A study of strong convergence of differential equations based on Euler’s algorithm

Differential equations have important applications in many fields, such as chemistry, biology, epidemiology, and finance. Most analytic solutions of differential equations are difficult to obtain. Therefore numerical solutions of differential equations become an important tool. The truncated Euler method is proposed in this paper, and we investigate how the truncated EM solution of the derived SDDE converges strongly under the local Lipschitz condition and the one-sided linear growth condition after relaxation. The basic strong convergence theorem is set up and the new notation X(t,x;s) is introduced as an analytic solution of the stochastic differential equation. Establish the assumption that the coefficients of the drift term and the coefficients of the diffusion term of the stochastic differential equation satisfy a contractionary monotonicity condition in order to prove, by induction, that the exact solution of this stochastic differential equation is bounded for a long time. Different examples are given to compare the simulated deviations between the numerical and analytical solutions of Euler’s algorithm and the modified Euler’s algorithm for the initial value problem of fractional order differential equations and to analyze the convergence of the two numerical solutions. The truncated Euler method is applied to highly nonlinear time-transformed stochastic differential equations by means of the dyadic principle, and it is proved that the order of convergence of the strong convergence of the truncated Euler-Maruyama method for time-transformed stochastic differential equations is min(α,γ,12−ɛ)\min \left( {\alpha ,\gamma ,{1 \over 2} - \varepsilon } \right).