The log-likelihood gradient for infinite dimensional stochastic systems

Using a covariance operator approach, we derive an explicit expression for the log-likelihood ratio gradient for system parameter estimation for continuous time infinite dimensional stochastic systems. The gradient formula includes the smoother estimates and derivatives of system operators, with no derivatives of estimates or covariance operators. The unbounded operators typically found in partial differential equations place limitations on the expression for the gradient. An example of a random heat equation is considered