401.7253.
27. NEVEU, J. (1975). Discrete-Parameter Martingales, Revised ed. North-Holland, Amsterdam. MR0402915
28. NUTZ, M. (2012). Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17 no. 24, 7. MR2950190
29. NUTZ, M. (2012). A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17 no. 23, 23. MR2900464
30. PARDOUX, É . and PENG, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55-61. MR1037747
31. PENG, S. (2007). G-expectation, G-Brownian motion and related stochastic calculus of Itǒ type. In Stochastic Analysis and Applications. Abel Symp. 2 541-567. Springer, Berlin. MR2397805
32. PENG S. 2010). Nonlinear expectations and stochastic calculus under uncertainty. Preprint. Available at arXiv: 1002.4546.
33. PROTTER, P. (2000). Stochastic Integration and Differential Equations. Springer, Berlin.
34. ROYER, M. (2006). Backward stochastic differential equations with jumps and related nonlinear expectations. Stochastic Process. Appl. 116 1358-1376. MR2260739
35. SONER, H. M., TOUZI, N. and ZHANG, J. (2011). Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 1844-1879. MR2842089
36. SONER, H. M., TOUZI, N. and ZHANG, J. (2012). Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 149-190. MR2925572
37. TANG, S. J. and LI, X. J. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 1447-1475. MR1288257