Ergodic theory
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.
Literature
O. Peters and W. Klein
Partial ensemble averages in geometric Brownian motion.
arXiv:1209.4517 (2012).
O. Peters
The time resolution of the St Petersburg paradox.
Phil. Trans. R. Soc. A 369, 369, 1956, 4913–4931 (2011).
arXiv:1011.4404
doi:10.1098/rsta.2011.0065
O. Peters
Optimal leverage from non-ergodicity.
Quant. Fin. 11, 11, 1593–1602 (2011).
arXiv:0902.2965v2
doi:/10.1080/14697688.2010.51333
Economics
Mathematical treatments of randomness started with gambling problems and from there moved quickly to more general economic settings in the 17th century. Physics did not seriously consider randomness before the mid-19th century. However, when physicists entered the discussion with much more clearly defined problems, our understanding took a leap forward.
Mostly, this is the leap into ergodic theory and the recognition that the mathematics developed in the 17th and 18th century was only appropriate for the very special class of ergodic systems. Economics models — for instance any that exhibit growth — are usually not ergodic. Examples are the random walk or geometric Brownian motion. But economics stuck to its 17th-century methods rather than adopt new techniques in the 19th and 20th century. We apply the new techniques to old problems and find new answers. Some of these answers are of tremendous importance to the current state of economics and to many economic and political debates.
This work has attracted much interest from the investment community.
Literature
O. Peters and A. Adamou
Stochastic Market Efficiency.
Santa Fe Institute Working Paper #2013-06-022 (2013).
arXiv:1101.4548
www.santafe.edu/media/workingpapers/13-06-022.pdf
O. Peters
Menger 1934 revisited.
arXiv:1110.1578 (2011).
O. Peters
The time resolution of the St Petersburg paradox.
Phil. Trans. R. Soc. A 369, 369, 1956, 4913–4931 (2011).
arXiv:1011.4404
doi:10.1098/rsta.2011.0065
O. Peters
Optimal leverage from non-ergodicity.
Quant. Fin. 11, 11, 1593–1602 (2011).
arXiv:0902.2965v2
doi:/10.1080/14697688.2010.513338
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