Convergence And Stability Of Balanced Implicit Method Or Stochastic Differential Equations With Poisson Jumps

In the study of semi-implicit Euler-Maruyama method,implicit part is only restrictedto drift term,which is completely adapted for stiff systems with small stochastic noise in-tensity or additive noise.But the method is unsuitable for solving stiff stochastic differentialequations,in which stochastic term (or diffusion term) plays an essential role,since explicitmethods of diffusion term work unreliably for not too small time step sizes. Milstein pro-posed an implicit balanced method in1998,which is a special method with a kind of balancebetween approximating stochastic terms in numerical scheme.By choosing appropriate pa-rameters involved in the scheme,we can make it satisfy the corresponding demand.Milsteinconcluded that the balanced implicit method has better behavior than the Euler-Maruyamamethod,by the aid of numerical experiment.And in comparison with the explicit Euler-Maruyama method,the balanced method is rather general,which covers the semi-implicitEuler-Maruyama method.Not only the diffusion term of stochastic differential equations with Passion jumps butalso the additive noise N (t) is important in systems. The paper mainly studied mean-squareconvergence and stability of balanced implicit method for the equations. The paper makesuse of Ito lemma,one-step method,Taylor formula, stochastic analysis and so on, section1is introduction which introduces the development status of stochastic differential equations.the paper gives the proof of existence and uniqueness of the solution of the equation (2.2) insection2, and the theorem of Euler-Maruyama method for the equation which is consistentwith order P1=2in the mean,consistent with order P2=1in the mean-square. In section3,The paper uses the balanced implicit method to (2.2) on the basis of the above, provesthat the mean-square convergence of the equation.Lastly,the paper will give the satisfiedcondition of mean-square stability of balanced implicit method for n dimensional linearstochastic differential equation.