Optimal liquidation under stochastic liquidity (Q1691443)

This paper deals with a model where temporary market imbalances involve their own stochasticity. The price impact is transient, it could be persistent but eventually vanishes over time, and it is nonlinear. The price process \(S=(S_{t})_{t\geq0}=(f(Y_{t})\bar S_{t})_{t\geq0}\) observed in the market deviates by the factor \(f(Y_{t})\) from the fundamental price \(\bar S_{t}\) that would be previal in the absence of large traders. The impact function \(f\) is positive and increasing. The stochastic impact process \(Y\) is of controlled Ornstein-Uhlenbeck type, namely it is driven by a Brownian motion and the large trader's holdings in the risky asset. In the proposed multiplicative model with transient impact the authors take the fundamental price \(\bar S\) to be an exponential Brownian motion and permit non-zero correlation with the stochastic volume effect process \(Y\). In this setup, the authors study the optimal liquidation problem for an infinite time horizon as a singular stochastic control problem of finite fuel type and construct its explicit solution. The main result gives the optimal strategy as the local time process of a diffusion reflected obliquely at a curved free boundary in \(\mathbb R^2\). The authors solve the singular control problem by explicitly constructing the value function as a classical solution of the HJB variational inequality.