Mathematical Formulation of an Optimal Execution Problem with Uncertain Market Impact

TL;DR

This paper develops a realistic model for optimal trade execution considering uncertain market impact, deriving a continuous-time stochastic control framework involving Levy processes.

Contribution

It introduces a novel stochastic market impact model combining deterministic and noise components, and derives a continuous-time limit as a stochastic control problem.

Findings

Continuous-time value function characterized by Levy process

Discrete-to-continuous model derivation

Enhanced realism in market impact modeling

Abstract

We study an optimal execution problem with uncertain market impact to derive a more realistic market model. We construct a discrete-time model as a value function for optimal execution. Market impact is formulated as the product of a deterministic part increasing with execution volume and a positive stochastic noise part. Then, we derive a continuous-time model as a limit of a discrete-time value function. We find that the continuous-time value function is characterized by a stochastic control problem with a Levy process.