## Abstract and Applied Analysis

### Backward Stochastic Differential Equations Coupled with Value Function and Related Optimal Control Problems

#### Abstract

We get a new type of controlled backward stochastic differential equations (BSDEs), namely, the BSDEs, coupled with value function. We prove the existence and the uniqueness theorem as well as a comparison theorem for such BSDEs coupled with value function by using the approximation method. We get the related dynamic programming principle (DPP) with the help of the stochastic backward semigroup which was introduced by Peng in 1997. By making use of a new, more direct approach, we prove that our nonlocal Hamilton-Jacobi-Bellman (HJB) equation has a unique viscosity solution in the space of continuous functions of at most polynomial growth. These results generalize the corresponding conclusions given by Buckdahn et al. (2009) in the case without control.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 262713, 17 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273262

Digital Object Identifier
doi:10.1155/2014/262713

Mathematical Reviews number (MathSciNet)
MR3191027

Zentralblatt MATH identifier
07022039

#### Citation

Hao, Tao; Li, Juan. Backward Stochastic Differential Equations Coupled with Value Function and Related Optimal Control Problems. Abstr. Appl. Anal. 2014 (2014), Article ID 262713, 17 pages. doi:10.1155/2014/262713. https://projecteuclid.org/euclid.aaa/1412273262

#### References

• R. Buckdahn, B. Djehiche, J. Li, and S. Peng, “Mean-field backward stochastic differential equations: a limit approach,” The Annals of Probability, vol. 37, no. 4, pp. 1524–1565, 2009.
• T. Chan, “Dynamics of the McKean-Vlasov equation,” The Annals of Probability, vol. 22, no. 1, pp. 431–441, 1994.
• P. Kotelenez, “A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation,” Probability Theory and Related Fields, vol. 102, no. 2, pp. 159–188, 1995.
• J.-M. Lasry and P.-L. Lions, “Mean field games,” Japanese Journal of Mathematics, vol. 2, no. 1, pp. 229–260, 2007.
• P. D. Pra and F. D. Hollander, “McKean-Vlasov limit for interacting random processes in random media,” Journal of Statistical Physics, vol. 84, no. 3-4, pp. 735–772, 1996.
• A.-S. Sznitman, “Topics in propagation of chaos,” in École d'Été de Probabilités de Saint-Flour XIX–-1989, vol. 1464 of Lecture Notes in Mathematics, pp. 165–251, Springer, Berlin, Germany, 1991.
• R. Buckdahn, J. Li, and S. Peng, “Mean-field backward stochastic differential equations and related partial differential equations,” Stochastic Processes and Their Applications, vol. 119, no. 10, pp. 3133–3154, 2009.
• L. S. Pontryagin, V. G. Boltyanskii, and R. V. Gamkrelidze, The Mathematical Theory of Optimal Processes, Wiley, New York, NY, USA, 1962.
• R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, USA, 1957.
• H. J. Kushner, “On the stochastic maximum principle: fixed time of control,” Journal of Mathematical Analysis and Applications, vol. 11, pp. 78–92, 1965.
• H. J. Kushner, “Necessary conditions for continuous parameter stochastic optimization problems,” SIAM Journal on Control and Optimization, vol. 10, no. 3, pp. 550–565, 1972.
• A. Bensoussan, “Lectures on stochastic control,” in Nonlinear Filtering and Stochastic Control, vol. 972 of Lecture Notes in Mathematics, pp. 1–62, Springer, Berlin, Germany, 1982.
• U. G. Haussmann, A Stochastic Maximum Principle for Optimal Control of Diffusions, vol. 151 of Pitman Research Notes in Mathematics Series, Longman, Harlow, UK, 1986.
• S. G. Peng, “A general stochastic maximum principle for optimal control problems,” SIAM Journal on Control and Optimization, vol. 28, no. 4, pp. 966–979, 1990.
• S. J. Tang and X. J. Li, “Necessary conditions for optimal control of stochastic systems with random jumps,” SIAM Journal on Control and Optimization, vol. 32, no. 5, pp. 1447–1475, 1994.
• X. Y. Zhou, “Stochastic near-optimal controls: necessary and sufficient conditions for near-optimality,” SIAM Journal on Control and Optimization, vol. 36, no. 3, pp. 929–947, 1998.
• A. Bensoussan, K. C. J. Sung, S. C. P. Yam, and S. P. Yung, “Linear-quadraticčommentComment on ref. [4?]: Please update the information of this reference, if possible. mean field games,”. In press.
• R. Buckdahn, B. Djehiche, and J. Li, “A general stochastic maximum principle for SDEs of mean-field type,” Applied Mathematics and Optimization, vol. 64, no. 2, pp. 197–216, 2011.
• J. Li, “Stochastic maximum principle in the mean-field controls,” Automatica, vol. 48, no. 2, pp. 366–373, 2012.
• T. Meyer-Brandis, B. ${\text{\O}}$ksendal, and X. Y. Zhou, “A mean-field stochastic maximum principle via Malliavin calculus,” Stochastics an International Journal of Probability and Stochastic Processes, vol. 84, no. 5-6, pp. 643–666, 2012.
• J. M. Yong, “A linear-quadratic optimal control problem for mean-field stochastic differential equations,” Tech. Rep., 2011.
• R. Buckdahn and J. Li, “Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations,” SIAM Journal on Control and Optimization, vol. 47, no. 1, pp. 444–475, 2008.
• S. G. Peng, “A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation,” Stochastics and Stochastics Reports, vol. 38, no. 2, pp. 119–134, 1992.
• S. G. Peng, “Backward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJB equations,” in Topics on Stochastic Analysis, J. A. Yan, S. G. Peng, S. Z. Fang, and L. M. Wu, Eds., pp. 85–138, Science Press, Beijing, China, 1997 (Chinese).
• J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43, Springer, New York, NY, USA, 1999.
• N. El Karoui, S. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,” Mathematical Finance, vol. 7, no. 1, pp. 1–71, 1997.
• É. Pardoux and S. G. Peng, “Adapted solution of a backward stochastic differential equation,” Systems & Control Letters, vol. 14, no. 1, pp. 55–61, 1990.
• M. G. Crandall, H. Ishii, and P.-L. Lions, “User's guide to viscosity solutions of second order partial differential equations,” Bulletin of the American Mathematical Society, vol. 27, no. 1, pp. 1–67, 1992.
• G. Barles, R. Buckdahn, and E. Pardoux, “Backward stochastic differential equations and integral-partial differential equations,” Stochastics and Stochastics Reports, vol. 60, no. 1-2, pp. 57–83, 1997.
• S. G. Peng, “Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,” Stochastics and Stochastics Reports, vol. 37, no. 1-2, pp. 61–74, 1991. \endinput