Some methods of parameter estimation for stochastic differential equations

We propose three methods of parameter estimation based on discrete observation data for stochastic differential equations (SDEs). The first method is designed for linear stochastic differential equations (SDEs). For these equations we deduce distribution of certain operation of the exact solution and assume that the relevant operation of the observed data obey this distribution, from which we estimate the unknown parameters in the drift and diffusion coefficients. In the second method, we suppose that certain operation of the observation data and that of the numerical solution arising from the Euler-Maruyama scheme for the SDEs of Ito sense obey the same distribution, from which the unknown parameters can be estimated. We use the third method for SDEs of Stratonovich sense. For these equations we derive the distribution of relevant operation of the numerical solution produced by the midpoint scheme and let the same operation of the data obey this distribution to get estimation of the unknown parameters. Numerical results show validity of the proposed methods, and illustrate that the estimation error produced by the Euler-Maruyama scheme is about of order O(h0.5) while that by the midpoint scheme is about of order O(h), with h being the time step size of the numerical methods. Furthermore, the numerical results show that our methods are more accurate than the existing EM-MLE estimator.