Convergence of jump processes with stochastic intensity to Brownian motion with inert drift

TL;DR

This paper proves that a class of jump processes with stochastic, state-dependent intensities converges to Brownian motion with inert drift under scaling, confirming a conjecture in the one-dimensional case.

Contribution

It establishes the convergence of jump processes with stochastic intensities to Brownian motion with inert drift, extending previous theoretical results.

Findings

Convergence of jump processes to BMID under scaling

Confirmation of Burdzy and White's conjecture in 1D

Extension of BMID theory to stochastic intensity processes

Abstract

Consider a random walker on the nonnegative lattice, moving in continuous time, whose positive transition intensity is proportional to the time the walker spends at the origin. In this way, the walker is a jump process with a stochastic and adapted jump intensity. We show that, upon Brownian scaling, the sequence of such processes converges to Brownian motion with inert drift (BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in 2007. This confirms a conjecture of Burdzy and White in 2008 in the one-dimensional setting.