January  2016, 1: 9 doi: 10.1186/s41546-016-0007-y

A branching particle system approximation for a class of FBSDEs

1 School of Mathematics, Shandong University, Jinan 250100, People's Republic of China;

2 Department of Mathematics, Hebei Normal University, Shijiazhuang 050024, People's Republic of China;

3 Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macao, Special Administrative Region of China

Received  April 06, 2016 Revised  August 10, 2016

Fund Project: support by National Science Foundation of China NSFC 11501164. Xiong acknowledges research support by Macao Science and Technology Fund FDCT 076/2012/A3 and MultiYear Research Grants of the University of Macau No. MYRG2014-00015-FST and MYRG2014-00034-FST.

In this paper, a new numerical scheme for a class of coupled forwardbackward stochastic differential equations (FBSDEs) is proposed by using branching particle systems in a random environment. First, by the four step scheme, we introduce a partial differential Eq. (PDE) used to represent the solution of the FBSDE system. Then, infinite and finite particle systems are constructed to obtain the approximate solution of the PDE. The location and weight of each particle are governed by stochastic differential equations derived from the FBSDE system. Finally, a branching particle system is established to define the approximate solution of the FBSDE system. The branching mechanism of each particle depends on the path of the particle itself during its short lifetime =n-2α, where n is the number of initial particles and α < $\frac{1}{2}$ is a fixed parameter. The convergence of the scheme and its rate of convergence are obtained.
Citation: Dejian Chang, Huili Liu, Jie Xiong. A branching particle system approximation for a class of FBSDEs. Probability, Uncertainty and Quantitative Risk, doi: 10.1186/s41546-016-0007-y
References:
[1]

Bally, V:Approximation scheme for solutions of BSDE, Backward stochastic differential equations, (Paris, 1995-1996), Pitman Res. Notes Math. Ser., vol. 364, pp. 177-191. Longman, Harlow (1997),

[2]

Bouchard, B, Touzi, N:Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stoch. Process. Appl 111, 175-206 (2004),

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Briand, P, Delyon, B, Mémin, J:Donsker-type theorem for BSDEs. Electron. Comm. Probab 6, 1-14(2001),

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Chevance, D:Numerical methods for backward stochastic differential equations, Numerical methods in finance, Publ. Newton Inst., pp. 232-244. Cambridge Univ. Press, Cambridge (1997),

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Crisan, D:Numerical methods for solving the stochastic filtering problem, Numerical methods and stochastics (Toronto, ON, 1999), Fields Inst. Commun., 34, Amer. Math. Soc., pp. 1-20, Providence, RI (2002),

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Crisan, D, Lyons, T:Nonlinear filtering and measure-valued processes. Probab. Theory Related Fields 109, 217-244 (1997),

[7]

Crisan, D, Xiong, J:Numerical solution for a class of SPDEs over bounded domains. Stochastics 86(3), 450-472 (2014),

[8]

Cvitanić, J, Ma, J:Hedging options for a large investor and forward-backward SDE's. Ann. Appl. Probab 6, 370-398 (1996),

[9]

Cvitanić, J, Zhang, J:The steepest descent method for forward-backward SDEs. Electron. J. Probab 10, 1468-1495 (2005),

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Del Moral, P:Non-linear filtering:interacting particle resolution. Markov Process. Related Fields 2 4, 555-581 (1996),

[11]

Delarue, F, Menozzi, S:A forward-backward stochastic algorithm for quasi-linear PDEs. Ann. Appl.Probab 16, 140-184 (2006),

[12]

Douglas, J Jr., Ma, J, Protter, P:Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab 6, 940-968 (1996),

[13]

El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997),

[14]

Föllmer, H, Schied, A:Convex measures of risk and trading constraints. Finance Stoch. 6 (2002), 429-447(1999). Springer-Verlag, New York Friedman, A:Stochastic Differential Equations and Applications. Vol. 1., Probability and Mathematical Statistics, vol. 28. Academic Press, New York-London (1975),

[15]

Henry-Labordère, P, Tan, X, Touzi, N:A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process. Appl 124(2), 1112-1140 (2014),

[16]

Kurtz, T, Xiong, J:Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl 83, 103-126 (1999),

[17]

Kurtz, T, Xiong, J:Numerical solutions for a class of SPDEs with application to filtering, Stochastics in finite and infinite dimensions:in honor of Gopinath Kallianpur. Trends Math., pp. 233-258. Birkhuser Boston, Boston, MA (2001),

[18]

Liu, H, Xiong, J:A branching particle system approximation for nonlinear stochastic filtering. Sci. China Math 56, 1521-1541 (2013),

[19]

Ma, J, Protter, P, San Martin, J, Torres, S:Numerical method for backward stochastic differential equations. Ann. Appl. Probab 12, 302-316 (2002),

[20]

Ma, J, Protter, P, Yong, J:Solving forward-backward stochastic differential equations explicitly-a four step scheme. Probab. Theory Related Fields 98 3, 339-359 (1994),

[21]

Ma, J, Shen, J, Zhao, Y:On numerical approximations of forward-backward stochastic differential equations. SIAM J. Numer. Anal 46, 2636-2661 (2008),

[22]

Ma, J, Yong, J:Forward-backward stochastic differential equations and their applications. Springer-Verlag, Berlin (1999),

[23]

Ma, J, Zhang, J:Representations and regularities for solutions to BSDEs with reflections. Stochast.Process. Appl 115, 539-569 (2005),

[24]

Milstein, GN, Tretyakov, MV:Numerical algorithms for forward-backward stochastic differential equations. SIAM J. Sci. Comput 28, 561-582 (2006),

[25]

Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett 14, 55-61 (1990),

[26]

Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians. Volume I, pp. 393-432. Hindustan Book Agency, New Delhi (2010),

[27]

Rosazza Gianin, E:Risk measures via g-expectations. Insurance Math. Econom 39, 19-34 (2006),

[28]

Xiong, J:An Introduction to Stochastic Filtering Theory. Oxford Graduate Texts in Mathematics, 18.Oxford University Press, Oxford (2008),

[29]

Xiong, J, Zhou, X:Mean-variance portfolio selection under partial information. SIAM J. Control Optim 46, 156-175 (2007),

[30]

Yong, J, Zhou, X:Stochastic Controls. Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999),

[31]

Zhang, J:A numerical scheme for BSDEs. Ann. Appl. Probab 14, 459-488 (2004),

show all references

References:
[1]

Bally, V:Approximation scheme for solutions of BSDE, Backward stochastic differential equations, (Paris, 1995-1996), Pitman Res. Notes Math. Ser., vol. 364, pp. 177-191. Longman, Harlow (1997),

[2]

Bouchard, B, Touzi, N:Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stoch. Process. Appl 111, 175-206 (2004),

[3]

Briand, P, Delyon, B, Mémin, J:Donsker-type theorem for BSDEs. Electron. Comm. Probab 6, 1-14(2001),

[4]

Chevance, D:Numerical methods for backward stochastic differential equations, Numerical methods in finance, Publ. Newton Inst., pp. 232-244. Cambridge Univ. Press, Cambridge (1997),

[5]

Crisan, D:Numerical methods for solving the stochastic filtering problem, Numerical methods and stochastics (Toronto, ON, 1999), Fields Inst. Commun., 34, Amer. Math. Soc., pp. 1-20, Providence, RI (2002),

[6]

Crisan, D, Lyons, T:Nonlinear filtering and measure-valued processes. Probab. Theory Related Fields 109, 217-244 (1997),

[7]

Crisan, D, Xiong, J:Numerical solution for a class of SPDEs over bounded domains. Stochastics 86(3), 450-472 (2014),

[8]

Cvitanić, J, Ma, J:Hedging options for a large investor and forward-backward SDE's. Ann. Appl. Probab 6, 370-398 (1996),

[9]

Cvitanić, J, Zhang, J:The steepest descent method for forward-backward SDEs. Electron. J. Probab 10, 1468-1495 (2005),

[10]

Del Moral, P:Non-linear filtering:interacting particle resolution. Markov Process. Related Fields 2 4, 555-581 (1996),

[11]

Delarue, F, Menozzi, S:A forward-backward stochastic algorithm for quasi-linear PDEs. Ann. Appl.Probab 16, 140-184 (2006),

[12]

Douglas, J Jr., Ma, J, Protter, P:Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab 6, 940-968 (1996),

[13]

El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997),

[14]

Föllmer, H, Schied, A:Convex measures of risk and trading constraints. Finance Stoch. 6 (2002), 429-447(1999). Springer-Verlag, New York Friedman, A:Stochastic Differential Equations and Applications. Vol. 1., Probability and Mathematical Statistics, vol. 28. Academic Press, New York-London (1975),

[15]

Henry-Labordère, P, Tan, X, Touzi, N:A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process. Appl 124(2), 1112-1140 (2014),

[16]

Kurtz, T, Xiong, J:Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl 83, 103-126 (1999),

[17]

Kurtz, T, Xiong, J:Numerical solutions for a class of SPDEs with application to filtering, Stochastics in finite and infinite dimensions:in honor of Gopinath Kallianpur. Trends Math., pp. 233-258. Birkhuser Boston, Boston, MA (2001),

[18]

Liu, H, Xiong, J:A branching particle system approximation for nonlinear stochastic filtering. Sci. China Math 56, 1521-1541 (2013),

[19]

Ma, J, Protter, P, San Martin, J, Torres, S:Numerical method for backward stochastic differential equations. Ann. Appl. Probab 12, 302-316 (2002),

[20]

Ma, J, Protter, P, Yong, J:Solving forward-backward stochastic differential equations explicitly-a four step scheme. Probab. Theory Related Fields 98 3, 339-359 (1994),

[21]

Ma, J, Shen, J, Zhao, Y:On numerical approximations of forward-backward stochastic differential equations. SIAM J. Numer. Anal 46, 2636-2661 (2008),

[22]

Ma, J, Yong, J:Forward-backward stochastic differential equations and their applications. Springer-Verlag, Berlin (1999),

[23]

Ma, J, Zhang, J:Representations and regularities for solutions to BSDEs with reflections. Stochast.Process. Appl 115, 539-569 (2005),

[24]

Milstein, GN, Tretyakov, MV:Numerical algorithms for forward-backward stochastic differential equations. SIAM J. Sci. Comput 28, 561-582 (2006),

[25]

Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett 14, 55-61 (1990),

[26]

Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians. Volume I, pp. 393-432. Hindustan Book Agency, New Delhi (2010),

[27]

Rosazza Gianin, E:Risk measures via g-expectations. Insurance Math. Econom 39, 19-34 (2006),

[28]

Xiong, J:An Introduction to Stochastic Filtering Theory. Oxford Graduate Texts in Mathematics, 18.Oxford University Press, Oxford (2008),

[29]

Xiong, J, Zhou, X:Mean-variance portfolio selection under partial information. SIAM J. Control Optim 46, 156-175 (2007),

[30]

Yong, J, Zhou, X:Stochastic Controls. Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999),

[31]

Zhang, J:A numerical scheme for BSDEs. Ann. Appl. Probab 14, 459-488 (2004),

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