Validating numerical solutions of high-dimensional backward SDEs arising from finance

验证金融领域高维后向 SDE 的数值解

• 批准号: 79152879
• 负责人: Professor Dr. Christian Bender
• 金额: --
• 依托单位: Fachrichtung Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2011-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/79152879?language=en
• 关键词: Validating; numerical; solutions; dimensional; backward

Backward stochastic differential equations (BSDEs) are a powerful tool to solve problems arising in mathematical finance, e.g. in the pricing of financial derivatives, the hedging of financial risks, and optimal investment problems. Moreover, they yield stochastic representation formulas for semi-linear parabolic Cauchy problems. Therefore the numerical solvability of BSDEs is a problem of high practical relevance, and it is particularly challenging, if a BSDE depends on a high-dimensional system of random sources. In recent years several Monte-Carlo-algorithms for BSDEs based on stochastic meshes or on quantization techniques have been developed. A serious drawback of these algorithms is that they produce point estimators for the solution only, whose quality is not validated. The aim of this project is to add upper (resp. lower) biased terms to these algorithms, which theoretically vanish in the limit. In the practically relevant pre-limit situations the difference between the corresponding ‘upper’ and ‘lower’ solutions may serve as indicator of the success of the numerical procedure. In particular, the absolute size of the biased terms can be monitored in the single discretization steps, which allows for the development of adaptive algorithms that apply more expensive estimators in critical steps. Apart from an error analysis of the additional biased terms for generic estimators of conditional expectations, a more detailed one for least-squares Monte-Carlo estimators is planned.

倒向随机微分方程（BSDEs）是解决数学金融中出现的问题的有力工具，例如金融衍生品定价、金融风险对冲和最优投资问题。并给出了半线性抛物型柯西问题的随机表示公式。因此，BSDE的数值可解性是一个具有高度实际意义的问题，如果BSDE依赖于随机源的高维系统，则特别具有挑战性。近年来，人们开发了几种基于随机网格或量化技术的BSDEs蒙特卡罗算法。这些算法的一个严重缺点是，它们只产生解的点估计，其质量没有得到验证。这个项目的目的是增加上层建筑的质量。这些算法的低偏项，理论上在极限中消失。在实际相关的前极限情况下，相应的“上”解和“下”解之间的差异可以作为数值过程成功的指标。特别是，偏项的绝对大小可以在单个离散化步骤中监测，这允许自适应算法的发展，在关键步骤中应用更昂贵的估计器。除了对条件期望的一般估计器的附加偏项进行误差分析外，还计划对最小二乘蒙特卡罗估计器进行更详细的分析。