We study local asymptotic properties of likelihood ratios of certain Heston models. We distinguish three cases: subcritical, critical and supercritical models. For the drift parameters, local asymptotic normality is proved in the subcritical case, only local asymptotic quadraticity is shown in the critical case, while in the supercritical case not even local asymptotic quadraticity holds. For certain submodels, local asymptotic normality is proved in the critical case, and local asymptotic mixed normality is shown in the supercritical case. As a consequence, asymptotically optimal (randomized) tests are constructed in cases of local asymptotic normality. Moreover, local asymptotic minimax bound, and hence, asymptotic efficiency in the convolution theorem sense are concluded for the maximum likelihood estimators in cases of local asymptotic mixed normality.

For the affine stochastic delay differential equationthe local asymptotic properties of the likelihood function are studied. Local asymptotic normality is proved in case of a ∈ − π 2 2 , 0 , local asymptotic mixed normality is shown if a ∈ 0, ∞ , periodic local asymptotic mixed normality is valid if a ∈ −∞, − π 2 2 , and only local asymptotic quadraticity holds at the points − π 2 2 and 0. Applications to the asymptotic behaviour of the maximum likelihood estimator a T of a based on (X(t)) t∈[0,T ] are given as T → ∞.

Assume that we observe a stochastic process (X(t)) t∈ [−r,T ] , which satisfies the linear stochastic delay differential equationwhere a is a finite signed measure on [−r, 0]. The local asymptotic properties of the likelihood function are studied. Local asymptotic normality is proved in case of v * ϑ < 0, local asymptotic quadraticity is shown if v * ϑ = 0, and, under some additional conditions, local asymptotic mixed normality or periodic local asymptotic mixed normality is valid if v * ϑ > 0, where v * ϑ is an appropriately defined quantity. As an application, the asymptotic behaviour of the maximum likelihood estimator ϑ T of ϑ based on (X(t)) t∈[−r,T ] can be derived as T → ∞.

Res z=λ (f (z)) residue of the function f at λ LAQ local asymptotic quadratic, see Denition 2.1.4 LAMN local asymptotic mixed normal, see Denition 2.1.5 LAN local asymptotic normal, see Denition 2.1.6 PLAMN periodic local asymptotic mixed normal, see Denition 2.1.7 MLE maximum likelihood estimator N (µ, σ 2) the normal (or Gaussian) distribution with expected value µ ∈ R and variance σ 2 ∈ R ++ N d (µ, Σ) the d-dimensional normal (or Gaussian) distribution with expected value µ ∈ R d and covariance matrix Σ ∈ R d×d v

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