# 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
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