Rigorous Multi-Asset Optimal Execution With Bayesian Learning of the Drift

Liquidity issues have been increasingly addressed recently, especially with regards to optimal execution of large orders. In practice, agents are uncertain about the future dynamics of the assets, and face the risk of model misspecification, which could make the execution strategies non optimal. In this paper, we address the problem of uncertainty faced by an agent wishing to execute large orders on multiple assets. The agent only knows the distribution of the future drift of the assets composing her portfolio. We provide a solution to the problem of portfolio optimal execution where prices have drifted Bachelier dynamics with an unknown drift. The agent uses Bayesian learning to update her estimation of the drift, while she maximizes the expected exponential utility of her final wealth. We consider the specific case where the prior is Gaussian, and the costs are quadratic. We use stochastic optimal control tools to show how the problem simplifies into a matrix Riccati ODE with time-dependent coefficients, and use a-priori estimates to solve the control problem