Backward Stochastic Differential Equations (BSDEs) have been widely employed in various areas of social and natural sciences, such as the pricing and hedging of financial derivatives, stochastic optimal control problems, optimal stopping problems and gene expression. Most BSDEs cannot be solved analytically and thus numerical methods must be applied in order to approximate their solutions. There have been a variety of numerical methods proposed over the past few decades as well as many more currently being developed. For the most part, they exist in a complex and scattered manner with each requiring different and similar assumptions and conditions. The aim of the present work is thus to systematically survey various numerical methods for BSDEs, and in particular, compare and categorize them, for further developments and improvements. To achieve this goal, we focus primarily on the core features of each method on the basis of an exhaustive collection of 289 references: the main assumptions, the numerical algorithm itself, key convergence properties and advantages and disadvantages, in order to provide a full up-to-date coverage of numerical methods for BSDEs, with insightful summaries of each and a useful comparison and categorization.
翻译:社会科学和自然科学的各个领域,例如金融衍生物的定价和套期保值、最佳最佳控制问题、最佳遏制问题和基因表达方式等,广泛采用了落后的统计等不同方法,大部分无法通过分析解决,因此,必须采用数字方法来估计解决办法。在过去几十年里,提出了各种数字方法,而且目前正在制订更多的数字方法。在大部分情况下,它们以复杂和分散的方式存在,每个方法都需要不同和相似的假设和条件。因此,目前工作的目的是系统调查生物衍生物的各种数字方法,特别是比较和分类,以便进一步发展和改进。为实现这一目标,我们主要侧重于每种方法的核心特征,其依据是289种详尽的参考资料:主要假设、数字算法本身、关键趋同特性和利弊,以便全面涵盖BSDE的数值方法,并附有对每一种方法的深刻概述和有用的比较和分类。