Some Structure-preserving Numerical Methods For Stochastic Differential Equations

Stochastic differential equations are widely used to model phenomena in many fields such as physics,economics,biology,and so on.Since most stochastic differential equations cannot be solved exactly,the study of numerical methods for stochastic differential equations has played a more and more important role in recent years.Numerical methods that can preserve the intrinsic properties such as geometrical or physical properties of the original systems are usually called structure-preserving methods,which have drawn a lot of attention recently for their good performance especially in a long-term numerical simulation.Since conserved quantity and symplecticity are two of the most important intrinsic properties,this dissertation mainly devotes to constructing numerical methods that preserve the conserved quantity or the symplecticity of some stochastic differential equations.The main work is as follows:Stochastic differential equations with a conserved quantity are considered.A class of discrete gradient methods based on the skew-gradient form is constructed,and the sufficient condition of convergence order 1 in the mean square sense is given.Then a class of linear projection methods is constructed.The relation of the two classes of methods for preserving a conserved quantity is proved,that is,the constructed linear projection methods can be considered as a subset of the constructed discrete gradient methods.Numerical experiments verify the analytical results and show the efficiency of proposed numerical methods.Partitioned stochastic differential equations with a conserved quantity are considered.A stochastic partitioned averaged vector field method is proposed and analyzed.We prove this numerical method is able to preserve the conserved quantity of the original system automatically.Then the convergence analysis is carried out in detail and we derive the method is convergent with order 1 in the mean-square sense.Finally some numerical examples are reported to verify the effectiveness of the proposed method.Two classes of stochastic differential equations of single integrand type are considered.In the first case,arbitrary high-order energy-preserving numerical methods for stochastic canonical Hamiltonian systems are studied.A class of stochastic parametric Runge-Kutta methods is obtained by use of W-transformation.Increments of Wiener processes are replaced by some truncated random variables.We prove the replacement doesn’t change the convergence order under some conditions.The methods turn out to be symplectic for any given parameter.It is shown that there exists a suitable parameter at each step such that the energy-preserving property holds,and the energy-preserving methods retain the order of the underlying stochastic Gauss-Runge-Kutta methods.Numerical results illustrate the effectiveness of energy-preserving methods when applied to stochastic canonical Hamiltonian systems.In the second case,arbitrary high-order energy-preserving numerical methods for stochastic Poisson systems are studied.A class of explicit parametric stochastic Runge-Kutta methods based on perturbed collocation methods with truncated random variables is developed for solving stochastic Poisson systems.Similar to that in the first case,we prove there exists a suitable parameter at each step such that the energy conservation holds.The derived energy-preserving method turns out to retain the mean-square convergence order of the original stochastic Runge-Kutta method.Numerical results illustrate the effectiveness and the convergence results of the constructed methods.A novel way of constructing symplectic stochastic partitioned Runge-Kutta methods for stochastic Hamiltonian systems is presented.A class of continuous-stage stochastic partitioned Runge-Kutta methods for partitioned stochastic differential equations is proposed.The order conditions of the continuous-stage stochastic partitioned Runge-Kutta methods are derived via the stochastic B-series theory.The symplectic conditions of the continuous-stage stochastic partitioned Runge-Kutta methods when applied to stochastic Hamiltonian systems are analyzed.Then we prove applying any quadrature formula to a symplectic continuous-stage stochastic partitioned Runge-Kutta method will result in a symplectic stochastic partitioned Runge-Kutta method.In this way,various symplectic stochastic partitioned Runge-Kutta methods can be easily constructed by using different quadrature formulas.A concrete symplectic continuous-stage stochastic partitioned Runge-Kutta method of order 1 is constructed according to the symplectic conditions and order conditions,and several retrieved stochastic partitioned Runge-Kutta methods are obtained.Numerical experiments are presented to verify the theoretical results and show the effectiveness of the derived methods.