基于相互作用粒子法的非线性FBSDEs的微扰展开技术
Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method
摘要：
在本文中，我们提出了一个有效的蒙特卡罗实现的非线性FBSED作为一个相互作用的粒子系统的思想启发的分支扩散方法。这将是特别有用的，调查大型和复杂的系统，因此，它是一个很好的补充，我们以前的工作提出了一个通用的非线性FBSDES的分析扰动程序。出现了多个种类的粒子，其中第一个遵循原来的底层状态的扩散，和其他的Malliavin衍生物与分级结构。分支点的数量由扰动的阶数限制，这有望使该方案的数值强度降低。所提出的方法可以应用于半线性问题，如美国和丹麦的选择，信用价值调整（CVA），甚至完全非线性问题，如最优投资组合问题，在不完全和/或约束市场，从大投资者的反馈，以及各种风险措施的分析。
英文摘要：
In this paper, we propose an efficient Monte Carlo implementation of non-linear FBSDEs as a system of interacting particles inspired by the ideas of branching diffusion method. It will be particularly useful to investigate large and complex systems, and hence it is a good complement of our previous work presenting an analytical perturbation procedure for generic non-linear FBSDEs. There appear multiple species of particles, where the first one follows the diffusion of the original underlying state, and the others the Malliavin derivatives with a grading structure. The number of branching points are capped by the order of perturbation, which is expected to make the scheme less numerically intensive. The proposed method can be applied to semi-linear problems, such as American and Bermudan options, Credit Value Adjustment (CVA), and even fully non-linear issues, such as the optimal portfolio problems in incomplete and/or constrained markets, feedbacks from large investors, and also the analysis of various risk measures.
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