Optimal control for estimation in partially observed elliptic and hypoelliptic stochastic differential equations

Abstract : Multi-dimensional Stochastic Differential Equations (SDEs) are a powerful tool to describe dynamics of several fields (pharmacokinetic, neurosciences, ecology, etc). The estimation of the parameters of these systems has been widely studied. We focus in this paper in the case of partial observations, only a one-dimensional observation is available. We consider two families of SDE, the elliptic family with a full-rank diffusion coefficient and the hypoelliptic family with a degenerate diffusion coefficient. The estimation for the second class is much more difficult and only few references have proposed estimation strategies in that case. Here, we adopt the framework of the optimal control theory to derive an estimation contrast (or cost function) based on the best control sequence mimicking the (unobserved) Brownian motion. We propose a full data-driven approach to estimate the parameters of the drift and of the diffusion coefficient. The estimation reveals to be very stable in a simulation study conducted on different examples (Harmonic Oscillator, FitzHugh-Nagumo, Lotka-Volterra).
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Submitted on : Monday, October 23, 2017 - 9:55:19 AM
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Quentin Clairon, Adeline Samson. Optimal control for estimation in partially observed elliptic and hypoelliptic stochastic differential equations. 2017. ⟨hal-01621241⟩

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