Optimizing a variable-rate diffusion to hit an infinitesimal target at a set time

TL;DR

This paper investigates an optimal control problem for a time-changed Bessel process with bounded diffusion rates, aiming to maximize the probability of hitting an infinitesimal target at a fixed future time, revealing that the optimal exponent satisfies a transcendental equation.

Contribution

It introduces a novel stochastic optimization framework for variable-rate diffusion processes and characterizes the optimal hitting probability exponent via a transcendental equation.

Findings

Optimal exponent depends on diffusion rate ratio and process dimension.

The problem reduces to solving a transcendental equation.

Provides insights into controlling diffusion to maximize hitting probabilities.

Abstract

I consider a stochastic optimization problem for a time-changed Bessel process whose diffusion rate is constrained to be between two positive values $r_{1}<r_{2}$. The problem is to find an optimal adapted strategy for the choice of diffusion rate in order to maximize the chance of hitting an infinitesimal region around the origin at a set time in the future. More precisely, the parameter associated with "the chance of hitting the origin" is the exponent for a singularity induced at the origin of the final time probability density. I show that the optimal exponent solves a transcendental equation depending on the ratio $\frac{r_{2}}{r_{1}}$ and the dimension of the Bessel process.