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

倒向随机微分方程(BSDE)是解决金融数学中衍生产品定价、金融风险套期保值、最优投资等问题的有力工具。此外，它们还给出了半线性抛物柯西问题的随机表示式。因此，倒向随机微分方程的数值可解性是一个具有很高实际意义的问题，如果一个倒向随机微分方程依赖于高维随机源系统，那么它尤其具有挑战性。近年来，已有几种基于随机网格或量化技术的蒙特卡罗算法用于求解BSDE。这些算法的一个严重缺陷是它们只为解产生点估计器，其质量没有得到验证。本项目的目的是增加上部(分别为。较低)偏向这些算法的项，其理论上在极限中消失。在实际相关的预极限情况下，相应的“上”和“下”解之间的差异可以作为数值过程成功的指标。具体地说，可以在单个离散化步骤中监控有偏项的绝对大小，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进行更详细的误差分析。

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.