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Approximating some Volterra type stochastic integrals with applications to parameter estimation. (English) Zbl 1075.60532

Summary: 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 Skorokhod 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 the 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.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
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