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International Journal for Uncertainty Quantification
Fator do impacto: 3.259 FI de cinco anos: 2.547 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Imprimir: 2152-5080
ISSN On-line: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2019028300
pages 143-159

ADJOINT FORWARD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY JUMP DIFFUSION PROCESSES AND ITS APPLICATION TO NONLINEAR FILTERING PROBLEMS

Feng Bao
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
Yanzhao Cao
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849; School of Mathematics, Sun Yat Sun University, China
Hongmei Chi
Department of Computer and Information Sciences, Florida A&M University, Tallahassee, Florida 32306, USA

RESUMO

Forward backward stochastic differential equations (FBSDEs) were first introduced as a probabilistic interpretation for the Kolmogorov backward equation, and the solution of FBSDEs is equivalent to the solution of quasilinear partial differential equations. In this work, we introduce the adjoint relation between a generalized FBSDE system driven by jump diffusion processes and its time inverse adjoint FBSDE system under the probabilistic framework without translating them into their corresponding PDEs. The "exact solution" of a nonlinear filtering problem is derived as an application.

Referências

  1. Øksendal, B., Stochastic Differential Equations, 6th ed., Berlin: Springer, 2013.

  2. Pardoux, E. and Peng, S., Adapted Solution of a Backward Stochastic Differential Equation, Systems Control Lett., 14(1):55– 61, 1990.

  3. Pardoux, E. and Peng, P., Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Dquations, in Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991), Lect. Notes Control Inf. Sci., vol. 176, Berlin: Springer.

  4. Delong, Ł., Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications, European Actuarial Academy (EAA) Series, Berlin: Springer, 1998.

  5. Kavallaris, N. and Suzuki, T., Non-Local Partial Differential Equations for Engineering and Biology, vol. 31 of Mathematics for Industry (Tokyo), Cham, Switzerland: Springer International Publishing, 2017.

  6. Metzler, R. and Klafter, J., The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach, Phys. Rep., 339(1):77, 2000.

  7. Barles, G., Buckdahn, R., and Pardoux, E., Backward Stochastic Differential Equations and Integral-Partial Differential Equations, Stochastics: Int. J. Prob. Stochastic Proc., 60(1-2):57–83, 1997.

  8. Archibald, R., Bao, F., and Maksymovych, P., Backward SDE Filter for Jump Diffusion Processes and Its Applications in Material Sciences, arXiv preprint arXiv:1805.11038, 2018.

  9. Du, Q., Ju, L., Tian, L., and Zhou, K., A Posteriori Error Analysis of Finite Element Method for Linear Nonlocal Diffusion and Peridynamic Models, Math. Comput., 82(284):1889–1922, 2013.

  10. He, J., Duan, J., and Gao, H., A Nonlocal Fokker-Planck Equation for Non-Gaussian Stochastic Dynamical Systems, Appl. Math. Lett., 49:1–6, 2015.

  11. Djouadi, S.M., Maroulas, V., Pan, X., and Xiong, J., Consistency and Asymptotics of a Poisson Intensity Least-Squares Estimator for Partially Observed Jump-Diffusion Processes, Stat. Probab. Lett., 123:8–16, 2017.

  12. Maroulas, V., Pan, X., and Xiong, J., Statistical Inference for the Intensity in a Partially Observed Jump Diffusion, J. Math. Anal. Appl., 472(1):1–10, 2019.

  13. Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Mathematics Monograph Series, vol. 3, Beijing, China: Science Press Beijing, 2006.

  14. Sow, A., BSDE with Jumps and Non-Lipschitz Coefficients: Application to Large Deviations, Braz. J. Probab. Stat., 28(1):96– 108, 2014.

  15. Tang, T., Yuan, H., and Zhou, T., Hermite Spectral Collocation Methods for Fractional PDES in Unbounded Domain, Commun. Comput. Phys., 24:1143–1168, 2018.

  16. Zhao,W., Zhang,W., and Zhang, G., Second-Order Numerical Schemes for Decoupled Forward-Backward Stochastic Differential Equations with Jumps, J. Comput. Math., 35(2):213–244, 2017.

  17. Bao, F., Cao, Y., and Zhao, W., Numerical Solutions for Forward Backward Doubly Stochastic Differential Equations and Zakai Equations, Int. J. Uncertainty Quantif., 1(4):351–367, 2011.

  18. Bao, F., Cao, Y., and Zhao, W., A Backward Doubly Stochastic Differential Equation Approach for Nonlinear Filtering Problems, Commun. Comput. Phys., 23(5):1573–1601, 2018.

  19. Bao, F., Cao, Y., and Zhao, W., A First Order Semi-Discrete Algorithm for Backward Doubly Stochastic Differential Equations, Discrete Contin. Dyn. Syst. Ser. B, 20(5):1297–1313, 2015.

  20. Bao, F. and Maroulas, V., Adaptive Meshfree Backward SDE Filter, SIAM J. Sci. Comput., 39(6):2664–2683, 2017.

  21. Pardoux, E. and Protter, P., A Two-Sided Stochastic Integral and Its Calculus, Probab. Theory Relat. Fields, 76(1):15–49, 1987.

  22. Bao, F., Cao, Y., and Han, X., Forward Backward Doubly Stochastic Differential Equations and Optimal Filtering of Diffusion Processes, arXiv preprint arxiv:1509.06352, 2016.

  23. Sun, X. and Duan, J., Fokker-Planck Equations for Nonlinear Dynamical Systems Driven by Non-Gaussian L´evy Processes, J. Math. Phys., 53(7):072701, 2012.

  24. Bao, F., Cao, Y., and Han, X., An Implicit Algorithm of Solving Nonlinear Filtering Problems, Commun. Comput. Phys., 16:382–402, 2014.

  25. Anderson, B. and Moore, J., Optimal Filtering, Dover Books on Electrical Engineering, Mineola, NY: Dover Publications, 2005.

  26. Gordon, N., Salmond, D., and Smith, A., Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation, IEE Proc.- F, 140(2):107–113, 1993.

  27. Zhang, G., Zhao, W., Webster, C., and Gunzburger, M., Numerical Methods for a Class of Nonlocal Diffusion Problems with the Use of Backward SDEs, Comput. Math. Appl., 71(11):2479–2496, 2016.

  28. Fu, Y., Zhao, W., and Zhou, T., Multistep Schemes for Forward Backward Stochastic Differential Equations with Jumps, J. Sci. Comput., 69(2):651–672, 2016.


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