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In this paper we investigate the finite sample performances of five estimation methods for a continuous-time stochastic process from discrete observations. Applying these methods to two examples of stochastic differential equations, one with linear drift and state-dependent diffusion coefficients and the other with nonlinear drift and constant diffusion coefficients, Monte Carlo experiments are carried out to evaluate the finite sample performance of each method. The Monte Carlo results indicate that the differences between the methods are large when the discrete-time interval is large. In addition, these differences are noticeable in estimations of the diffusion coefficients.

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Abstract. This paper investigates the rate of convergence of an alternative approximation method for stochastic differential equations. The rates of convergence of the one-step and multi-step approximation errors are proved to be O((∆t) 2 ) and O(∆t) in the Lp sense respectively, where ∆t is discrete time interval. The rate of convergence of the one-step approximation error is improved as compared with methods assuming the value of Brownian motion to be known only at discrete time. Through numerical experiments, the rate of convergence of the multi-step approximation error is seen to be much faster than in the conventional method.