An explicit solution of a non-linear quadratic constrained stochastic control problem with an application to optimal liquidation in dark pools with adverse selection

摘要

We study a constrained stochastic control problem with jumps; the jump timesof the controlled process are given by a Poisson process. The cost functionalcomprises quadratic components for an absolutely continuous control and thecontrolled process and an absolute value component for the control of the jumpsize of the process. We characterize the value function by a "polynomial" ofdegree two whose coefficients depend on the state of the system; thesecoefficients are given by a coupled system of ODEs. The problem hence reducesfrom solving the Hamilton Jacobi Bellman (HJB) equation (i.e., a PDE) tosolving an ODE whose solution is available in closed form. The state space isseparated by a time dependent boundary into a continuation region where theoptimal jump size of the controlled process is positive and a stopping regionwhere it is zero. We apply the optimization problem to a problem faced byinvestors in the financial market who have to liquidate a position in a riskyasset and have access to a dark pool with adverse selection.