A model of optimal portfolio selection under liquidity risk and price impact

We study a financial model with one risk-free and one risky asset subject to liquidity risk and price impact. In this market, an investor may transfer funds between the two assets at any discrete time. Each purchase or sale policy decision affects the price of the risky asset and incurs some fixed transaction cost. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to a solvency constraint. This is formulated as an impulse control problem under state constraint and we prove that the value function is characterized as the unique constrained viscosity solution to the associated quasi-variational Hamilton-Jacobi-Bellman inequality.