Parameter estimation in stochastic oscillators based on transition probability function decomposition

The problem of parameter estimation in stochastic oscillators is considered using a decomposition of the transition probability density in terms of Hermite polynomials. Building upon recent theoretical results on transition function expansions, it is constructed a quasi-likelihood framework that enables consistent inference for drift and diffusion parameters in discretely observed stochastic differential equations. Three representative case studies are considered. Across these examples, the proposed methodology demonstrates accurate recovery of oscillator parameters, with estimators showing asymptotic normality in most cases and highlighting practical challenges when degeneracy arises in diffusion terms. The results underscore the potential of transition probability decomposition as a general tool for parameter estimation in noisy oscillatory systems spanning physics, biology, and engineering applications.