A parallel between Brownian bridges and gamma bridges
type: Item
instance of: scholarly article
title: A parallel between Brownian bridges and gamma bridges
zbMATH Open document ID: 1074.60054
author: Michel Émery
author: Marc Yor
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2005-04-04T00:00:00Z
review text: Let \(\left\{ \gamma _{t},t\geq 0\right\} \) be the Gamma processes with natural filtration \(G\). Then \(\{a\gamma_u/ \gamma_t,u\in[0,t]\} \) is the Gamma bridge starting at zero and ending at \(a\) with the filtration \[ G_t^*\equiv \{\sigma (\gamma_u/ \gamma _t;\;u\in[0,t])\} . \] It is proved that the Gamma filtration \(G_{t}^{\ast }\) is generalized by the Gamma process \[ \gamma _{t}^{\ast }=\sum\limits_{s\in \left( 0,t\right] }u_{s}(\Delta \gamma _{s}/\gamma _{s-}) \] where \(u_{s}\left( x\right) \) is the function implicitly defined by \[ \int\limits_{u_{s}\left( x\right) }^{\infty }\frac{e^{-z}}{z} dz=\int\limits_{x}^{\infty }\frac{dy}{y\left( 1+y\right) ^{s}},\quad x>0,~s>0. \] The harness property, the independence of \(G_{t}^{\ast }\) of \(\sigma _{t}^{+}\equiv \sigma \left\{ \gamma _{u},u\in \left[ t,\infty \right) \right\} \) and the homogenity of increments without independence of the process \( \ln \{\gamma _{1}/t\), \(t>0\} \) are proved emphasizing their similiarity to that of the Brownian bridge.
reviewed by: Nijolė Kalinauskaitė
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 2151098
zbMATH Keywords: jump processes
zbMATH Keywords: processes with independent increments
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145475488
OpenAlex ID: W4297423375
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DOI: 10.2977/PRIMS/1145475488
description: scientific article; zbMATH DE number 2151098