Approximating some Volterra type stochastic integrals with applications to parameter estimation

We consider Volterra type processes which are Gaussian processes admitting representation as a Volterra type stochastic integral with respect to the standard Brownian motion, for instance the fractional Brownian motion. Gaussian processes can be represented as a limit of a sequence of processes in the associated reproducing kernel Hilbert space and as a special case of this representation, we derive Karhunen–Loéve expansions for Volterra type processes. In particular, a wavelet decomposition for the fractional Brownian motion is obtained. We also consider a Skorohod type stochastic integral with respect to a Volterra type process and using the Karhunen–Loéve expansions we show how it can be approximated. Finally, we apply the results to estimation of drift parameters in stochastic models driven by Volterra type processes using a Girsanov transformation and we prove consistency, the rate of convergence and asymptotic normality of the derived maximum likelihood estimators.