Lauri Viitasaari

Stochastic Analysis of Gaussian Processes via Fredholm Representation

We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations.

Adapted integral representations of random variables

We study integral representations of random variables with respect to general Holder continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that an arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distribution. If in addition the random variable is a final value of an adapted Holder continuous process, then it can be represented as a proper integral. It is also shown that in the particular case of mixed fractional Brownian motion, any adapted random variable can be represented as a proper integral.

Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion

In this article, a uniform discretization of stochastic integrals

Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

\int _{0}^{1} f^{\prime }_-(B_t)\mathrm d B_t

Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian–fractional Brownian model

∫01f−′(Bt)dBt, where

Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind

B