We analyze linear-quadratic (LQ) stochastic control problems that arise in optimal trade execution in models of Obizhaeva-Wang type. Extending previous literature, order book depth and resilience are both allowed to be stochastic processes. Moreover, the target position can be a random variable, and we can include a risk term with stochastic target process.
In discrete time, we tind via the dynamic programming principle that the optimal trade sizes and the minimal costs are characterized by a process Y, which is defined by backward recursion, and by, for general targets, a further process ψ. We moreover investigate properties of our model such as savings in the long-time horizon, existence of profitable round trips, and premature closure of the position.
In continuous time, we go beyond the usual finite-variation strategies, and present two approaches. In the first one, we set up and solve a relevant control problem where we consider càdlàg semimartingales as execution strategies, while in the second one, we start from a typical formulation for finite-variation strategies, extend this continuously to progressively measurable strategies, and solve the extended problem via reduction to a standard LQ stochastic control problem and subsequent application of relevant literature. The counterpart of the process Y from discrete time now is the solution of a quadratic backward stochastic differential equation (BSDE), and ψ becomes the solution of a linear BSDE. It turns out that optimal strategies indeed can have infinite variation.
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