Stationary distributions for stochastic differential equations with random effects and statistical applications
2013
Laredo, Catherine | Genon-Catalot, Valentine
Let (X(t), t ≥ 0) be de ned by a stochastic di erential equation including a random effect φ in the drift and di usion coe cients. We characterize the stationary distributions of the joint process ((φ, X(t)), t ≥ 0) which are non unique and prove limit theorems and central limit theorems for functionals of the sample path (X(t), t ∈ [0, T ]) as T tends to in nity. This allows to build several estimators of √ random variable φ which are the consistent and asymptotically mixed normal with rate T . Examples are given ful lling the assumptions of the limit theorems. Parametric estimation of the distribution of the random e ect from N i.i.d. processes (Xj (t), t ∈ √ T ]), j = 1, . . . , N is considered. Parametric estimators are built and proved to be N -consistent and asymptotically Gaussian as both N and T = T (N ) tend to in nity with T (N )/N tending to in nity.
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