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 [PSS11], 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.

We prove continuity of a controlled SDE solution in Skorokhod's M 1 and J 1 topologies and also uniformly, in probability, as a non-linear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that M 1 -continuity is the key to ensure that proceeds and wealth processes from (self-financing) càdlàg trading strategies are determined as the continuous extensions for those from continuous strategies. We demonstrate by examples how continuity properties are useful to solve different stochastic control problems on optimal liquidation and to identify asymptotically realizable proceeds.

We study a multiplicative transient price impact model for an illiquid financial market, where trading causes price impact which is multiplicative in relation to the current price, transient over time with finite rate of resilience, and non-linear in the order size. We construct explicit solutions for the optimal control and the value function of singular optimal control problems to maximize expected discounted proceeds from liquidating a given asset position. A free boundary problem, describing the optimal control, is solved for two variants of the problem where admissible controls are monotone or of bounded variation.Comment: To appear in Applied Mathematics and Optimization. Model assumptions relaxed; corrections and improvements on referees' suggestion

We show a probabilistic functional limit result for one-dimensional diffusion processes that are reflected at an elastic boundary which is a function of the reflection local time. Such processes are constructed as limits of a sequence of diffusions which are discretely reflected by small jumps at an elastic boundary, with reflection local times being approximated by ε-step processes. The construction yields the Laplace transform of the inverse local time for reflection. Processes and approximations of this type play a role in finite fuel problems of singular stochastic control.