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Diffusions of multiplicative cascades

Alberts, Tom and Rifkind, Ben (2014) Diffusions of multiplicative cascades. Stochastic Processes and their Applications, 124 (2). pp. 1141-1169. ISSN 0304-4149. http://resolver.caltech.edu/CaltechAUTHORS:20140313-090648465

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Abstract

A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. Our process has a Markov property: at any given time it is a cascade of the process at any earlier time by random variables that are independent of the past. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. For Gaussian independent increment processes we develop the infinite-dimensional stochastic calculus that describes the evolution of the measure process, and use it to compute the optimal Hölder exponent in the Wasserstein distance on measures. We also discuss applications of this process to the model of tree polymers.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.spa.2013.10.007 DOIArticle
http://www.sciencedirect.com/science/article/pii/S0304414913002603PublisherArticle
http://arxiv.org/abs/1205.2413arXivDiscussion Paper
Additional Information:© 2013 Elsevier B.V. Received 26 May 2013; received in revised form 29 September 2013; accepted 27 October 2013; Available online 4 November 2013. We thank Bálint Virág for several helpful comments and suggestions, and Sourav Chatterjee for pointing out the connection with [5]. We also thank anonymous referees for careful reading of the paper and many helpful suggestions which led to a much improved presentation.
Subject Keywords:Measure-valued Markov process; Infinite dimensional stochastic calculus; Tree polymers
Classification Code:MSC: primary 82B27; 82B44; secondary 60G42
Record Number:CaltechAUTHORS:20140313-090648465
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20140313-090648465
Official Citation:Tom Alberts, Ben Rifkind, Diffusions of multiplicative cascades, Stochastic Processes and their Applications, Volume 124, Issue 2, February 2014, Pages 1141-1169, ISSN 0304-4149, http://dx.doi.org/10.1016/j.spa.2013.10.007. (http://www.sciencedirect.com/science/article/pii/S0304414913002603)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:44291
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:13 Mar 2014 22:12
Last Modified:13 Mar 2014 22:12

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