An improved log-likelihood gradient for continuous time stochastic systems with deterministic input

We derive a log-likelihood gradient formula to use in calculating the maximum likelihood estimate of the unknown parameter θ in a system defined in terms of independent Gaussian white noise processes with identity spectral density matrices, and a known deterministic input. The initial mean and covariance may also depend on θ by presenting a method that needs no error covariances, and hence no Riccati equation solutions. We allow the range of the state noise covariance to depend on θ, deterministic inputs, and the initial state may have a nonzero mean, which depends on θ