# American Institute of Mathematical Sciences

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:
 [1] F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793. doi: 10.1214/aoap/1177005363. [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] 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. [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. [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. [6] M. Crandall, H. 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. [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. [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. [9] I. Ekren, N. 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. [10] I. Ekren, N. 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. [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. [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. [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. [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. [15] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Mathematical Society, 1996. doi: 10.1090/gsm/012. [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. [17] J. Ma, P. 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. [18] J. Ma, Z. Wu, D. 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. [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. [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. [21] J. Ma, J. 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. [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. [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. [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. [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. [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. [27] H. M. Soner, N. 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. [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 [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. [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. [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. [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. [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. [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.

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##### References:
 [1] F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793. doi: 10.1214/aoap/1177005363. [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] 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. [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. [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. [6] M. Crandall, H. 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. [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. [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. [9] I. Ekren, N. 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. [10] I. Ekren, N. 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. [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. [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. [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. [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. [15] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Mathematical Society, 1996. doi: 10.1090/gsm/012. [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. [17] J. Ma, P. 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. [18] J. Ma, Z. Wu, D. 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. [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. [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. [21] J. Ma, J. 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. [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. [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. [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. [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. [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. [27] H. M. Soner, N. 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. [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 [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. [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. [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. [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. [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. [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.
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