Optimal control of martingales in a radially symmetric environment

TL;DR

This paper solves a stochastic control problem for multidimensional martingales in a radially symmetric setting, deriving explicit optimal strategies that switch between radial and tangential motions based on the process's radius.

Contribution

It provides an explicit solution and optimal strategy characterization for controlling multidimensional martingales with fixed quadratic variation in a symmetric environment.

Findings

Optimal strategies switch between radial and tangential motion based on radius.

Value function exhibits smooth fit at switching points.

Results extend to cost functions that are infinite at the origin.

Abstract

We study a stochastic control problem for continuous multidimensional martingales with fixed quadratic variation. In a radially symmetric environment, we are able to find an explicit solution to the control problem and find an optimal strategy. We show that it is optimal to switch between two strategies, depending only on the radius of the controlled process. The optimal strategies correspond to purely radial and purely tangential motion. It is notable that the value function exhibits smooth fit even when switching to tangential motion, where the radius of the optimal process is deterministic. Under sufficient regularity on the cost function, we prove optimality via viscosity solutions of a Hamilton-Jacobi-Bellman equation. We extend the results to cost functions that may become infinite at the origin. Extra care is required to solve the control problem in this case, since it is not clear how to define the optimal strategy with deterministic radius at the origin. Our results generalise some problems recently considered in Stochastic Portfolio Theory and Martingale Optimal Transport.