×

On Métivier-Pellaumail inequality, Emery topology and pathwise formulae in stochastic calculus. (English) Zbl 0722.60048

This paper addresses the question how to formulate convergence of semimartingales and their integrals in an efficient manner so as to arrive simply at a.s. convergent approximations of solutions of SDE etc.
From the introduction: We introduce a notion of a dominating process of a semimartingale, which is a modification of the notion of a control process introduced by M. Métivier [Semimartingales: A course on stochastic processes (1982; Zbl 0503.60054)]. The rest of the paper tries to establish that the dominating process is a very useful tool in the study of stochastic integrals and SDE’s. The claim is that a dominating process does for a semimartingale what the total variation process does for a process with bounded variation paths. After establishing elementary properties, we give a new metric for the Emery topology and then introduce the notion of fast convergence in the Emery topology, which we denote by \(\to^{*}\). We then prove stability of solution to SDE’s in the sense of \(\to^{*}\) which implies stability results in Emery topology. This notion of \(\to^{*}\) is particularly useful for pathwise formulae for stochastic integrals, solutions to SDE’s, multiplicative integrals, etc.

MSC:

60H05 Stochastic integrals
60G44 Martingales with continuous parameter
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G07 General theory of stochastic processes

Citations:

Zbl 0503.60054
PDFBibTeX XMLCite