Mean Reversion with a Variance Threshold

Starting from a multivariate data set, we study several techniques to isolate affine combinations of the variables with a maximum amount of mean reversion, while constraining the variance to be larger than a given threshold. We show that many of the optimization problems arising in this context can be solved exactly using semidefinite programming and some variant of the \mathcalS-lemma. In finance, these methods are used to isolate statistical arbitrage opportunities, i.e. mean reverting portfolios with enough variance to overcome market friction. In a more general setting, mean reversion and its generalizations are also used as a proxy for stationarity, while variance simply measures signal strength.