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In recent years, academics, regulators, and market practitioners haveincreasingly addressed liquidity issues. Amongst the numerous problemsaddressed, the optimal execution of large orders is probably the one that hasattracted the most research works, mainly in the case of single-assetportfolios. In practice, however, optimal execution problems often involvelarge portfolios comprising numerous assets, and models should consequentlyaccount for risks at the portfolio level. In this paper, we address multi-assetoptimal execution in a model where prices have multivariate Ornstein-Uhlenbeckdynamics and where the agent maximizes the expected (exponential) utility ofher PnL. We use the tools of stochastic optimal control and simplify theinitial multidimensional Hamilton-Jacobi-Bellman equation into a system ofordinary differential equations (ODEs) involving a Matrix Riccati ODE for whichclassical existence theorems do not apply. By using \textit{a priori} estimatesobtained thanks to optimal control tools, we nevertheless prove an existenceand uniqueness result for the latter ODE, and then deduce a verificationtheorem that provides a rigorous solution to the execution problem. Usingexamples based on data from the foreign exchange and stock markets, weeventually illustrate our results and discuss their implications for bothoptimal execution and statistical arbitrage.

Multi-asset optimal execution and statistical arbitrage strategies under Ornstein-Uhlenbeck dynamics