# American Institute of Mathematical Sciences

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), [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),

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),
 [1] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [2] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020264 [3] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [4] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [5] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [6] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, doi: 10.3934/krm.2020050 [7] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [8] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, doi: 10.3934/era.2020073 [9] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [10] Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020016 [11] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [12] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [13] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [14] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020273 [15] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, doi: 10.3934/ipi.2020056 [16] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020241 [17] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [18] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [19] Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, doi: 10.3934/era.2020080 [20] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

Impact Factor: