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摘要翻译：
在本工作中，我们对一般非线性FBSDEs提出了一个简单的解析逼近格式。通过将感兴趣的系统看作非线性发生器和反馈项扰动下的线性解耦FBSDE,我们证明了它可以实现任意高阶的递归逼近，其中每个阶所需的计算量相当于标准欧式未定权益的计算量。我们还将摄动方法应用到PDE框架中，遵循所谓的四步格式。该方法将原来的非线性偏微分方程转化为一系列标准的抛物线线性偏微分方程。由于这两种方法的等价性，将渐近展开式应用于相应的概率模型，也可以得到非线性偏微分方程的近似解析解。给出了两个简单的例子来说明微扰是如何工作的，并表明了它相对于已知数值技术的精度。本文所提出的方法可用于各种至今未能进行分析处理的重要问题。
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英文标题：
《Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme》
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作者：
Masaaki Fujii, Akihiko Takahashi
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最新提交年份：
2012
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分类信息：
一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Quantitative Finance 数量金融学
二级分类：Pricing of Securities 证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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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.
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PDF链接：
https://arxiv.org/pdf/1106.0123
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