Mean reversion with a variance threshold

Starting from a sample path of a multivariate stochastic process, we study several techniques to isolate linear combinations of the variables with a maximal amount of mean reversion, while constraining the variance of the combination 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 a variant of the S-lemma. In finance, these methods can be used to isolate statistical arbitrage opportunities, i.e. mean reverting baskets with enough variance to overcome market friction. In a more general setting, mean reversion and its generalizations can also be used as a proxy for stationarity, while variance simply measures signal strength.