Relative entropy-regularized robust optimal order execution

The problem of order execution is cast as a relative entropy-regularized robust optimal control problem in this article. The order execution agent's goal is to maximize an objective functional associated with his profit-and-loss of trading and simultaneously minimize the inventory risk associated with market's liquidity and uncertainty. We model the market's liquidity and uncertainty by the principle of least relative entropy associated with the market trading rate. The problem of order execution is made into a relative entropy-regularized stochastic differential game. Standard argument of dynamic programming yields that the value function of the differential game satisfies a relative entropy-regularized Hamilton–Jacobi–Isaacs (rHJI) equation. Under the assumptions of linear-quadratic model with Gaussian prior, the rHJI equation reduces to a system of Riccati and linear differential equations. Further imposing constancy of the corresponding coefficients, the system of differential equations can be solved in closed form, resulting in analytical expressions for optimal strategy and trajectory as well as the posterior distribution of market trading rate. Numerical examples illustrating the optimal strategies and the comparisons with conventional trading strategies are conducted.