  We introduce a price impact model which accounts for finite market depth,
tightness and resilience. Its coupled bid- and ask-price dynamics induce convex
liquidity costs. We provide existence of an optimal solution to the classical
problem of maximizing expected utility from terminal liquidation wealth at a
finite planning horizon. In the specific case when market uncertainty is
generated by an arithmetic Brownian motion with drift and the investor exhibits
constant absolute risk aversion, we show that the resulting singular optimal
stochastic control problem readily reduces to a deterministic optimal tracking
problem of the optimal frictionless constant Merton portfolio in the presence
of convex costs. Rather than studying the associated Hamilton-Jacobi-Bellmann
PDE, we exploit convex analytic and calculus of variations techniques allowing
us to construct the solution explicitly and to describe the free boundaries of
the action- and non-action regions in the underlying state space. As expected,
it is optimal to trade towards the frictionless Merton position, taking into
account the initial bid-ask spread as well as the optimal liquidation of the
accrued position when approaching terminal time. It turns out that this leads
to a surprisingly rich phenomenology of possible trajectories for the optimal
share holdings.
