Abstract
In this paper, a new numerical scheme for a class of coupled forward-backward 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 e=n −2α , where n is the number of initial particles and ${\alpha }<\frac {1}{2}$ is a fixed parameter. The convergence of the scheme and its rate of convergence are obtained.
Highlights
Since the work of Pardoux and Peng (1990), forward-backward stochastic differential equations (FBSDEs) have been extensively studied and have found important applications in many fields, including finance, risk measure, stochastic control and so on (cf. Cvitanicand Ma (1996); El Karoui et al (1997); Ma and Yong (1999); Xiong and Zhou (2007), and Yong and Zhou (1999))
The location and weight of each particle are governed by stochastic differential equations derived from the FBSDE system
A branching particle system is established to define the approximate solution of the FBSDE
Summary
Since the work of Pardoux and Peng (1990), forward-backward stochastic differential equations (FBSDEs) have been extensively studied and have found important applications in many fields, including finance, risk measure, stochastic control and so on (cf. Cvitanicand Ma (1996); El Karoui et al (1997); Ma and Yong (1999); Xiong and Zhou (2007), and Yong and Zhou (1999)). 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. A branching particle system is established to define the approximate solution of the FBSDE
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