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

倒向随机微分方程（BSDEs）是解决金融数学问题的有力工具，例如金融衍生品定价、金融风险对冲和最优投资问题。此外，他们产生半线性抛物柯西问题的随机表示公式。因此，BSDES的数值可解性是一个具有高度实际意义的问题，如果BSDES依赖于随机源的高维系统，则特别具有挑战性。近年来，几个蒙特-卡罗算法的基础上随机网格或量化技术的BSDES已经开发。这些算法的一个严重缺点是，它们产生的点估计的解决方案，其质量没有验证。该项目的目的是增加上层（分别为。较低的）偏见的条款，这些算法，理论上消失的限制。在实际相关的预限制的情况下，相应的“上”和“下”的解决方案之间的差异可以作为数值过程的成功的指标。特别是，可以在单个离散化步骤中监测偏置项的绝对大小，这允许开发在关键步骤中应用更昂贵的估计器的自适应算法。除了对条件期望的一般估计量的附加有偏项进行误差分析外，还计划对最小二乘蒙特-卡罗估计量进行更详细的分析。

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.