September  2018, 8(3&4): 1021-1049. doi: 10.3934/mcrf.2018044

Forward backward SDEs in weak formulation

1. 

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

2. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

* Corresponding author: J. Zhang

Received  December 2017 Revised  February 2018 Published  September 2018

Fund Project: Wang is supported by Distinguished Middle-Aged and Young Scientist Encourage and Reward Foundation of Shandong Province (ZR2017BA033); Zhang is supported by NSF grant #1413717, and Zhang would like to thank Daniel Lacker for very helpful discussion on Section 2.3.2

Although having been developed for more than two decades, the theory of forward backward stochastic differential equations is still far from complete. In this paper, we take one step back and investigate the formulation of FBSDEs. Motivated from several considerations, both in theory and in applications, we propose to study FBSDEs in weak formulation, rather than the strong formulation in the standard literature. That is, the backward SDE is driven by the forward component, instead of by the Brownian motion. We establish the Feyman-Kac formula for FBSDEs in weak formulation, both in classical and in viscosity sense. Our new framework is efficient especially when the diffusion part of the forward equation involves the $Z$-component of the backward equation.

Citation: Haiyang Wang, Jianfeng Zhang. Forward backward SDEs in weak formulation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1021-1049. doi: 10.3934/mcrf.2018044
References:
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[23]

S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37 (1999), 825-843.  doi: 10.1137/S0363012996313549.  Google Scholar

[24]

T. Pham and J. Zhang, Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation, SIAM J. Control Optim., 52 (2014), 2090-2121.  doi: 10.1137/120894907.  Google Scholar

[25]

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H. M. SonerN. Touzi and J. Zhang, Wellposedness of second order backward SDEs, Probab. Theory Related Fields, 153 (2012), 149-190.  doi: 10.1007/s00440-011-0342-y.  Google Scholar

[28]

D. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2006. xii+338 pp  Google Scholar

[29]

B. Tsirelson, An example of a stochastic differential equation having no strong solution, Theory of Probability and Its Applications, 20 (1975), 416-418.  doi: 10.1137/1120049.  Google Scholar

[30]

J. Yong, Finding adapted solutions of forward-backward stochastic differential equations: method of continuation, Probab. Theory Related Fields, 107 (1997), 537-572.  doi: 10.1007/s004400050098.  Google Scholar

[31]

J. Yong and X. Y. Zhou, Stochastic controls. Hamiltonian systems and HJB equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. xxii+438 pp. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[32]

J. Zhang, The wellposedness of FBSDEs, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 927-940 (electronic). doi: 10.3934/dcdsb.2006.6.927.  Google Scholar

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J. Zhang, Backward Stochastic Differential Equations - from Linear to Fully Nonlinear Theory, Springer, New York, 2017. doi: 10.1007/978-1-4939-7256-2.  Google Scholar

[34]

W. A. Zheng, Tightness results for laws of diffusion processes application to stochastic mechanics, Ann. Inst. H. Poincare Probab. Statist., 21 (1985), 103-124.   Google Scholar

show all references

References:
[1]

F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793.  doi: 10.1214/aoap/1177005363.  Google Scholar

[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.  Google Scholar

[3]

M. T. Barlow, One-dimensional stochastic differential equations with no strong solution, J. London Math. Soc., 26 (1982), 335-347.  doi: 10.1112/jlms/s2-26.2.335.  Google Scholar

[4]

A. Cherny and H.-J. Engelbert, Singular Stochastic Differential Equations, Lecture Notes in Mathematics, 1858. Springer-Verlag, Berlin, 2005. ⅷ+128 pp. doi: 10.1007/b104187.  Google Scholar

[5]

C. Costantini and T. Kurtz, Viscosity methods giving uniqueness for martingale problems, Electron. J. Probab., 20 (2015), 27 pp. doi: 10.1214/EJP.v20-3624.  Google Scholar

[6]

M. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[7]

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

[8]

F. Delarue and G. Guatteri, Weak existence and uniqueness for forward-backward SDEs, Stochastic Process. Appl., 116 (2006), 1712-1742.  doi: 10.1016/j.spa.2006.05.002.  Google Scholar

[9]

I. EkrenN. Touzi and J. Zhang, Viscosity solutions of fully nonlinear parabolic path dependent PDEs: part Ⅰ, Ann. Probab., 44 (2016), 1212-1253.  doi: 10.1214/14-AOP999.  Google Scholar

[10]

I. EkrenN. Touzi and J. Zhang, Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part Ⅱ, Ann. Probab., 44 (2016), 2507-2553.  doi: 10.1214/15-AOP1027.  Google Scholar

[11]

N. El Karoui and S. J. Huang, A general result of existence and uniqueness of backward stochastic differential equations, Backward Stochastic Differential Equations, 364 (1997), 27-36, N. El Karoui and L. Mazliak, eds., Longman, Harlow.  Google Scholar

[12]

N. Halidias and P. E. Kloeden, A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient, Journal of Applied Mathematics and Stochastic Analysis, 2006 (2006), Article ID 73257, 6 pages. doi: 10.1155/JAMSA/2006/73257.  Google Scholar

[13]

S. Hamadene and J.-P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems Control Lett., 24 (1995), 259-263.  doi: 10.1016/0167-6911(94)00011-J.  Google Scholar

[14]

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

[15]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Mathematical Society, 1996. doi: 10.1090/gsm/012.  Google Scholar

[16]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R. I. 1968.  Google Scholar

[17]

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

[18]

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.  Google Scholar

[19]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, 1702. Springer-Verlag, Berlin, 1999. xiv+270 pp.  Google Scholar

[20]

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.  Google Scholar

[21]

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

[22]

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001.  Google Scholar

[23]

S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37 (1999), 825-843.  doi: 10.1137/S0363012996313549.  Google Scholar

[24]

T. Pham and J. Zhang, Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation, SIAM J. Control Optim., 52 (2014), 2090-2121.  doi: 10.1137/120894907.  Google Scholar

[25]

P. Protter, Stochastic Integration and Differential Equations, Second edition. Version 2. 1. Corrected third printing. Stochastic Modeling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. xiv+419 pp. doi: 10.1007/978-3-662-10061-5.  Google Scholar

[26]

H., M. Soner and N. Touzi, Stochastic target problems, dynamic programming and viscosity solutions, SIAM Journal on Control and Optimization, 41 (2002), 404-424.  doi: 10.1137/S0363012900378863.  Google Scholar

[27]

H. M. SonerN. Touzi and J. Zhang, Wellposedness of second order backward SDEs, Probab. Theory Related Fields, 153 (2012), 149-190.  doi: 10.1007/s00440-011-0342-y.  Google Scholar

[28]

D. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2006. xii+338 pp  Google Scholar

[29]

B. Tsirelson, An example of a stochastic differential equation having no strong solution, Theory of Probability and Its Applications, 20 (1975), 416-418.  doi: 10.1137/1120049.  Google Scholar

[30]

J. Yong, Finding adapted solutions of forward-backward stochastic differential equations: method of continuation, Probab. Theory Related Fields, 107 (1997), 537-572.  doi: 10.1007/s004400050098.  Google Scholar

[31]

J. Yong and X. Y. Zhou, Stochastic controls. Hamiltonian systems and HJB equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. xxii+438 pp. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[32]

J. Zhang, The wellposedness of FBSDEs, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 927-940 (electronic). doi: 10.3934/dcdsb.2006.6.927.  Google Scholar

[33]

J. Zhang, Backward Stochastic Differential Equations - from Linear to Fully Nonlinear Theory, Springer, New York, 2017. doi: 10.1007/978-1-4939-7256-2.  Google Scholar

[34]

W. A. Zheng, Tightness results for laws of diffusion processes application to stochastic mechanics, Ann. Inst. H. Poincare Probab. Statist., 21 (1985), 103-124.   Google Scholar

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