Ergodic theory is about the relationship between time averages and ensemble averages. A typical problem in ergodic theory may be to show that a given system of equations (“a model”) is ergodic. Once this has been shown, it is then safe to use expectation values instead of time averages because they are guaranteed to be identical.
Our research in ergodic theory is a little unusual because we’re interested in non-ergodic systems. What happens, where do we have to be careful, if a stochastic process is not ergodic? These processes challenge our intuition with conundrums such as this: under a non-ergodic dynamic the expectation value of my wealth in some gamble may be exponentially increasing while I’m guaranteed to go bankrupt.
A common strategy for this type of problem is diversification, but often that only delays the eventual failure. This has applications in financial mathematics but also in evolutionary biology where different types of diversification strategies are successful in different types of environments. We aim to explain these phenomena.
O. Peters and W. Klein
Partial ensemble averages in geometric Brownian motion.
The time resolution of the St Petersburg paradox.
Phil. Trans. R. Soc. A 369, 369, 1956, 4913–4931 (2011).
Optimal leverage from non-ergodicity.
Quant. Fin. 11, 11, 1593–1602 (2011).