doi: 10.3934/naco.2022010
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. MaP. 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. MaZ. WuD. 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.

show all references

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. MaP. 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. MaZ. WuD. 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.

[1]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115

[2]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control and Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613

[3]

Jiongmin Yong. Forward-backward stochastic differential equations: Initiation, development and beyond. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022011

[4]

Jiongmin Yong. Forward-backward evolution equations and applications. Mathematical Control and Related Fields, 2016, 6 (4) : 653-704. doi: 10.3934/mcrf.2016019

[5]

Fabio Paronetto. Elliptic approximation of forward-backward parabolic equations. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1017-1036. doi: 10.3934/cpaa.2020047

[6]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

[7]

Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 773-795. doi: 10.3934/dcdss.2021044

[8]

G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783

[9]

Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks and Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941

[10]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002

[11]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control and Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013

[12]

Xiao Ding, Deren Han. A modification of the forward-backward splitting method for maximal monotone mappings. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 295-307. doi: 10.3934/naco.2013.3.295

[13]

Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1893-1906. doi: 10.3934/cpaa.2021051

[14]

Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054

[15]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929

[16]

Yinggu Chen, Said HamadÈne, Tingshu Mu. Mean-field doubly reflected backward stochastic differential equations. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022012

[17]

Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043

[18]

Flavia Smarrazzo, Andrea Terracina. Sobolev approximation for two-phase solutions of forward-backward parabolic problems. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1657-1697. doi: 10.3934/dcds.2013.33.1657

[19]

Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations and Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023

[20]

Richard Archibald, Feng Bao, Yanzhao Cao, He Zhang. A backward SDE method for uncertainty quantification in deep learning. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022062

 Impact Factor: 

Article outline

[Back to Top]