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Multistep Schemes for Forward Backward Stochastic Differential Equations with Jumps

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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Correspondence to Tao Zhou.

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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

Keywords

  • Multistep scheme
  • Jump-diffusion process
  • Forward backward stochastic differential equation with jumps

Mathematics Subject Classification

  • 60H35
  • 65C20