Reflected and doubly reflected bsdes with jumps: A priori estimates and comparison

Annals of Applied Probability

Volumn 18, Issue 5, 2008, Pages 2041-2069

  Crépey, Stéphane ^a   Matoussi, Anis ^b  

^a UNIVERSITÉ D EVRY VAL D ESSONNE   (France)

^b UNIVERSITÉ DU MAINE   (France)

Author keywords

A priori estimates; Comparison theorem; Convertible bonds; Finance; Jumps; Markovian BSDEs; Reflected BSDEs

Indexed keywords

EID: 63049101064     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/08-AAP517     Document Type: Article

References (32)

1. BICHTELER, K. (2002). Stochastic Integration with Jumps. Encyclopedia of Mathematics and Its Applications 89. Cambridge Univ. Press, Cambridge. MR1906715
2. BARLES, G., BUCKDAHN, R. and PARDOUX, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics 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. BECHERER, D. and SCHWEIZER, M. (2005). Classical solutions to reaction-diffusion systems for hedging problems with interacting Itô and point processes. Ann. Appl. Probab. 15 1111-1144. MR2134099
5. BIELECKI, T. R., CRÉPEY, S., JEANBLANC, M. and RUTKOWSKI, M. (2008). Arbitrage pricing of defaultable game options with applications to convertible bonds. Quantitative Finance. To appear.
6. BIELECKI, T. R., CRÉPEY, S., JEANBLANC, M. and RUTKOWSKI, M. (2007). Valuation and hedging of defaultable game options in a hazard process model. Submitted.
7. BIELECKI, T. R., CRÉPEY, S., JEANBLANC, M. and RUTKOWSKI, M. (2008). Defaultable options in a Markovian intensity model of credit risk. Math. Finance. To appear.
8. BIELECKI, T. R., CRÉPEY, S., JEANBLANC, M. and RUTKOWSKI, M. (2007). Convertible bonds in a defaultable diffusion model. Submitted.
9. BIELECKI, T. R., CRÉPEY, S., JEANBLANC, M. and RUTKOWSKI, M. (2007). Valuation of basket credit derivatives in the credit migrations environment. In Handbook of Financial Engineering (J. Birge and V Linetsky, eds.). North-Holland, Amsterdam.
10. CRÉPEY, S. (2008). Markovian reflected and doubly reflected BSDEs in a jump-diffusion setting with regimes. Submitted.
11. CRÉPEY, S. (2008). About the pricing equation in finance. Submitted.
12. CRÉPEY, S. and MATOUSSI, A. About the greeking equation in finance. Work in preparation.
13. CVITANIĆ, J. and KARATZAS, I. (1996). Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 2024-2056. MR1415239
14. DARLING, R. W. R. and PARDOUX, E. (1997). Backwards SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Probab. 25 1135-1159. MR1457614
15. DELLACHERIE, C. and MEYER, P.-A. (1980). Probabilités et Potentiel, Chapitres Va VIII, Théorie des Martingales. Hermann, Paris.
16. EL KAROUI, N., HAMADÈNE, S. and MATOUSSI, A. (2008). Backward stochastic differential equations and applications. In Paris-Princeton Lecture Notes on Mathematical Finance (R. Carmona, ed.). Springer, Berlin. To appear.
17. 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. Probab. 25 702-737. MR1434123
18. ESSAKY, E.H., OUKNINE, Y. and HARRAJ, N. (2005). Backward stochastic differential equations with two reflecting barriers and jumps. Stock Anal. Appl. 23 921-938. MR2158885
19. ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. MR838085
20. HAMADÈNE, S. (2006). Mixed zero-sum differential game and American game options. SIAM J. Control. Optim. 45 496-518.
21. HAMADÈNE, S. and OUKNINE, Y (2003). Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Probab. 8 no. 2, 20 pp. (electronic). MR1961164
22. HAMADÈNE, S. and HASSANI, M. (2006). BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game. Electron. J. Probab. 11 no. 5, 121-145 (electronic). MR2217812
23. HAMADÈNE, S., LEPELTIER, J.-P. and MATOUSSI, A. (1997). Double barriers reflected backward SDE's with continuous coefficients. In Pitman Research Notes in Mathematics Series 364 115-128. Longman, Harlow.
24. JACOD, J. (2003). Calcul Stochastique et Problèmes de Martingales. Springer, Berlin.
25. JACOD, J. and SHIRYAEV, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin. MR1943877
26. LEPELTIER, J.-P. and SAN MARTÍN, J. (2004). Backward SDEs with two barriers and continuous coefficient: An existence result. J. Appl. Probab. 41 162-175. MR2036279
27. LEPELTIER, J.-P. and XU, M. (2005). Penalization method for reflected backward stochastic differential equations with one r.c.1.1. barrier. Statist. Probab. Lett. 75 58-66. MR2185610
28. MÉMIN, J. and SLOMINSKI, L. (1991). Condition UT et stabilité en loi des solutions d'équations différentielles stochastiques. Séminaire de Probabilités de Strasbourg 25 162-177.
29. PRIGENT, J.-L. (2003). Weak Convergence of Financial Markets. Springer, Berlin. MR2036683
30. PROTTER, P. E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin. MR2020294
31. ROYER, M. (2006). Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process. Appl. 116 1358-1376. MR2260739
32. 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