Affine stochastic functional differential equations and local asymptotic properties of their parameter estimations (Q2784241)

In this thesis, the author studies so-called stochastic delay (or functional) differential equations of the form NEWLINE\[NEWLINEdX(t)=(\int_{-r}^0 X(t+s)da(s))dt+\sigma dW(t),\;t\geq 0.NEWLINE\]NEWLINE Here, \(r>0\) is the memory length, \(a\) is a function of bounded variation and \(W\) a standard Brownian motion. First, analytical properties are established: representation as a series of Ornstein-Uhlenbeck processes, and characterisation of the spectral set of the associated semigroup operator. Secondly, the function \(a=a_\theta\) is supposed to be parameter-dependent. Then the likelihood function and properties of the maximum likelihood estimator are derived: local asymptotic normality (LAN), local asymptotic mixed normality (LAMN) and periodic local asymptotic normality (PLAMN), the latter of which has not occurred before. Which of these properties holds for a certain parameter value \(\theta_0\) depends on the spectral set of the associated semigroup operator. The methods applied range from complex and harmonic analysis to stochastic analysis and Le Cam theory, but are each time carefully introduced.