  We solve explicitly a two-dimensional singular control problem of finite fuel
type for infinite time horizon. The problem stems from the optimal liquidation
of an asset position in a financial market with multiplicative and transient
price impact. Liquidity is stochastic in that the volume effect process, which
determines the inter-temporal resilience of the market in spirit of Predoiu,
Shaikhet and Shreve (2011), is taken to be stochastic, being driven by own
random noise. The optimal control is obtained as the local time of a diffusion
process reflected at a non-constant free boundary. To solve the HJB variational
inequality and prove optimality, we need a combination of probabilistic
arguments and calculus of variations methods, involving Laplace transforms of
inverse local times for diffusions reflected at elastic boundaries.
