The Annals of Probability

Anticipated backward stochastic differential equations

Shige Peng and Zhe Yang

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Abstract

In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.

Article information

Source
Ann. Probab., Volume 37, Number 3 (2009), 877-902.

Dates
First available in Project Euclid: 19 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1245434023

Digital Object Identifier
doi:10.1214/08-AOP423

Mathematical Reviews number (MathSciNet)
MR2537524

Zentralblatt MATH identifier
1186.60053

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations 93E03: Stochastic systems, general

Keywords
Anticipated backward stochastic differential equation backward stochastic differential equation adapted process

Citation

Peng, Shige; Yang, Zhe. Anticipated backward stochastic differential equations. Ann. Probab. 37 (2009), no. 3, 877--902. doi:10.1214/08-AOP423. https://projecteuclid.org/euclid.aop/1245434023


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References

  • [1] Beneš, V. E. (1970). Existence of optimal strategies based on specified information, for a class of stochastic decision problems. SIAM J. Control Optim. 8 179–188.
  • [2] Beneš, V. E. (1971). Existence of optimal stochastic control laws. SIAM J. Control Optim. 9 446–472.
  • [3] Cao, Z. G. and Yan, J.-A. (1999). A comparison theorem for solutions of backward stochastic differential equations. Adv. Math. (China) 28 304–308.
  • [4] Cvitanić, J. and Karatzas, I. (1996). Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 2024–2056.
  • [5] Dellacherie, C. (1972). Capacités et Processus Stochastiques. Ergebnisse der Mathematik und ihrer Grenzgebiete 67. Springer, Berlin.
  • [6] 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.
  • [7] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [8] Hu, Y. and Peng, S. (2006). On the comparison theorem for multidimensional BSDEs. C. R. Math. Acad. Sci. Paris Ser. I 343 135–140.
  • [9] Lepeltier, J.-P. and Martín, J. S. (2004). Backward SDEs with two barriers and continuous coefficient: An existence result. J. Appl. Probab. 41 162–175.
  • [10] Lin, Q. Q. (2001). A comparison theorem for backward stochastic differential equations. J. Huazhong Univ. Sci. Tech. 29 1–3.
  • [11] Liu, J. and Ren, J. (2002). Comparison theorem for solutions of backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 56 93–100.
  • [12] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Related Fields 113 473–499.
  • [13] Peng, S. (2004). Nonlinear expectations, nonlinear evaluations and risk measures. In Stochastic Methods in Finance. Lecture Notes in Math. 1856 165–253. Springer, Berlin.
  • [14] Peng, S. and Xu, M. Y. (2005). The smallest g-supermartingale and reflected BSDE with single and double L2 obstacles. Ann. Inst. H. Poincaré Probab. Statist. 41 605–630.
  • [15] Situ, R. (1999). Comparison theorem of solutions to BSDE with jumps, and viscosity solution to a generalized Hamilton–Jacobi–Bellman equation. In Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998) 275–282. Kluwer Academic, Boston, MA.
  • [16] Zhang, T. S. (2003). A comparison theorem for solutions of backward stochastic differential equations with two reflecting barriers and its applications. In Probabilistic Methods in Fluids 324–331. World Science, River Edge, NJ.