# American Institute of Mathematical Sciences

doi: 10.3934/naco.2022010
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## On path-dependent multidimensional forward-backward SDEs

 1 CMAP, Ecole Polytechnique, France 2 CEREMADE, Université Paris Dauphine-PSL, France

*Corresponding author: Nizar Touzi

This paper is handled by Jianfeng Zhang as guest editor.

Received  January 2022 Revised  April 2022 Early access May 2022

This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time $t$ can depend on the whole path of the forward process up to time $t$. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling field on small time interval. Then by introducing the characteristic BSDE, we show that a global decoupling field can be constructed by patching local solutions together as long as the solution of the characteristic BSDE remains bounded. Finally, we provide a stability result for path-dependent forward-backward SDEs.

Citation: Kaitong Hu, Zhenjie Ren, Nizar Touzi. On path-dependent multidimensional forward-backward SDEs. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022010
##### References:
 [1] F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793. [2] J. Cvitanic and J. F. Zhang, The steepest descent method for forward-backward sdes, Electron. J. Probab., 10 (2005), 1468-1495.  doi: 10.1214/EJP.v10-295. [3] J. Cvitanic and J. F. Zhang, Contract Theory in Continuous-Time Models, Springer, 2012. doi: 10.1007/978-3-642-14200-0. [4] F. Delarue, On the existence and uniqueness of solutions to fbsdes in a non-degenerate case, Stochastic Process, 99 (2002), 209-286.  doi: 10.1016/S0304-4149(02)00085-6. [5] A. Fromm and P. Imkeller, Existence, uniqueness and regularity of decoupling fields to multidimensional fully coupled FBSDEs, arXiv e-prints, October 2013. [6] Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probability Theory and Related Fields, 103 (1995), 273-283.  doi: 10.1007/BF01204218. [7] M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 28 (2000), 558-602.  doi: 10.1214/aop/1019160253. [8] J. Ma and J. M. Yong, On linear, degenerate backward stochastic partial differential equations, Probability Theory and Related Fields, 113 (1999), 135-170.  doi: 10.1007/s004400050205. [9] J. Ma and J. M. Yong, Forward-Backward Stochastic Differential Equations and their Applications, Springer, Berlin, Heidelberg, 2007. doi: 10.1007/s002459900100. [10] J. Ma, P. Protter and J. M. Yong, Solving forward-backward stochastic differential equations explicitly — a four step scheme, Probability Theory and Related Fields, 98 (1994), 339-359.  doi: 10.1007/BF01192258. [11] J. Ma, Z. Wu, D. T. Zhang and J. F. Zhang, On well-posedness of forward-backward SDEs–a unified approach, Ann. Appl. Probab., 25 (2015), 2168-2214.  doi: 10.1214/14-AAP1046. [12] J. Ma and J. M. Yong, Adapted solution of a degenerate backward spde, with applications, Stochastic Processes and Their Applications, 70 (1997), 59-84.  doi: 10.1016/S0304-4149(97)00057-4. [13] E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, In Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Information Sciences (eds. B. L. Rozovskii and R. B. Sowers), 176 (1992), Springer, Berlin, Heidelberg. doi: 10.1007/BFb0007334. [14] E. Pardoux and S. J. Tang, Forward-backward stochastic differential equations and quasilinear parabolic pdes, Probability Theory and Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001. [15] S. G. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM Journal on Control and Optimization, 37 (1999), 825-843.  doi: 10.1137/S0363012996313549. [16] Z. Wu and Z. Y. Yu, Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations, Stochastic Processes and Their Applications, 124 (2014), 3921-3947.  doi: 10.1016/j.spa.2014.07.013. [17] J. M. Yong, Finding adapted solutions of forward-backward stochastic differential equations: method of continuation, Probability Theory and Related Fields, 107 (1997), 537-572.  doi: 10.1007/s004400050098. [18] J. M. Yong, Forward-backward stochastic differential equations with mixed initial-terminal conditions, Trans. Amer. Math. Soc., 362 (2010), 1047-1096.  doi: 10.1090/S0002-9947-09-04896-X. [19] Z. Y. Yu, Equivalent cost functionals and stochastic linear quadratic optimal control problems, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 78-90.  doi: 10.1051/cocv/2011206. [20] J. F. Zhang, The wellposedness of FBSDEs, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 927-940.  doi: 10.3934/dcdsb.2006.6.927. [21] J. F. Zhang, The wellposedness of FBSDEs (ⅱ), arXiv: 1708.05785, 2017.

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##### References:
 [1] F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793. [2] J. Cvitanic and J. F. Zhang, The steepest descent method for forward-backward sdes, Electron. J. Probab., 10 (2005), 1468-1495.  doi: 10.1214/EJP.v10-295. [3] J. Cvitanic and J. F. Zhang, Contract Theory in Continuous-Time Models, Springer, 2012. doi: 10.1007/978-3-642-14200-0. [4] F. Delarue, On the existence and uniqueness of solutions to fbsdes in a non-degenerate case, Stochastic Process, 99 (2002), 209-286.  doi: 10.1016/S0304-4149(02)00085-6. [5] A. Fromm and P. Imkeller, Existence, uniqueness and regularity of decoupling fields to multidimensional fully coupled FBSDEs, arXiv e-prints, October 2013. [6] Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probability Theory and Related Fields, 103 (1995), 273-283.  doi: 10.1007/BF01204218. [7] M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 28 (2000), 558-602.  doi: 10.1214/aop/1019160253. [8] J. Ma and J. M. Yong, On linear, degenerate backward stochastic partial differential equations, Probability Theory and Related Fields, 113 (1999), 135-170.  doi: 10.1007/s004400050205. [9] J. Ma and J. M. Yong, Forward-Backward Stochastic Differential Equations and their Applications, Springer, Berlin, Heidelberg, 2007. doi: 10.1007/s002459900100. [10] J. Ma, P. Protter and J. M. Yong, Solving forward-backward stochastic differential equations explicitly — a four step scheme, Probability Theory and Related Fields, 98 (1994), 339-359.  doi: 10.1007/BF01192258. [11] J. Ma, Z. Wu, D. T. Zhang and J. F. Zhang, On well-posedness of forward-backward SDEs–a unified approach, Ann. Appl. Probab., 25 (2015), 2168-2214.  doi: 10.1214/14-AAP1046. [12] J. Ma and J. M. Yong, Adapted solution of a degenerate backward spde, with applications, Stochastic Processes and Their Applications, 70 (1997), 59-84.  doi: 10.1016/S0304-4149(97)00057-4. [13] E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, In Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Information Sciences (eds. B. L. Rozovskii and R. B. Sowers), 176 (1992), Springer, Berlin, Heidelberg. doi: 10.1007/BFb0007334. [14] E. Pardoux and S. J. Tang, Forward-backward stochastic differential equations and quasilinear parabolic pdes, Probability Theory and Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001. [15] S. G. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM Journal on Control and Optimization, 37 (1999), 825-843.  doi: 10.1137/S0363012996313549. [16] Z. Wu and Z. Y. Yu, Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations, Stochastic Processes and Their Applications, 124 (2014), 3921-3947.  doi: 10.1016/j.spa.2014.07.013. [17] J. M. Yong, Finding adapted solutions of forward-backward stochastic differential equations: method of continuation, Probability Theory and Related Fields, 107 (1997), 537-572.  doi: 10.1007/s004400050098. [18] J. M. Yong, Forward-backward stochastic differential equations with mixed initial-terminal conditions, Trans. Amer. Math. Soc., 362 (2010), 1047-1096.  doi: 10.1090/S0002-9947-09-04896-X. [19] Z. Y. Yu, Equivalent cost functionals and stochastic linear quadratic optimal control problems, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 78-90.  doi: 10.1051/cocv/2011206. [20] J. F. Zhang, The wellposedness of FBSDEs, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 927-940.  doi: 10.3934/dcdsb.2006.6.927. [21] J. F. Zhang, The wellposedness of FBSDEs (ⅱ), arXiv: 1708.05785, 2017.
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