Stratonovich stochastic differential equations driven by general semimartingales
type: Item
instance of: scholarly article
title: Stratonovich stochastic differential equations driven by general semimartingales
zbMATH Open document ID: 0823.60046
author: Thomas G. Kurtz
author: Etienne Pardoux
author: Philip Protter
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1995-04-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77513
review text: Let \(Z\) be a semimartingale, possibly with jumps, and \(f \in C^ 1\). The s.d.e. \[ X_ t = X_ 0 + \int^ t_ 0 f(X_ s) \circ dZ_ s \tag{1} \] is understood as \[ \begin{multlined} X_ t = X_ 0 + \int^ t_ 0 f(X_{s^ -})dZ_ s + {1\over 2} \int^ t_ 0 f'f(X_ s) d[Z, Z]^ c_ s +\\ + \sum_{0 < s \leq t} \{\varphi (f \Delta Z_ s, X_{s^ -}) - X_{s^ -} - f(X_{s^ -})\Delta Z_ s\}, \end{multlined}\tag{2} \] where the first integral is the Itô integral, \([Z, Z]^ c\) is the continuous part of the quadratic variation of \(Z\) and \(\varphi(g,x)\) is the solution at \(u = 1\) of the o.d.e. \({dy\over du} (u) = g(y(u))\), \(y(0) = x\). There is no known definition of a stochastic integral leading to this interpretation. However, if \(X\) satisfies (2) for some \(Z\) and \(f\), then \(\int^ t_ 0 g(X_ s) \circ dZ_ s\) \((g \in C^ 1)\) can be consistently defined and a change of variable formula holds: \[ \Psi \in C^ 2 \Rightarrow \Psi(X_ t) = \Psi(X_ 0) + \int^ t_ 0 \Psi'(X_ s) f(X_ s) \circ dZ_ s. \] It is also shown that: a) under Lipschitz conditions, there exists a càdlàg solution to (1), which is unique and is a semimartingale; b) the flow \(x \mapsto X_ t(x, \omega)\) is a diffeomorphism if \(f\) is \(C^ \infty\) and all derivatives of \(f\) and \(f'f\) are bounded; c) if \(Z\) is a Lévy process, \(X_ 0\) and \(Z\) are independent and \(f\), \(f'f\) are Lipschitz, the solution has the strong Markov property. The main result is the following: If \(Z\) is substituted by the continuous process \({1\over h} \int^ t_{t-h}Z_ s ds\), \(X^ h\) is the corresponding solution, and \(\gamma_ h\) is a suitably defined time change, the process \(Y^ h_ t := X^ h_{\gamma^{-1}_ h(t)}\) converges in the u.c.p. topology as \(h \to 0\) to a process \(Y\) such that \(X_ t = Y_{\gamma_ 0(t)}\). And for countably many \(t > 0\), \(X^ h_ t \to X_ t\) in probability. This approximation result justifies the interpretation (2) from the modeling viewpoint, since the approximants of \(Z\) are continuous proc…
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 740313
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: semimartingale
zbMATH Keywords: change of variable formula holds
zbMATH Keywords: Stratonovich equation
reviewed by: Aureli Alabert
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 740313