Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach

TL;DR

This paper derives asymptotic lower bounds for optimal target tracking problems modeled by Itô semi-martingales, using a linear programming approach to relate control costs to Brownian motion control.

Contribution

It introduces a novel linear programming framework to establish lower bounds for tracking problems involving stochastic target dynamics.

Findings

Lower bounds depend on the cost structure.

Explicit expressions for bounds are provided in various examples.

The approach links stochastic control to deterministic linear programming.

Abstract

We consider the problem of tracking a target whose dynamics is modeled by a continuous It\=o semi-martingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control of Brownian motion, which is characterized as a deterministic linear programming problem. A comprehensive list of examples with explicit expressions for the lower bounds is provided.