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January 2007 Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes
Rosanna Coviello, Francesco Russo
Ann. Probab. 35(1): 255-308 (January 2007). DOI: 10.1214/009117906000000566


In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb{F}$-semimartingale M and a finite cubic variation process ξ which has the structure Q+R, where Q is a finite quadratic variation process and R is strongly predictable in some technical sense: that condition implies, in particular, that R is weak Dirichlet, and it is fulfilled, for instance, when R is independent of M. The method is based on a transformation which reduces the diffusion coefficient multiplying ξ to 1. We use generalized Itô and Itô–Wentzell type formulae. A similar method allows us to discuss existence and uniqueness theorem when ξ is a Hölder continuous process and σ is only Hölder in space. Using an Itô formula for reversible semimartingales, we also show existence of a solution when ξ is a Brownian motion and σ is only continuous.


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Rosanna Coviello. Francesco Russo. "Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes." Ann. Probab. 35 (1) 255 - 308, January 2007.


Published: January 2007
First available in Project Euclid: 19 March 2007

zbMATH: 1127.60055
MathSciNet: MR2303950
Digital Object Identifier: 10.1214/009117906000000566

Primary: 60H05 , 60H10
Secondary: 60G18 , 60G20

Keywords: Finite cubic variation , fractional Brownian motion , Hölder processes , Itô–Wentzell formula , Stochastic differential equation , weak Dirichlet processes

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 1 • January 2007
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