Parameter Estimation in a Linear Stochastic Differential Equation

This paper is concerned with the problem of parameter estimation in a vector linear differential equation, with constant coefficients, of the form: $$ d{X_t} = \theta {X_t}\,dt + G\,d{\xi _t};\;t\,\underline \geqslant \,0;\;{X_0}\underline = \,0 $$ where (ξ t : t ≧ 0) is a standard Wiener process and G is a non singular matrix, when one observes a trajectory of the solution process. We first show, using a strong limit theorem for Brownian motion, that we can assume the matrix GG′ is completely known with the meaning that, with probability one, it may be computed on every finite time interval. Then we show that the statistical structure for the problem of estimation of the matrix θ, when GG’ is known, is dominated and we compute the likelihood function with help of Girsanov’s results [1]. We determine the maximum likelihood estimate of θ and study its asymptotic properties.