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doi: 10.3934/naco.2022011
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Forward-backward stochastic differential equations: Initiation, development and beyond

Deparment of Mathematics, University of Central Florida, Orlando, FL 32816, USA

This paper is dedicated to Professor Jin Ma on the occasion of his 65th birthday

Received  December 2021 Revised  April 2022 Early access May 2022

Fund Project: The author is supported by NSF grant DMS-1812921

Jin Ma has made fundamental contributions to the theory of forward-backward stochastic differential equations (FBSDEs, for short). In this paper, as one of his main collaborators, I will recall the initiation, the path of the development for FBSDEs, as well as some interesting historic stories.

Citation: Jiongmin Yong. Forward-backward stochastic differential equations: Initiation, development and beyond. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022011
References:
[1]

F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793. 

[2]

F. Antonelli and J. Ma, Weak solutions of forward-backward SDE's, Stochastic Anal. Appl., 21 (2003), 493-514.  doi: 10.1081/SAP-120020423.

[3]

R. BellmanR. Kalaba and G. M. Wing, Invariant imbedding and the reduction of two-point boundary value problems to initial value problems, Proc. Nat. Acad. Sci. U.S.A., 46 (1960), 1646-1649.  doi: 10.1073/pnas.46.12.1646.

[4]

A. Bensoussan, Leture on Stochastic Control, Lecture Notes in Math, 972 (1981), Springer-Verlag, Berlin.

[5]

J. M. Bismut, Analyse Convexe et Probabilités, Thése, Faculté des Sciences de Paris, Paris, 1973. doi: 10.1016/0022-247X(73)90170-4.

[6]

J. M. Bismut, An introductory approach to dualty in optimal stochastic control, SIAM Rev., 20 (1978), 62-78.  doi: 10.1137/1020004.

[7]

J. ChenJ. Ma and H. Yin, Forward-backward SDEs with discontinuous coefficients, Stoch. Anal. Appl., 36 (2018), 274-294.  doi: 10.1080/07362994.2017.1399799.

[8]

J. Cvitanić and J. Ma, Hedging options for a large investor and forward-backward SDE's, Ann. Appl. Probab., 6 (1996), 370-398.  doi: 10.1214/aoap/1034968136.

[9]

J. Cvitanić and J. Ma, Reflected forward-backward SDEs and obstacle problems with boundary conditions, J. Appl. Math. Stochastic Anal., 14 (2001), 113-138.  doi: 10.1155/S1048953301000090.

[10]

J. CvtanicD. Possamai and N. Touzi, Moral hazard in dynamic risk management, Management Sci., 63 (2017), 3328-3346. 

[11]

F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stoch. Proc. Appl., 99 (2002), 209-286.  doi: 10.1016/S0304-4149(02)00085-6.

[12]

J. DouglasJ. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968.  doi: 10.1214/aoap/1034968235.

[13]

K. Du and S. Chen, Backward stochastic partial differential equations with quadratic growth, J. Math. Anal. Appl., 419 (2014), 447-468.  doi: 10.1016/j.jmaa.2014.04.050.

[14]

K. DuS. Tang and Q. Zhang, $W^{m, p}$-slution ($p\ge2$) of linear degenerate backward stochastic partial differential equations in the whole space, J. Diff. Eqn., 254 (2013), 2877-2904.  doi: 10.1016/j.jde.2013.01.013.

[15]

D. Duffie and L. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394.  doi: 10.2307/2951600.

[16]

K. Du and Q. Zhang, Semi-linear degenerate backward stochastic partial differential equations and associated forward-backward stochastic differential equations, Stoch. Proc. Appl., 123 (2013), 1616-1637.  doi: 10.1016/j.spa.2013.01.005.

[17]

D. DuffieJ. Ma and J. Yong, Black's consol rate conjecture, Ann. Appl. Probab., 5 (1995), 356-382. 

[18]

W. EJ. Han and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Comm. Math. Stats., 5 (2017), 349-380.  doi: 10.1007/s40304-017-0117-6.

[19]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[20]

R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics, 20 (1948), 367-387.  doi: 10.1103/revmodphys.20.367.

[21]

Y. Hu and J. Ma, Nonlinear Feynman-Kac formula and discrete-functional-type BSDEs with continuous coefficients, Stochastic Process. Appl., 112 (2004), 23-51.  doi: 10.1016/j.spa.2004.02.002.

[22]

Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory & Rel. Fields, 103 (1995), 273-283.  doi: 10.1007/BF01204218.

[23]

Y. HuJ. Ma and J. Yong, On semilinear degenerate backward stochastic partial differential equations, Probab. Theory & Rel. Fields, 123 (2002), 381-411.  doi: 10.1007/s004400100193.

[24]

M. Kac, On distributions of certain Wiener functionnals, Trans. AMS, 65 (1949), 1-13.  doi: 10.2307/1990512.

[25]

J. Ma, On state estimation with infinite duration, Control Theory Appl., (in Chinese), 4 (1987), 114–120.

[26]

J. MaP. ProtterJ. San Mart and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 302-316.  doi: 10.1214/aoap/1015961165.

[27]

J. MaP. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly — a four-step scheme, Probability Theory & Related Fields, 98 (1994), 339-359.  doi: 10.1007/BF01192258.

[28]

J. MaJ. Shen and Y. Zhao, On numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46 (2008), 2636-2661.  doi: 10.1137/06067393X.

[29]

J. Ma and Y. Wang, On variant reflected backward SDEs, with applications, J. Appl. Math. Stoch. Anal., 2009, Art. ID 854768, 26 pp. doi: 10.1155/2009/854768.

[30]

J. MaZ. WuD. Zhang and J. Zhang, On well-posedness of forward-backward SDEs — a unified approach, Ann. Appl. Probab., 25 (2015), 2168-2214.  doi: 10.1214/14-AAP1046.

[31]

J. MaH. Yin and J. Zhang, On non-Markovian forward-backward SDEs and backward stochastic PDEs, Stochastic Process. Appl., 122 (2012), 3980-4004.  doi: 10.1016/j.spa.2012.08.002.

[32]

J. Ma and J. Yong, Solvability of forward-backward SDEs and the nodal set of Hamilton-Jacobi-Bellman equations, Chin. Ann. Math. Ser. B, 16 (1995), 279-298. 

[33]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stoch. Proc. & Appl., 70 (1997), 59-84.  doi: 10.1016/S0304-4149(97)00057-4.

[34]

J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations, Prob. Theory & Rel. Fields, 113 (1999), 135-170.  doi: 10.1007/s004400050205.

[35]

J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702 (1999), Springer-Verlag, Berlin.

[36]

J. Ma and J. Yong, Approximate solvability of forward-backward stochastic differential equations, Appl. Math. Optim., 45 (2002), 1-22.  doi: 10.1007/s00245-001-0025-7.

[37]

J. MaJ. Yong and Y. Zhao, Four step scheme for general Markovian forward-backward SDEs, J. System Science and Complexity, 23 (2010), 546-571.  doi: 10.1007/s11424-010-0145-8.

[38]

J. Ma and J. Zhang, Representation theorems for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 1390-1418.  doi: 10.1214/aoap/1037125868.

[39]

J. Ma and J. Zhang, Path regularity for solutions of backward stochastic differential equations, Probab. Theory Related Fields, 122 (2002), 163-190.  doi: 10.1007/s004400100144.

[40]

J. Ma and J. Zhang, Representations and regularities for solutions to BSDEs with reflections, Stochastic Process. Appl., 115 (2005), 539-569.  doi: 10.1016/j.spa.2004.05.010.

[41]

J. Ma and J. Zhang, On weak solutions of forward-backward SDEs, Probab. Theory Related Fields, 151 (2011), 475-507.  doi: 10.1007/s00440-010-0305-8.

[42]

J. MaJ. Zhang and Z. Zheng, Weak solutions for forward-backward SDEs martingale problem approach, Ann. Probab., 36 (2008), 2092-2125.  doi: 10.1214/08-AOP0383.

[43]

E. Pardoux and S. Peng, Adapted solution of backward stochastic differential equations, Syst. Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

[44]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.

[45]

S. Peng, A nonlinear Feynman-Kac formula and applications, Control Theory, Stochastic Analysis and Applications (eds. S. Chen and J. Yong), World Scientific, (1991), 173–184.

[46]

Y. Sannikov, A continuous-time version of the Principal-Agent problem, Review Econ. Study, 75 (2008), 957-984.  doi: 10.1111/j.1467-937X.2008.00486.x.

[47]

J. Yong, Finding adapted solutions of forward-backward stochastic differential equations — method of continuation, Prob. Theory & Rel. Fields, 107 (1997), 537-572.  doi: 10.1007/s004400050098.

[48]

J. Zhang, A numerical scheme for backward stochasic differential equations, Ann. Appl. Probab., 14 (2004), 459-488.  doi: 10.1214/aoap/1075828058.

show all references

References:
[1]

F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793. 

[2]

F. Antonelli and J. Ma, Weak solutions of forward-backward SDE's, Stochastic Anal. Appl., 21 (2003), 493-514.  doi: 10.1081/SAP-120020423.

[3]

R. BellmanR. Kalaba and G. M. Wing, Invariant imbedding and the reduction of two-point boundary value problems to initial value problems, Proc. Nat. Acad. Sci. U.S.A., 46 (1960), 1646-1649.  doi: 10.1073/pnas.46.12.1646.

[4]

A. Bensoussan, Leture on Stochastic Control, Lecture Notes in Math, 972 (1981), Springer-Verlag, Berlin.

[5]

J. M. Bismut, Analyse Convexe et Probabilités, Thése, Faculté des Sciences de Paris, Paris, 1973. doi: 10.1016/0022-247X(73)90170-4.

[6]

J. M. Bismut, An introductory approach to dualty in optimal stochastic control, SIAM Rev., 20 (1978), 62-78.  doi: 10.1137/1020004.

[7]

J. ChenJ. Ma and H. Yin, Forward-backward SDEs with discontinuous coefficients, Stoch. Anal. Appl., 36 (2018), 274-294.  doi: 10.1080/07362994.2017.1399799.

[8]

J. Cvitanić and J. Ma, Hedging options for a large investor and forward-backward SDE's, Ann. Appl. Probab., 6 (1996), 370-398.  doi: 10.1214/aoap/1034968136.

[9]

J. Cvitanić and J. Ma, Reflected forward-backward SDEs and obstacle problems with boundary conditions, J. Appl. Math. Stochastic Anal., 14 (2001), 113-138.  doi: 10.1155/S1048953301000090.

[10]

J. CvtanicD. Possamai and N. Touzi, Moral hazard in dynamic risk management, Management Sci., 63 (2017), 3328-3346. 

[11]

F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stoch. Proc. Appl., 99 (2002), 209-286.  doi: 10.1016/S0304-4149(02)00085-6.

[12]

J. DouglasJ. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968.  doi: 10.1214/aoap/1034968235.

[13]

K. Du and S. Chen, Backward stochastic partial differential equations with quadratic growth, J. Math. Anal. Appl., 419 (2014), 447-468.  doi: 10.1016/j.jmaa.2014.04.050.

[14]

K. DuS. Tang and Q. Zhang, $W^{m, p}$-slution ($p\ge2$) of linear degenerate backward stochastic partial differential equations in the whole space, J. Diff. Eqn., 254 (2013), 2877-2904.  doi: 10.1016/j.jde.2013.01.013.

[15]

D. Duffie and L. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394.  doi: 10.2307/2951600.

[16]

K. Du and Q. Zhang, Semi-linear degenerate backward stochastic partial differential equations and associated forward-backward stochastic differential equations, Stoch. Proc. Appl., 123 (2013), 1616-1637.  doi: 10.1016/j.spa.2013.01.005.

[17]

D. DuffieJ. Ma and J. Yong, Black's consol rate conjecture, Ann. Appl. Probab., 5 (1995), 356-382. 

[18]

W. EJ. Han and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Comm. Math. Stats., 5 (2017), 349-380.  doi: 10.1007/s40304-017-0117-6.

[19]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[20]

R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics, 20 (1948), 367-387.  doi: 10.1103/revmodphys.20.367.

[21]

Y. Hu and J. Ma, Nonlinear Feynman-Kac formula and discrete-functional-type BSDEs with continuous coefficients, Stochastic Process. Appl., 112 (2004), 23-51.  doi: 10.1016/j.spa.2004.02.002.

[22]

Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory & Rel. Fields, 103 (1995), 273-283.  doi: 10.1007/BF01204218.

[23]

Y. HuJ. Ma and J. Yong, On semilinear degenerate backward stochastic partial differential equations, Probab. Theory & Rel. Fields, 123 (2002), 381-411.  doi: 10.1007/s004400100193.

[24]

M. Kac, On distributions of certain Wiener functionnals, Trans. AMS, 65 (1949), 1-13.  doi: 10.2307/1990512.

[25]

J. Ma, On state estimation with infinite duration, Control Theory Appl., (in Chinese), 4 (1987), 114–120.

[26]

J. MaP. ProtterJ. San Mart and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 302-316.  doi: 10.1214/aoap/1015961165.

[27]

J. MaP. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly — a four-step scheme, Probability Theory & Related Fields, 98 (1994), 339-359.  doi: 10.1007/BF01192258.

[28]

J. MaJ. Shen and Y. Zhao, On numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46 (2008), 2636-2661.  doi: 10.1137/06067393X.

[29]

J. Ma and Y. Wang, On variant reflected backward SDEs, with applications, J. Appl. Math. Stoch. Anal., 2009, Art. ID 854768, 26 pp. doi: 10.1155/2009/854768.

[30]

J. MaZ. WuD. Zhang and J. Zhang, On well-posedness of forward-backward SDEs — a unified approach, Ann. Appl. Probab., 25 (2015), 2168-2214.  doi: 10.1214/14-AAP1046.

[31]

J. MaH. Yin and J. Zhang, On non-Markovian forward-backward SDEs and backward stochastic PDEs, Stochastic Process. Appl., 122 (2012), 3980-4004.  doi: 10.1016/j.spa.2012.08.002.

[32]

J. Ma and J. Yong, Solvability of forward-backward SDEs and the nodal set of Hamilton-Jacobi-Bellman equations, Chin. Ann. Math. Ser. B, 16 (1995), 279-298. 

[33]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stoch. Proc. & Appl., 70 (1997), 59-84.  doi: 10.1016/S0304-4149(97)00057-4.

[34]

J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations, Prob. Theory & Rel. Fields, 113 (1999), 135-170.  doi: 10.1007/s004400050205.

[35]

J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702 (1999), Springer-Verlag, Berlin.

[36]

J. Ma and J. Yong, Approximate solvability of forward-backward stochastic differential equations, Appl. Math. Optim., 45 (2002), 1-22.  doi: 10.1007/s00245-001-0025-7.

[37]

J. MaJ. Yong and Y. Zhao, Four step scheme for general Markovian forward-backward SDEs, J. System Science and Complexity, 23 (2010), 546-571.  doi: 10.1007/s11424-010-0145-8.

[38]

J. Ma and J. Zhang, Representation theorems for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 1390-1418.  doi: 10.1214/aoap/1037125868.

[39]

J. Ma and J. Zhang, Path regularity for solutions of backward stochastic differential equations, Probab. Theory Related Fields, 122 (2002), 163-190.  doi: 10.1007/s004400100144.

[40]

J. Ma and J. Zhang, Representations and regularities for solutions to BSDEs with reflections, Stochastic Process. Appl., 115 (2005), 539-569.  doi: 10.1016/j.spa.2004.05.010.

[41]

J. Ma and J. Zhang, On weak solutions of forward-backward SDEs, Probab. Theory Related Fields, 151 (2011), 475-507.  doi: 10.1007/s00440-010-0305-8.

[42]

J. MaJ. Zhang and Z. Zheng, Weak solutions for forward-backward SDEs martingale problem approach, Ann. Probab., 36 (2008), 2092-2125.  doi: 10.1214/08-AOP0383.

[43]

E. Pardoux and S. Peng, Adapted solution of backward stochastic differential equations, Syst. Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

[44]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.

[45]

S. Peng, A nonlinear Feynman-Kac formula and applications, Control Theory, Stochastic Analysis and Applications (eds. S. Chen and J. Yong), World Scientific, (1991), 173–184.

[46]

Y. Sannikov, A continuous-time version of the Principal-Agent problem, Review Econ. Study, 75 (2008), 957-984.  doi: 10.1111/j.1467-937X.2008.00486.x.

[47]

J. Yong, Finding adapted solutions of forward-backward stochastic differential equations — method of continuation, Prob. Theory & Rel. Fields, 107 (1997), 537-572.  doi: 10.1007/s004400050098.

[48]

J. Zhang, A numerical scheme for backward stochasic differential equations, Ann. Appl. Probab., 14 (2004), 459-488.  doi: 10.1214/aoap/1075828058.

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