Semimartingales on Rays, Walsh Diffusions, and Related Problems of Control and Stopping

TL;DR

This paper introduces and analyzes Walsh diffusions, a class of planar processes with a change-of-variable formula, exploring their behavior, explosion criteria, and applications to optimal control and stopping problems.

Contribution

It develops a new class of processes called semimartingales on rays, establishes existence and uniqueness results for Walsh diffusions, and applies these to control and stopping problems.

Findings

Established existence and uniqueness of Walsh diffusions.

Developed criteria for explosion in finite time.

Applied results to optimal control and stopping strategies.

Abstract

We introduce a class of continuous planar processes, called "semimartingales on rays", and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such planar processes are diffusions which choose, once they reach the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results up to an explosion time for these "Walsh diffusions", study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of control with discretionary stopping involving Walsh diffusions.