非线性倒向随机微分方程的摄动解析逼近
Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme
摘要：
在这项工作中，我们提出了一个简单的解析近似方案的通用非线性FBSDES。通过将感兴趣的系统视为具有非线性生成元和反馈项的线性解耦FBB-Bundle扰动，我们证明了可以对任意高阶进行递归逼近，其中每一阶所需的计算与标准欧式未定权益的计算等价。我们还采用了微扰方法的PDE框架下，所谓的四步计划。该方法可以将原来的非线性偏微分方程转化为一系列标准的抛物线性偏微分方程。由于这两种方法的等价性，也可以通过将渐近展开应用于相应的概率模型来获得非线性偏微分方程的近似解析解。提供了两个简单的例子来演示如何摄动工程，并显示其相对于已知的数值技术的精度。本文所提出的方法对于迄今为止还没有得到解析处理的各种重要问题都是有用的。
英文摘要：
In this work, we have presented a simple analytical approximation scheme for generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown that it is possible to carry out a recursive approximation to an arbitrarily higher order, where the required calculations in each order are equivalent to those for standard European contingent claims. We have also applied the perturbative method to the PDE framework following the so-called Four Step Scheme. The method is found to render the original non-linear PDE into a series of standard parabolic linear PDEs. Due to the equivalence of the two approaches, it is also possible to derive approximate analytic solution for the non-linear PDE by applying the asymptotic expansion to the corresponding probabilistic model. Two simple examples are provided to demonstrate how the perturbation works and show its accuracy relative to known numerical techniques. The method presented in this paper may be useful for various important problems which have eluded analytical treatment so far.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
