Walsh Spider Diffusions as Time Changed Multi-parameter Processes

TL;DR

This paper introduces a novel pathwise construction of Walsh spider diffusions using time changes of multi-parameter processes, inspired by multi-armed bandit strategies, providing explicit equations for the time allocation on graph edges.

Contribution

It presents a new method to construct Walsh spider diffusions via explicit time changes of multi-parameter processes, linking stochastic processes on graphs with allocation strategies.

Findings

Unique time change associated with any infinitesimal generator on a star graph.

Explicit derivation of the time allocation equations for the process.

Connection between diffusion processes on graphs and multi-armed bandit inspired strategies.

Abstract

Inspired by allocation strategies in multi-armed bandit model, we propose a pathwise construction of Walsh spider diffusions. For any infinitesimal generator on a star shaped graph, there exists a unique time change associated with a multi-parameter process such that the time change of this multi-parameter process is the desired diffusion. The time change has an interpretation of time allocation of the process on each edge, and it can be derived explicitly from a family of equations.