A new formula for the log-likelihood gradient for continuous-time stochastic systems

Using a finitely additive white noise approach, we obtain an explicit expression for the gradient of the log-likelihood ratio for system parameter estimation for continuous-time linear stochastic systems with noisy observations. Our gradient formula includes the smoother estimates of the state vector, and derivatives of only the system matrices, and not the estimates or error covariances. A scheme to calculate the log-likelihood gradient without solving a Riccati equation is described when only one matrix and the initial covariance depend on the unknown parameter