An approximate maximum likelihood estimator of drift parameters in a multidimensional diffusion model
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For a fixed T and k ≥ 2, a k-dimensional vector stochastic differential equation dX_t=μ(X_t, θ)dt+ν(X_t)dW_t, is studied over a time interval [0,T]. Vector of drift parameters θ is unknown. The dependence in θ is in general nonlinear. We prove that the difference between approximate maximum likelihood estimator of the drift parameter θ_n≡θ_n,T obtained from discrete observations (X_iΔ_n, 0 ≤ i ≤ n) and maximum likelihood estimator θ̂≡θ̂_T obtained from continuous observations (X_t, 0≤ t≤ T), when Δ_n=T/n tends to zero, converges stably in law to the mixed normal random vector with covariance matrix that depends on θ̂ and on path (X_t, 0 ≤ t≤ T). The uniform ellipticity of diffusion matrix S(x)=ν(x)ν(x)^T emerges as the main assumption on the diffusion coefficient function.
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