In this paper we construct and analyse a multi-asset model to be used for long-term statistical arbitrage strategies. A key feature of the model is that all assets have \textit{co-integration}, which, if sustained, allows for long-term positive profits with low probability of losses. Optimal portfolios are found by solving a Hamilton-Jacobi-Bellman equation, to which we can introduce portfolio constraints such as market neutral or dollar neutral. Under specific conditions of the parameters, we can prove there is long-term stability for an optimal portfolio with stable growth rate. Historical prices of the S\&P500 constituents can be tested for co-integration and our model calibrated for analysis, from which we find that co-integration strategies require a terminal investment horizon sufficiently far into the future in order for the optimal portfolios to gain from co-integration. The data also demonstrates that statistical arbitrage portfolios will have improved in-sample Sharpe ratios compared to multivariate Merton portfolios, and that statistical arbitrage portfolios are naturally immune to market fluctuations.
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