A new type of controlled fully coupled forward-backward stochastic differential equations is discussed, namely those involving the value function. With a new iteration method, we prove an existence and uniqueness theorem of a solution for this type of equations. Using the notion of extended “backward semigroup”, we prove that the value function satisfies the dynamic programming principle and is a viscosity solution of the associated nonlocal Hamilton−Jacobi−Bellman equation.

DOI: 10.1051/cocv/2015016

Keywords: Fully coupled FBSDE involving value function, dynamic programming principle, fully coupled mean-field FBSDE, viscosity solution, nonlocal Hamilton−Jacobi−Bellman equation

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@article{COCV_2016__22_2_519_0, author = {Hao, Tao and Li, Juan}, title = {Fully coupled forward-backward {SDEs} involving the value function and associated nonlocal {Hamilton\ensuremath{-}Jacobi\ensuremath{-}Bellman} equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {519--538}, publisher = {EDP-Sciences}, volume = {22}, number = {2}, year = {2016}, doi = {10.1051/cocv/2015016}, zbl = {1338.60146}, mrnumber = {3491781}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015016/} }

TY - JOUR AU - Hao, Tao AU - Li, Juan TI - Fully coupled forward-backward SDEs involving the value function and associated nonlocal Hamilton−Jacobi−Bellman equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 519 EP - 538 VL - 22 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015016/ DO - 10.1051/cocv/2015016 LA - en ID - COCV_2016__22_2_519_0 ER -

%0 Journal Article %A Hao, Tao %A Li, Juan %T Fully coupled forward-backward SDEs involving the value function and associated nonlocal Hamilton−Jacobi−Bellman equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 519-538 %V 22 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015016/ %R 10.1051/cocv/2015016 %G en %F COCV_2016__22_2_519_0

Hao, Tao; Li, Juan. Fully coupled forward-backward SDEs involving the value function and associated nonlocal Hamilton−Jacobi−Bellman equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 519-538. doi : 10.1051/cocv/2015016. http://archive.numdam.org/articles/10.1051/cocv/2015016/

Backward-forward stochastic differential equations. Ann. Appl. Probab. 3 (1993) 777–793. | DOI | MR | Zbl

,Some stochastic particle methods for nonlinear parabolic PDEs. ESAIM: Proc. 15 (2005) 18–57. | DOI | MR | Zbl

,A stochastic particle method for the McKean−Vlasov and the Burgers equation. Math. Comput. 66 (1997) 157–192. | DOI | MR | Zbl

and ,Stochastic differential games and viscosity solutions of Hamilton−Jacobi−Bellman−Isaacs equations. SIAM J. Control Optim. 47 (2008) 444–475. | DOI | MR | Zbl

and ,Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Proc. Appl. 119 (2009) 3133–3154. | DOI | MR | Zbl

, and ,Mean-field backward stochastic differential equations: A limit approach. Ann. Probab. 37 (2009) 1524–1565. | DOI | MR | Zbl

, , and ,Dynamics of the McKean−Vlasov equation. Ann. Probab. 22 (1994) 431–441. | DOI | MR | Zbl

,User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. | DOI | MR | Zbl

, and ,Backward stochastic differential equations coupled with value function and related optimal control problems. Abstr. Appl. Anal. 2014 (2014) 262713. | MR | Zbl

and ,Solution of forward-backward stochastic differential equations. Probab. Theory Relat. Fields 103 (1995) 273–283. | DOI | MR | Zbl

and ,A class of quasilinear stochastic partial differential equations of McKean−Vlasov type with mass conservation. Probab. Theory Relat. Fields 102 (1995) 159–188. | DOI | MR | Zbl

,Fully coupled mean-field forward-backward stochastic differential equations and stochastic maximum principle. Abstr. Appl. Anal. 2014 (2014) 839467. | MR | Zbl

, and ,Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton−Jacobi−Bellman equations. SIAM J. Control Optim. 52 (2014) 1622–1662. | DOI | MR | Zbl

and ,Stochastic differential games for fully coupled FBSDEs with jumps. Appl Math Optim. 71 (2015) 411–448. | DOI | Zbl

and ,Sloving forward-backward stochastic differential equations explicitly-a four step scheme. Probab. Theory Relat. Fields 98 (1994) 339–359. | DOI | MR | Zbl

, and ,S.G. Peng, BSDE and Stochastic Optimizations, in Topics on Stochastic Analysis, edited by J.A. Yan, S.G. Peng, S.Z. Fang and L.M. Wu. Science Press, Beijing (1997) 85–138 (in Chinese).

Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37 (1999) 825–843. | DOI | MR | Zbl

and ,McKean−Vlasov limit for interacting random processes in random media. J. Stat. Phys. 84 (1996) 735–772. | DOI | MR | Zbl

and ,A stochastic particle method with random weights for the computation of statistical solutions of McKean−Vlasov equations. Ann. Appl. Probab. 13 (2003) 140–180. | DOI | MR | Zbl

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