Asymptotic behavior of the transition density for jump type processes in small time
type: Item
instance of: scholarly article
title: Asymptotic behavior of the transition density for jump type processes in small time
zbMATH Open document ID: 0818.60074
author: Yasushi Ishikawa
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1995-08-07T00:00:00Z
review text: Let \(x_ t\) be the Markov process of pure jump type which satisfies the following stochastic differential equation \[ x_ t(x) = x + \sum_{s \leq t} \gamma \bigl( x_{s-} (x),\;\Delta z(s) \bigr), \] where \(z(t)\) is an \(R^ d\)-valued Lévy process of pure jump type with the Lévy measure \(h(d \zeta)\), \(\Delta z(t) = Z(t) - z(t-)\) and \(\gamma(x,\zeta)\) is a nondegenerate bounded function from \(R^ d \times R^ d\) to \(R^ d\). Under some assumptions on the Lévy measure \(h(d \zeta)\), this paper provides an estimate on the density \(p_ t (x,y)\) of the Markov process \(x_ t\) when the time parameter is small. More precisely, it is proven that as \(t \to 0\), \[ p_ t (x,y) \sim C \bigl( x,y, \alpha (x,y) \bigr) t^{\alpha (x,y)}, \] where \(\alpha (x,y)\) can be interpreted roughly as the minimum number of jumps by which the trajectory can reach \(y\) from \(x\) \((x \neq y)\). This result is a refinement of an earlier results by \textit{R. Léandre} [Sémin. probabilités XXI, Lect. Notes Math. 1247, 81-99 (1987; Zbl 0616.60078)]. This paper makes use of the theory of Malliavin calculus of jump type and the technique of Léandre's paper.
reviewed by: Renming Song
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 729872
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: Lévy process
zbMATH Keywords: Lévy measure
zbMATH Keywords: Malliavin calculus
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225674
OpenAlex ID: W1992963225
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DOI: 10.2748/TMJ/1178225674
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description: scientific article; zbMATH DE number 729872