Smoothness of the distribution of the supremum of a multi-dimensional diffusion process
type: Item
instance of: scholarly article
title: Smoothness of the distribution of the supremum of a multi-dimensional diffusion process
zbMATH Open document ID: 1271.60063
author: Masafumi Hayashi
author: Arturo Kohatsu-Higa
published in: Potential Analysis
publication date: 2013-02-15T00:00:00Z
review text: Consider a multi-dimensional diffusion process \((X_t^i;\;0\leq t\leq T,\,1\leq i\leq d)\) and assume that the associated vector fields are commutative. Under some assumption, one can prove that almost surely, the first component \(X^1\) attains its maximum \(F^1=\sup_{0\leq t\leq T}X_t^1=X_{\theta^1}^1\) at a unique random time \(\theta^1\). The authors apply Malliavin's calculus and prove that the random vector \(F=(X^i_{\theta^1};\;1\leq i\leq d)\) has a smooth density on \((X_0^1,\infty)\times\mathbb{R}^{d-1}\).
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 6136699
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: maximum process
zbMATH Keywords: stochastic differential equations
reviewed by: Jean Picard
MaRDI profile type: Publication
full work available at URL: s11118-011-9263-8
OpenAlex ID: W2165339062
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DOI: 10.1007/S11118-011-9263-8
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description: scientific article; zbMATH DE number 6136699