Monte Carlo Algorithms For Forward Backward Stochastic Differential Equations And Related Topics

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.