# Advanced Methods in Mathematical Finance

1-4 September 2015
Angers - France
Europe/Paris timezone

## Random Periodic Processes, Periodic Measures and Ergodicity

4 Sep 2015, 17:40
40m
Angers - France

### Speaker

Prof. Huaizhong Zhao (Loughborough University)

### Description

An ergodic theorem and a mean ergodic theorem in the random periodic regime on a Polish space is proved. The idea of Poincaré sections is introduced and under the strong Feller and irreducible assumptions on Poincaré sections, the weak convergence of the transition probabilities at the discrete time of integral multiples of the period is obtained. Thus the Khas'minskii-Doob type theorem is established and the ergodicity of the invariant measure, which is the mean of the periodic measure over a period interval, is obtained. The Krylov-Bogoliubov type theorem for the existence of periodic measures by considering the Markovian semigroup on a Poincaré section at discrete times of integral multiples of the period is also proved. It is proved that three equivalent criteria give necessary and sufficient conditions to classify between random periodic and stationary regimes. The three equivalent criteria are given in terms of three different notion respectively, namely Poincaré sections, angle variable and infinitesimal generator of the induced linear transformation of the canonical dynamical system associated with the invariant measure. It is proved that infinitesimal generator has only two simple eigenvalues, which are $0$ and the quotient of $2\pi$ by the minimal period, while the classical Koopman-von Neumann theorem says that the generator has only one simple eigenvalue $0$ in the stationary and mixing case. The equivalence" of random periodic processes and periodic measures is established. The strong law of large numbers (SLLN) is also proved for random periodic processes. This is a joint work with Chunrong Feng.

### Primary author

Prof. Huaizhong Zhao (Loughborough University)

 Slides