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Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs

  • Tao Kong (a1), Weidong Zhao (a1) and Tao Zhou (a2)

In this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

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*Corresponding author. Email addresses: (T. Kong), (W. Zhao), (T. Zhou)
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[1] C. Bender and J. Zhang , Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 18(2008), pp. 143177.

[5] D. Crisan and K. Manolarakis , Solving backward stochastic differential equations using the cubature method, SIAM J. Math. Finance, (3)2012, pp. 534571.

[6] F. Delarue and S. Menozzi , A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab., 16(2006), pp. 140184.

[7] F. Delarue , and S. Menozzi , An interpolated stochastic algorithm for quasi-linear pdes. Mathematics of Computation 77, 261 (2008), 125158.

[16] N. G. Milstein and M. V. Tretyakov , Discretization of Forward-Backward Stochastic Differential Equations And Related Quasi-linear Parabolic Equations, SIAM J. Numer. Anal, 27(2007), 2434.

[18] E. Pardoux and S. Tang , Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Relat. Fields, 114 (1999), pp. 123150.

[21] H. M. Soner , N. Touzi , and J. Zhang , Wellposedness of second order backward SDEs, Probab. Theory Relat. Fields, Vol. 153 (2012), pp:149190.

[25] W. Zhao , L. Chen and S. Peng , A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), pp. 15631581.

[26] W.Zhao , Y. Fu , and T. Zhou , New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36 (4), pp. A17311751, 2014.

[27] W. Zhao , G. Zhang and L. Ju , A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), pp. 13691394.

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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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