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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 
References:
[1] 
Bally, V:Approximation scheme for solutions of BSDE, Backward stochastic differential equations, (Paris, 19951996), Pitman Res. Notes Math. Ser., vol. 364, pp. 177191. Longman, Harlow (1997), 
[2] 
Bouchard, B, Touzi, N:Discretetime approximation and MonteCarlo simulation of backward stochastic differential equations. Stoch. Process. Appl 111, 175206 (2004), 
[3] 
Briand, P, Delyon, B, Mémin, J:Donskertype theorem for BSDEs. Electron. Comm. Probab 6, 114(2001), 
[4] 
Chevance, D:Numerical methods for backward stochastic differential equations, Numerical methods in finance, Publ. Newton Inst., pp. 232244. 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. 120, Providence, RI (2002), 
[6] 
Crisan, D, Lyons, T:Nonlinear filtering and measurevalued processes. Probab. Theory Related Fields 109, 217244 (1997), 
[7] 
Crisan, D, Xiong, J:Numerical solution for a class of SPDEs over bounded domains. Stochastics 86(3), 450472 (2014), 
[8] 
Cvitanić, J, Ma, J:Hedging options for a large investor and forwardbackward SDE's. Ann. Appl. Probab 6, 370398 (1996), 
[9] 
Cvitanić, J, Zhang, J:The steepest descent method for forwardbackward SDEs. Electron. J. Probab 10, 14681495 (2005), 
[10] 
Del Moral, P:Nonlinear filtering:interacting particle resolution. Markov Process. Related Fields 2 4, 555581 (1996), 
[11] 
Delarue, F, Menozzi, S:A forwardbackward stochastic algorithm for quasilinear PDEs. Ann. Appl.Probab 16, 140184 (2006), 
[12] 
Douglas, J Jr., Ma, J, Protter, P:Numerical methods for forwardbackward stochastic differential equations. Ann. Appl. Probab 6, 940968 (1996), 
[13] 
El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 171 (1997), 
[14] 
Föllmer, H, Schied, A:Convex measures of risk and trading constraints. Finance Stoch. 6 (2002), 429447(1999). SpringerVerlag, New York Friedman, A:Stochastic Differential Equations and Applications. Vol. 1., Probability and Mathematical Statistics, vol. 28. Academic Press, New YorkLondon (1975), 
[15] 
HenryLabordère, P, Tan, X, Touzi, N:A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process. Appl 124(2), 11121140 (2014), 
[16] 
Kurtz, T, Xiong, J:Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl 83, 103126 (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. 233258. Birkhuser Boston, Boston, MA (2001), 
[18] 
Liu, H, Xiong, J:A branching particle system approximation for nonlinear stochastic filtering. Sci. China Math 56, 15211541 (2013), 
[19] 
Ma, J, Protter, P, San Martin, J, Torres, S:Numerical method for backward stochastic differential equations. Ann. Appl. Probab 12, 302316 (2002), 
[20] 
Ma, J, Protter, P, Yong, J:Solving forwardbackward stochastic differential equations explicitlya four step scheme. Probab. Theory Related Fields 98 3, 339359 (1994), 
[21] 
Ma, J, Shen, J, Zhao, Y:On numerical approximations of forwardbackward stochastic differential equations. SIAM J. Numer. Anal 46, 26362661 (2008), 
[22] 
Ma, J, Yong, J:Forwardbackward stochastic differential equations and their applications. SpringerVerlag, Berlin (1999), 
[23] 
Ma, J, Zhang, J:Representations and regularities for solutions to BSDEs with reflections. Stochast.Process. Appl 115, 539569 (2005), 
[24] 
Milstein, GN, Tretyakov, MV:Numerical algorithms for forwardbackward stochastic differential equations. SIAM J. Sci. Comput 28, 561582 (2006), 
[25] 
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett 14, 5561 (1990), 
[26] 
Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians. Volume I, pp. 393432. Hindustan Book Agency, New Delhi (2010), 
[27] 
Rosazza Gianin, E:Risk measures via gexpectations. Insurance Math. Econom 39, 1934 (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:Meanvariance portfolio selection under partial information. SIAM J. Control Optim 46, 156175 (2007), 
[30] 
Yong, J, Zhou, X:Stochastic Controls. Hamiltonian Systems and HJB Equations. SpringerVerlag, New York (1999), 
[31] 
Zhang, J:A numerical scheme for BSDEs. Ann. Appl. Probab 14, 459488 (2004), 
show all references
References:
[1] 
Bally, V:Approximation scheme for solutions of BSDE, Backward stochastic differential equations, (Paris, 19951996), Pitman Res. Notes Math. Ser., vol. 364, pp. 177191. Longman, Harlow (1997), 
[2] 
Bouchard, B, Touzi, N:Discretetime approximation and MonteCarlo simulation of backward stochastic differential equations. Stoch. Process. Appl 111, 175206 (2004), 
[3] 
Briand, P, Delyon, B, Mémin, J:Donskertype theorem for BSDEs. Electron. Comm. Probab 6, 114(2001), 
[4] 
Chevance, D:Numerical methods for backward stochastic differential equations, Numerical methods in finance, Publ. Newton Inst., pp. 232244. 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. 120, Providence, RI (2002), 
[6] 
Crisan, D, Lyons, T:Nonlinear filtering and measurevalued processes. Probab. Theory Related Fields 109, 217244 (1997), 
[7] 
Crisan, D, Xiong, J:Numerical solution for a class of SPDEs over bounded domains. Stochastics 86(3), 450472 (2014), 
[8] 
Cvitanić, J, Ma, J:Hedging options for a large investor and forwardbackward SDE's. Ann. Appl. Probab 6, 370398 (1996), 
[9] 
Cvitanić, J, Zhang, J:The steepest descent method for forwardbackward SDEs. Electron. J. Probab 10, 14681495 (2005), 
[10] 
Del Moral, P:Nonlinear filtering:interacting particle resolution. Markov Process. Related Fields 2 4, 555581 (1996), 
[11] 
Delarue, F, Menozzi, S:A forwardbackward stochastic algorithm for quasilinear PDEs. Ann. Appl.Probab 16, 140184 (2006), 
[12] 
Douglas, J Jr., Ma, J, Protter, P:Numerical methods for forwardbackward stochastic differential equations. Ann. Appl. Probab 6, 940968 (1996), 
[13] 
El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 171 (1997), 
[14] 
Föllmer, H, Schied, A:Convex measures of risk and trading constraints. Finance Stoch. 6 (2002), 429447(1999). SpringerVerlag, New York Friedman, A:Stochastic Differential Equations and Applications. Vol. 1., Probability and Mathematical Statistics, vol. 28. Academic Press, New YorkLondon (1975), 
[15] 
HenryLabordère, P, Tan, X, Touzi, N:A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process. Appl 124(2), 11121140 (2014), 
[16] 
Kurtz, T, Xiong, J:Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl 83, 103126 (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. 233258. Birkhuser Boston, Boston, MA (2001), 
[18] 
Liu, H, Xiong, J:A branching particle system approximation for nonlinear stochastic filtering. Sci. China Math 56, 15211541 (2013), 
[19] 
Ma, J, Protter, P, San Martin, J, Torres, S:Numerical method for backward stochastic differential equations. Ann. Appl. Probab 12, 302316 (2002), 
[20] 
Ma, J, Protter, P, Yong, J:Solving forwardbackward stochastic differential equations explicitlya four step scheme. Probab. Theory Related Fields 98 3, 339359 (1994), 
[21] 
Ma, J, Shen, J, Zhao, Y:On numerical approximations of forwardbackward stochastic differential equations. SIAM J. Numer. Anal 46, 26362661 (2008), 
[22] 
Ma, J, Yong, J:Forwardbackward stochastic differential equations and their applications. SpringerVerlag, Berlin (1999), 
[23] 
Ma, J, Zhang, J:Representations and regularities for solutions to BSDEs with reflections. Stochast.Process. Appl 115, 539569 (2005), 
[24] 
Milstein, GN, Tretyakov, MV:Numerical algorithms for forwardbackward stochastic differential equations. SIAM J. Sci. Comput 28, 561582 (2006), 
[25] 
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett 14, 5561 (1990), 
[26] 
Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians. Volume I, pp. 393432. Hindustan Book Agency, New Delhi (2010), 
[27] 
Rosazza Gianin, E:Risk measures via gexpectations. Insurance Math. Econom 39, 1934 (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:Meanvariance portfolio selection under partial information. SIAM J. Control Optim 46, 156175 (2007), 
[30] 
Yong, J, Zhou, X:Stochastic Controls. Hamiltonian Systems and HJB Equations. SpringerVerlag, New York (1999), 
[31] 
Zhang, J:A numerical scheme for BSDEs. Ann. Appl. Probab 14, 459488 (2004), 
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