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ENTROPY FLOW AND DE BRUIJN'S IDENTITY FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

Published online by Cambridge University Press:  17 December 2019

Michael C.H. Choi Open the ORCID record for Michael C.H. Choi [Opens in a new window] ,
Chihoon Lee  and
Jian Song Open the ORCID record for Jian Song [Opens in a new window]
Michael C.H. Choi
Affiliation:
Institute for Data and Decision Analytics, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, P.R. China and Shenzhen Institute of Artificial Intelligence and Robotics for Society E-mail: [email protected]
Chihoon Lee
Affiliation:
School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, USA and Institute for Data and Decision Analytics, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, P.R. China E-mail: [email protected]
Jian Song
Affiliation:
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, Shandong, 266237, China and School of Mathematics, Shandong University, Jinan, Shandong, 250100, China E-mail: [email protected]
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Abstract

Motivated by the classical De Bruijn's identity for the additive Gaussian noise channel, in this paper we consider a generalized setting where the channel is modelled via stochastic differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (0, 1). We derive generalized De Bruijn's identity for Shannon entropy and Kullback–Leibler divergence by means of Itô's formula, and present two applications. In the first application we demonstrate its equivalence with Stein's identity for Gaussian distributions, while in the second application, we show that for H ∈ (0, 1/2], the entropy power is concave in time while for H ∈ (1/2, 1) it is convex in time when the initial distribution is Gaussian. Compared with the classical case of H = 1/2, the time parameter plays an interesting and significant role in the analysis of these quantities.

Keywords

de bruijn's identityentropy powerfokker–planck equationfractional brownian motion
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 369 - 380
Copyright
© Cambridge University Press 2019

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