Consistency and Convergence of Non-parametric Estimation of Drift and Diffusion Coefficients in SDEs from Long Stationary Time-series




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We study the efficiency of non-parametric estimation of stochastic differential equations driven by Brownian motion (i.e. diffusions) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients for SDEs. These estimators involve time- and space-discretization parameters for computing discrete analogs of expected values from discretely-sampled stationary data. Number of observational points is the third important computational parameter. Next, we derive bounds for the asymptotic behavior of L2 errors for the drift and diffusion estimators. The asymptotic behavior is characterized when the number of observational points becomes infinite and observational time-step and bin size for spatial discretization of drift and diffusion coefficients tend to zero. Using our asymptotic analysis we are able to obtain practical guidelines for selecting computational parameters. Finally, we perform a series of numerical simulations which support our analytical investigation and illustrate practical guidelines for selecting near-optimal and computationally efficient values for computational parameters.



stochastic differential equations, drift and diffusion estimation, conditional expecta- tion, mean squared error, regression