【作者】 韩强；

【导师】 嵇少林；

【作者基本信息】 山东大学 ， 概率论与数理统计， 2022， 博士

【摘要】 本论文主要研究正倒向随机微分方程(FBSDEs)的蒙特卡罗(MC)数值算法。FBSDEs的理论及其应用大约蓬勃发展了三十年。关于这个领域有大量的文献资料可供参考。我们推荐读者阅读Pardoux和Peng的开创性的论文。这些文章将非线性FBSDEs的理论框架建立起来了。虽然FBSDEs在金融数学、随机控制、偏微分方程、保险精算、风险度量等领域中有着重要的应用,但是FBSDEs的解析解是很难求出的,甚至能求出解析解的FBSDEs的解的表达式是相当复杂的。这促使了众多学者对FBSDEs数值解的研究。随着对FBSDEs数值解的深入研究,需要进一步研究求数值解的新方法。这对深刻理解解的性质、进一步丰富计算方法和促进实际应用有重要的意义。在这种背景下,本文提出了几种基于多步格式、多阶段格式和多层蒙特卡罗(MLMC)的数值方法来计算FBSDEs的数值解。本论文的研究内容如下:在第一章中,我们介绍了本文的研究背景、研究动机和相关的概念等知识。同时,也说明了本文的研究是基于理论与实际需求的、是有意义的和必要的。在第二章中,基于Gobet et al.的文章[59],我们提出了求解倒向随机微分方程(BSDE)的多步预测—校正格式。此格式在尽可能的保证计算简单的情况下,提升数值解的精确性。我们严格证明了此数值格式的稳定性且给出误差估计。数值实验验证了所提多步预测—校正格式比Gobet et al.文章[59]中给出的MDP格式的计算结果更有效。在第三章中,基于It?-Taylor展式,我们提出了求解FBSDEs的新的多步预测—校正格式。我们严格证明了此数值格式的稳定性和高阶收敛性。我们还证明了多步格式是稳定的当且仅当Dahlquist根条件成立。数值例子验证了这类新的多步预测—校正格式使数值解与解析解的误差的精度进一步提升,数值结果比Chassagneux[28]提出的线性多步格式的计算结果更有效。在第四章中,提出求解BSDE的Runge-Kutta型的预测—校正格式。该格式是在Euler格式的基础上发展起来的并且具有易改变步长的优点。此外,我们分析了该格式的误差和收敛性。并通过数值例子验证所提数值格式的理论结果。在第五章中,对于BSDE,我们构造出一类新的关于Y和Z都是显式的多步格式,并且对该类格式的稳定性和收敛性进行了严格的分析。对于条件期望的计算,我们基于MLMC提出一种新的可降低计算复杂度的算法。数值结果验证了所提算法的稳定性、收敛性和所提算法相较于经典的MC方法可显著降低算法的计算复杂度。在第六章中,我们利用MLMC建立了二次增长型的倒向随机微分方程(qBSDE)数值格式中的条件数学期望的计算方法的理论框架。我们严格证明了所提MLMC方法可以降低qBSDE解的计算复杂度。与经典的MC方法相比,如果采用的都是改进的Euler格式且误差精度要求达到O(?),计算复杂度由O(?-2-2/(1-n))降到了 O(?-2/(1-n)),η∈(0,1)。最后通过数值实验验证了所提数值算法确实使得计算复杂度得以降低。更多还原

【Abstract】 This paper focuses on developing multi-step and multi-stage high order efficient numerical schemes for forward backward stochastic differential equations(FBSDEs).As we all know,the theory of backward stochastic differential equations(BSDEs)has flourished for nearly thirty years.It is impossible to list exhaustive references on this subject because there is tremendous amount of literature.Thus,we recommend readers to the seminal papers of Pardoux and Peng which establish the theory of nonlinear BSDEs.Although BSDEs have valuable applications in financial mathematics,stochastic control,partial differential equations,actuarial and financial,risk measures and so on,very few solutions of BSDEs have been explicitly known.Even if some analytical solutions of BSDEs are known,the explicit solutions are complex.This has prompted many scholars to study the numerical solutions of FBSDEs.With the further study of the numerical solutions of FBSDEs,new methods for the numerical solutions are required.It is of great significance to understand the properties of solutions,further enrich the computational methods and promote practical applications.Under this background,several algorithms based on multi-step,multi-stage and multilevel Monte Carlo(MLMC)are constructed.It is also shown that these algorithms are competitive compared to other available algorithms for FBSDEs.The research contents of this paper are as follows:In the first chapter,we introduce the research background,motivation and relevant concepts and so on.Simultaneously,we show that the research is based on the theory and practical requirements and is meaningful and necessary.In the second chapter,we design a multi-step predictor-corrector scheme for BSDEs.This scheme tries its best to retain the simplicity and improve its convergence rate as much as possible.We investigate the stability and rigorously deduce the error estimates of this scheme.Numerical experiments are compared with the scheme given by Gobet et al.[59]and are given to illustrate that the multi-step predictor-corrector scheme is an efficient probabilistic numerical method.In the third chapter,we propose the predictor-corrector type general linear multistep schemes for BSDEs.The stability and high order rate of convergence of the schemes are rigorously proved.We also present a sufficient and necessary condition for the stability of the general schemes.Numerical experiments are given to illustrate the theoretical results of the proposed methods.In the fourth chapter,we propose a family of Runge-Kutta type predictor-corrector(RKPC)schemes for BSDEs.These schemes develop from the Euler schemes and are easy to change the step-size.Moreover,the error estimations and the convergence of these schemes are also provided.Numerical experiments are provided to illustrate theoretical results.In the fifth chapter,for BSDEs,we construct a fully explicit multi-step timediscretization scheme and prove its stability and convergence rates.To approximate conditional expectations in our scheme,we design a new algorithm based on the MLMC method which can reduce the computational complexity.Numerical experiments are given to illustrate the theoretical results of the proposed methods.In the sixth chapter,we present that the MLMC method can be applied to reduce the computational complexity of approximating the solutions of BSDEs with generators of quadratic growth with respect to Z.Compared with Monte Carlo method,in the tamed Euler discretisation,the computational complexity to achieve an accuracy of O(ε)is reduced from ε(ε-2-2/(1-η))to O(ε-2/(1-η))with η∈(0,1).Moreover,we extend the computational complexity analysis to quite general cases so that it can be applied to a variety of numerical schemes.Numerical examples are given to illustrate significant computational savings.更多还原