On oscillating sticky Brownian motion

TL;DR

This paper introduces a new diffusion process called oscillating sticky Brownian motion, combining stickiness and oscillation, with explicit formulas for its stochastic properties and a construction method from two Brownian motions.

Contribution

It defines and analyzes a novel diffusion process with explicit stochastic differential equations, density functions, and a construction from Brownian motions, advancing understanding of complex stochastic behaviors.

Findings

Explicit stochastic differential equation, resolvent, and semigroup for the process

Closed-form trivariate density of position, local time, and occupation time

Construction method from two Brownian motions with drift and scaling

Abstract

Starting with a Brownian motion, we define and study a novel diffusion process by combining stickiness and oscillation properties. The associated stochastic differential equation, resolvent and semigroup are provided. Also the trivariate density of position, local time and occupation time of this diffusion is obtained explicitly. Furthermore, we give a construction of two Brownian motions with drift and scaling whose difference is an oscillating sticky Brownian motion, up to a multiplicative constant.