Abstract
In this work, we are concerned with multistep schemes for solving forward backward stochastic differential equations with jumps. The proposed multistep schemes admit many advantages. First of all, motivated by the local property of jump diffusion processes, the Euler method is used to solve the associated forward stochastic differential equation with jump, which reduce dramatically the entire computational complexity, however, the quantities of interests in the backward stochastic differential equations (with jump) are still of high order rate of convergence. Secondly, in each time step, only one jump is involved in the computational procedure, which again reduces dramatically the computational complexity. Finally, the method applies easily to partial-integral differential equations (and some nonlocal PDE models), by using the generalized Feynman–Kac formula. Several numerical experiments are presented to demonstrate the effectiveness of the proposed multistep schemes.
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References
Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60, 57–83 (1997)
Becherer, D.: Bounded solution to BSDE’s with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 2027–2054 (2006)
Bender, C., Zhang, J.: Time discretization and markovian iteration for coupled FBSDEs. Ann. Appl. Probab. 18, 143–177 (2008)
Bouchard, B., Elie, R.: Discrete time approximation of decoupled forward–backward SDE with jumps. Stoch. Process. Appl. 118, 53–75 (2008)
Cont, R., Tankov, R.: Financial modelling with jumpe processes. Chapman and Hall/CRC Press, London (2004)
Delong, L.: Backward stochastic differential equations with jumps and their actuarial and financial applications. BSDEs with jumps. Springer, New York (2013)
Du, Q., Gunzburger, M., Lebourq, R., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 4, 667–696 (2012)
Fu, Y., Yang, J., Zhao, W.: Prediction-correction scheme for decoupled forward backward stochastic differential equations with jumps. submitted (2015)
Fu, Y., Zhao, W.: An explicit second-order numerical scheme to solve decoupled forward backward stochastic equations. East Asian J. Appl. Math. 4, 368–385 (2014)
Kong, T., Zhao, W., Zhou, T.: High order numerical schemes for second order FBSDEs with applications to stochastic optimal control. arXiv:1502.03206 (2015)
Lemor, J.P., Gobet, E., Warin, X.: A regression-based monte carlo method for backward stochastic differential equations. Ann. Appl. Probab. 15(3), 2172–2202 (2005)
Li, J., Peng, S.: Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of hamilton–jacobi–bellman equations. Nonlinear Anal. Theory Methods Appl. 70, 1776–1796 (2009)
Milstein, G.N., Tretyakov, M.V.: Numerical algorithms for forward–backward stochastic differential equations. SIAM J. Sci. Comput. 28, 561–582 (2006)
Morlais, M.: A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem. Stoch. Process. Appl. 120, 1966–1996 (2010)
Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 56–61 (1990)
Ruijter, M., Oosterlee, C.W.: A fourier cosine method for an efficient computation of solutions to BSDEs. SIAM J. Sci. Comput. 37(2), A859–A889 (2015)
Tang, S., Li, X.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Sci. Comput. 32, 1447–1475 (1994)
Tang, T., Zhao, W., Zhou, T.: Highly accurate numerical schemes for forward backward stochastic differential equations based on deferred correction approach. submitted (2015)
Yang, J., Zhao, W.: Convergence of recent multistep schemes for a forward–backward stochastic differential equation. East Asian J. Appl. Math. 5(4), 387–404 (2015)
Zhang, G., Zhao, W., Webster, C., Gunzburger, M.: Numerical solution of backward stochastic differential equations with jumps for a class of nonlocal diffusion problems. arXiv preprint arXiv:1503.00213 (2015)
Zhao, W., Chen, L., Peng, S.: A new kind of accurate numerical method for backward stochastic differential equations. SIAM J. Sci. Comput. 28, 1563–1581 (2006)
Zhao, W., Fu, Y., Zhou, T.: New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations. SIAM J. Sci. Comput. 36, A1731–A1751 (2014)
Zhao, W., Zhang, G., Ju, L.: A stable multistep scheme for solving backward stochastic differential equations. SIAM J. Numer. Anal. 48, 1369–1394 (2010)
Zhao, W., Zhang, W., Zhang, G.: Numerical schemes for forward–backward stochastic differential equaiton to nonlinear partial integro-differential equations. submitted (2015)
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This work is partially supported by the National Natural Science Foundations of China under Grant Numbers 11571206, 91530118 and 11571351.
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Fu, Y., Zhao, W. & Zhou, T. Multistep Schemes for Forward Backward Stochastic Differential Equations with Jumps. J Sci Comput 69, 651–672 (2016). https://doi.org/10.1007/s10915-016-0212-y
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DOI: https://doi.org/10.1007/s10915-016-0212-y