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

TL;DR

This paper provides a closed-form solution to a complex stochastic control problem with jumps, applying it to optimize asset liquidation in dark pools with adverse selection, simplifying the solution process from PDEs to ODEs.

Contribution

It introduces a polynomial-based characterization of the value function, reducing the HJB PDE to a solvable system of ODEs, and applies this to financial liquidation problems.

Findings

Closed-form solution for the control problem using ODEs

Effective separation of the state space into continuation and stopping regions

Application to optimal liquidation in dark pools with adverse selection

Abstract

We study a constrained stochastic control problem with jumps; the jump times of the controlled process are given by a Poisson process. The cost functional comprises quadratic components for an absolutely continuous control and the controlled process and an absolute value component for the control of the jump size of the process. We characterize the value function by a "polynomial" of degree two whose coefficients depend on the state of the system; these coefficients are given by a coupled system of ODEs. The problem hence reduces from solving the Hamilton Jacobi Bellman (HJB) equation (i.e., a PDE) to solving an ODE whose solution is available in closed form. The state space is separated by a time dependent boundary into a continuation region where the optimal jump size of the controlled process is positive and a stopping region where it is zero. We apply the optimization problem to a problem faced by investors in the financial market who have to liquidate a position in a risky asset and have access to a dark pool with adverse selection.