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**Mean Reversion with a Variance Threshold**

**Authors:** *Marco Cuturi* and *Alexandre D'aspremont*

**Conference:** Proceedings of the 30th International Conference on Machine Learning (ICML-13)

**Year:** 2013

**Pages:** 271-279

**Abstract:** 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 $\mathcal{S}$-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.

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