Back
 AM  Vol.11 No.4 , April 2020
Non-Markovian Forward-Backward Stochastic Differential Equations with Discontinuous Coefficients
Abstract: In this paper, the existence and uniqueness of a weak solution in the sense of [1] and [2] has been shown for a class of fully coupled forward-backward SDE (FBSDE) such that the forward drift coefficient is allowed to be discontinuous with respect to the backward component of the solution. The novelty of this paper lies on the fact that the FBSDE is non-Markovian, i.e., the coefficients of the FBSDEs are allowed to be random. This type of FBSDEs is inspired by the regime shift model, where the short term interest rate switches between regimes according to the rate level. As a consequence, the discontinuity of the system becomes inevitable, making it violate the usual assumptions of most existing results for FBSDEs. We show the weak well-posedness of the FBSDE by an approximation scheme, along with the decoupling strategy.
Cite this paper: Hong, Y. (2020) Non-Markovian Forward-Backward Stochastic Differential Equations with Discontinuous Coefficients. Applied Mathematics, 11, 328-343. doi: 10.4236/am.2020.114024.
References

[1]   Antonelli, F. and Ma, J. (2003) Weak Solutions of Forward-Backward SDE’s. Stochastic Analysis and Applications, 21, 493-514.
https://doi.org/10.1081/SAP-120020423

[2]   Ma, J., Zhang, J. and Zheng, Z. (2008) Weak Solutions for Backward Stochastic Differential Equations, a Martingale Approach. The Annals of Probability, 36, 2092-2125.
https://doi.org/10.1214/08-AOP0383

[3]   Chen, J., Ma, J. and Yin, H. (2018) Forward-Backward Stochastic Differential Equations with Discontinuous Coefficients. Stochastic Analysis and Applications, 36, 274-294.
https://doi.org/10.1080/07362994.2017.1399799

[4]   Bali, T. (2003) Modeling the Stochastic Behavior of Short-Term Interest Rates: Pricing Implications for Discount Bonds. Journal of Banking and Finance, 27, 201-228.
https://doi.org/10.1016/S0378-4266(01)00216-3

[5]   Driffill, J., Kenc, T. and Sola, M. (2003) An Empirical Examination of Term Structure Models with Regime Shifts. EFMA 2003 Helsinki Meetings.
https://doi.org/10.2139/ssrn.393481

[6]   Bansal, R. and Zhou, H. (2002) Term Structure of Interest Rates with Regime Shifts. Journal of Finance, 57, 1997-2043.
https://doi.org/10.1111/0022-1082.00487

[7]   Dai, Q., Singleton, K. and Yang, W. (2007) Regime Shifts in a Dynamic Term Structure Model of U.S. Treasury Bond Yields. Review of Financial Studies, 20, 1669-1706.
https://doi.org/10.1093/rfs/hhm021

[8]   Evans, M. (2003) Real Risk, Inflation Risk, and the Term Structure. The Economic Journal, 113, 345-389.
https://doi.org/10.1111/1468-0297.00130

[9]   Landen, C. (2000) Bond Pricing in a Hidden Markov Model of the Short Rate. Finance and Stochastics, 4, 371-389.
https://doi.org/10.1007/PL00013526

[10]   Duffie, D., Ma, J. and Yong, J. (1995) Black’s Console Rate Conjecture. The Annals of Applied Probability, 5, 356-382.
https://doi.org/10.1214/aoap/1177004768

[11]   Ma, J., Wu, Z., Zhang, D. and Zhang, J. (2015) On Well-Posedness of Forward-Backward SDES: A Unified Approach. The Annals of Applied Probability, 25, 2168-2214.
https://doi.org/10.1214/14-AAP1046

[12]   Ma, J., Yin, H. and Zhang, J. (2014) On Non-Markovian Forward-Backward SDEs and Backward Stochastic PDEs. Stochastic Processes and Their Applications, 122, 3980-4004.
https://doi.org/10.1016/j.spa.2012.08.002

[13]   Pardoux, E. (1979) Stochastic Partial Differential Equations and Filtering of Diffusion Processes. Stochastics, 3, 127-167.
https://doi.org/10.1080/17442507908833142

[14]   Krylov, N.V. (1980) Controlled Diffusion Processes. Springer-Verlag, Berlin.
https://doi.org/10.1007/978-1-4612-6051-6

[15]   Kim, D. and Krylov, N.V. (2007) Parabolic Equations with Measurable Coefficients. Potential Analysis, 26, 345-361.
https://doi.org/10.1007/s11118-007-9042-8

[16]   Kuo, H.-H. (2000) Introduction to Stochastic Integration. Springer, Berlin.

[17]   Brossard, J. (2003) Deux notions équivalentes d’unicité en loi pour les équations différentielles stochastiques. In: Séminaire de Probabilités XXXVII, Volume 1832, Springer, Berlin, 246-250.
https://doi.org/10.1007/978-3-540-40004-2_10

 
 
Top