| The vast majority of research results on numerical methods for stochastic differential equations are based on the one-sided Lipschitz condition and coercive growth condition or more stringent coefficient conditions.Although the one-sided Lipschitz condition greatly broadens the research scope of numerical methods for stochastic differential equations,there are still many important models that do not satisfy this condition,such as Stochastic Lorenz equation,Stochastic Van der Pol oscillator model,Experimental psychology model,and the stochastic Lotka-Volterra competition model in this thesis.Currently,only a small amount of work focuses on the efficient simulation of these models.This thesis mainly concentrates on a class of d-dimensional stochastic Lotka-Volterra competition models with multi-dimensional noise and their strong approximation numerical methods.Under some appropriate assumptions,this stochastic model has a unique strong solution.Moreover,each component of the solution is positive and has long time bounded property.To approximate this model numerically,this thesis proposes a new linear-implicit Milstein method certified that the new method is positivity preserving,that is,each component of the numerical solution is positive.Based on the continuation of numerical methods and the perturbation theory of stochastic differential equations,this thesis proves that the newly proposed numerical method has a 0.5 strong convergence order.Furthermore,we demonstrate that the linear-implicit Milstein method succeeds the long time bounded property of the original equation.Finally,numerical experiments verify the above theoretical results.Compared with the positivity preserving algorithm in existing literature,numerical experiments also show that the linear-implicit Milstein method has significant advantages in terms of computational efficiency and long time bounded property,and is a better algorithm.4 Figures,1 Table,and 60 References. |
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