Estimating occupation time functionals

12.05.2017, 10.45-11.30  –  Haus 9, Raum 0.13
Forschungsseminar Wahrscheinlichkeitstheorie

Randolf Altmeyer

An occupation time functional is a Lebesgue integral of a function f(X_t) with X a stochastic process (X_t). Given
the observations at discrete times (X_{k T/n})_k, we study the approximation of the occupation time by the Rie -
mann-typeestimator.
When the process X is a continuous Itô semimartingale and the function f is smooth in the sense that its Fourier
transform is sufficiently integrable, we obtain stable central limit theorems with 1/n as rate of convergence. This
holds, in particular, for sufficiently "nice" X and weakly differentiable functions f. When X is a Brownian motion
and f is weakly differentiable, we show that the rate of convergence and the asymptotic variance are optimal among all possible estimators, while the rate does not improve for more smooth f. The methods for proving the central limit theorems also yield generalized Itô formulas which are of independent interest and which we will discuss briefly.
We further present general conditions on X to obtain rates of convergence when f lies in some fractional L^2-
Sobolev space of smoothness degree between 0 and 1, which explains rates obtained in the literature when f is a
Hölder function or an indicator function of a bounded set. The conditions on X apply to many important classes of
processes such as Markov processes, in particular diffusions and Lévy-processes, but also to non-Markovian processes, such as fractional Brownian motion. Again, for Brownian motion these rates are optimal.
[1]Altmeyer, R.,Chorowski,J. (2017) Estimating occupation ime function ,in preparation.
[2] Altmeyer, R., Chorowski, J. (2016) Estimating error for occupation time functionals of stationary Markov processes, arXiv preprint