Drifted Brownian motions governed by fractional tempered derivatives

TL;DR

This paper introduces fractional differential equations with tempered derivatives to model the distribution of drifted Brownian motions with reflection, providing new mathematical tools and representations for these stochastic processes.

Contribution

It presents novel fractional equations using tempered Riemann–Liouville derivatives and explores their properties, including Marchaud-type forms and Fourier transforms, for the first time in this context.

Findings

Derived fractional equations for reflecting drifted Brownian motions.

Established Marchaud-type forms for tempered derivatives.

Analyzed Fourier transforms of Riesz tempered fractional derivatives.

Abstract

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann--Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.