Stability of the Filter Equation for a Time-Dependent Signal on ℝd

Stability of the pathwise filter equation for a time-dependent signal process induced by a d-dimensional stochastic differential equation and a linear observation is studied, using a variational approach introduced in 16]. A lower bound for the rate of stability is identified in terms of the mass-gap of a parabolic ground state transform associated with the generator of the signal process and the square of the observation. The lower bound can be easily calculated a priori and provides hints on how precisely to measure the signal in order to reach a certain rate of stability. Ergodicity of the signal process is not needed.