Infinitesimal behaviour of a continuous local martingale. (English) Zbl 0764.60046

Delbaen, Freddy

Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 398-404 (1992).

[For the entire collection see Zbl 0754.00008.]
The author considers continuous vector valued local martingales \({\mathbf M}_ t\) which have the property that, for components \(M^ i\) and \(M^ j\), one has \(\lim_{t\to 0}{1\over t}\langle M^ i,M^ j\rangle_ t=a_{i,j}\), \(a_{i,j}\) being the entry \((i,j)\) of the positive definite matrix \(A\). The author proves that the process \({\mathbf M}_{st}/\sqrt{t}\) converges in law to \(\sqrt{A}{\mathbf W}_ s\), as \(t\to 0\), \({\mathbf W}\) being a standard Wiener process. The result is then applied to Markov processes \(X\), \(M\) being a martingale of the form \(f(X_ t)- f(x)-\int_ 0^ t Af(X_ s)ds\), and \(A\) being the infinitesimal generator of the process.

Reviewer: A.F.Gualtierotti (Lausanne)

MSC:

60G44	Martingales with continuous parameter
60F05	Central limit and other weak theorems
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60B05	Probability measures on topological spaces

Keywords:

continuous vector valued local martingales; Wiener process; infinitesimal generator

Citations:

Zbl 0754.00008