  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 execution risk and the market's
liquidity and uncertainty. We model the market's liquidity and uncertainty by
the principle of least relative entropy associated with the market volume. 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 volume. Numerical examples illustrating the optimal
strategies and the comparisons with conventional trading strategies are
conducted.
