Truncated Euler Method For Nonautonomous Stochastic Differential Equations And Its Applications

Stochastic differential equations(SDEs)have broad applications in many areas.Therefore,the numerical analyses and numerical methods are studied in more depth.The numerical methods are mainly divided into explicit methods and implicit ones.The classical explicit methods such as Euler-Maruyama method have been proved divergent for highly non-linear SDEs as well as the implicit methods have poor computational efficiency and practical applications.In this thesis,the truncated Euler-Maruyama method is put forward to approximate a class of highly non-linear and non-autonomous SDEs.This method not only avoids the risk of divergence,but also preserves the simple algorithm structure and relatively lower computational cost of the explicit method.In addition,the applications of this method in time-changed stochastic differential equations are studied.Firstly,the truncated Euler-Maruyama method is used to solve a class of highly non-linear and non-autonomous SDEs with the Holder continuity in the temporal variable and the super-linear growth in the state variable.And then the strong convergence rate of this method is proved to be min(?,?,1/2-?)which ?,? are the indexes of Holder continuity and ? is an arbitrarily small number.We find that when the smoothness of temporal variable is poor,such as ?=?=1/4.the strong convergence rate of the truncated Euler-Maruyama method is 1/4 which is poorer than the truncated Euler-Maruyama method in autonomous SDEs(1/2-?).Conversely,when the smoothness of temporal variable is appropriate,such as ?=?=3/4.the strong convergence rate is(1/2-?)which is the same as the strong convergence rate of truncated Euler-Maruyama method in autonomous SDEs.Secondly,the strong convergence rate of the truncated Euler-Maruyama method is applied to highly non-linear time-changed SDEs by duality principle.We study properties of time-changed SDEs and illustrate the one to one correspondence between classical and time-changed SDEs.Similarly,the strong convergence rate of the truncated Euler-Maruyama method in time-changed SDEs is proved to be min(?,?,1/2-?).Last but not least,we verify the correctness of the theoretical results by three numerical examples and broaden the usable range of the truncated Euler-Maruyama method.