The partially observed linear Gaussian system of stochastic differential equations with low noise in observations is considered. The coefficients of this system are supposed to depend on some unknown parameter. The problem of estimation of these parameters is considered and the possibility of the approximation of the filtering equations is discussed. An estimators are used for estimation of the quadratic variation of the derivative of the limit of the observed process. Then this estimator is used for nonparametric estimation of the integral of the square of volatility of unobservable component. This estimator is also used for construction of method of moments estimators in the case where the drift in observable component and the volatility of the state component depend on some unknown parameter. Then this method of moments estimator and Fisher-score device allow us to introduce the One-step MLE-process and adaptive Kalman–Bucy filter. The asymptotic efficiency of the proposed filter is discussed.

Motivated by the evolution of a population in a slowly varying random environment, we consider the 1D Anderson model on finite volume, with viscosity κ>0: (0.1)∂tu(t,x)=κΔu(t,x)+ξ(t,x)u(t,x),u(0,x)=u0(x),t>0,x∈T.The noise ξ is chosen constant on time intervals of length τ>0 and sampled independently after a time τ. We prove that the Lyapunov exponent λ(τ) is positive and near τ=0 follows a power law that depends on the regularity on the driving noise. As τ→∞ the Lyapunov exponent converges to the average top eigenvalue of the associated time-independent Anderson model. The proofs make use of a solid control of the projective component of the solution and build on the Furstenberg–Khasminskii and Boué–Dupuis formulas, as well as on Doob’s H-transform and on tools from singular stochastic PDEs.