Gradient of the log-likelihood ratio for infinite dimensional stochastic systems

Using a covariance operator approach, we derive an 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 limit how much the gradient formula can be simplified. A random heat equation is considered.