Adaptive Control and Online Parameter Estimation for Stochastic Systems

We propose a continuous-time online estimator derived from a likelihood-based formulation for stochastic differential equations. The estimator admits a compact stochastic differential representation and incorporates adaptive feedback through a lagged structure, ensuring measurability and well-posedness of the closed-loop system. Under standard regularity and monotonicity assumptions, we establish almost sure and mean-square convergence of the estimator to the true parameter. The convergence proof is based on a Lyapunov framework combined with an integrating factor technique, leading to a three-term decomposition that separates the effects of the initial condition, diffusion noise, and stochastic fluctuations. Importantly, the analysis does not require stationarity of the state process or time-scale separation.