Research On Numerical Approximations Of A Kind Of Stochastic Differential Equations With Jumps Under Super-linearly Growing Coefficients

As a class of important stochastic processes,the diffusion processes are solutions of stochastic differential equations(SDEs)driven by Gaussian noise and Poisson noise,i.e.,jump-diffusion SDEs have been widely used to model stochastic phenomena having both continuous and discrete movements in financial,actuarial and other areas.In this study,we investigate the mean-square convergence and stability of compensated stochastic theta methods(CSTMs)for jump-diffusion SDEs with the monotone condition and super-linearly growing drift,diffusion,jump coefficients.This thesis firstly shows some bound moments which can ensure upper mean-square error bounds for CSTMs with the parameters theta∈[1/2,1].Then,we can get the conclusion that the mean-square convergence order of CSTMs with theta∈[1/2,1] is arbitrarily close to1/2.Armed with more differentiability assumptions on the coefficients and It(?) formula,the mean-square convergence order of CSTMs can be improved to the standard order 1/2.Also,we extract mean-square convergence rates of CSTMs for SDEs driven by small noise,in particular,CSTM with theta=1/2 performs better in convergence than CSTMs with theta∈(1/2,1].Furthermore,the mean-square stability of the considered method was shown with the nonpositive monotonicity condition,which means CSTMs with theta ∈ [1/2,1] can inherit the mean-square stability of original equations for any given stepsize.7 picture,0 tables,55 references...