Convergence en loi de semimartingales et variation quadratique. (French) Zbl 0458.60037

Jacod, Jean

Seminaire de probabilites XV, Univ. Strasbourg 1979/80, Lect. Notes Math. 850, 547-560 (1981).

The main result is the following one: if a sequence \( \left(\mathrm{X}^{n}\right) \) of local martingales, whose jumps are uniformly bounded by a constant, converges in distribution to a process \( X \), then ( \( i \) ) the process \( X \) admits a quadratic variation process, and (ii) this quadratic variation process is the limit in distribution of the quadratic variation processes of the \( \mathrm{X}^{\mathrm{n}} \)’s. A similar result is proved when the \( \mathrm{X}^{\mathrm{n}} \)’s are semimartingales, under an additional assumption: namely, that the variations of the ”predictable with finite-variation part” of \( \mathrm{X}^{\mathrm{n}} \) stay bounded (in \( \mathrm{n} \) ) in measure. If this assumption is not satisfied, a counter-example is given.

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MSC:

60G44	Martingales with continuous parameter
60G48	Generalizations of martingales

Keywords:

quadratic variation; weak convergence of semimartingales; local martingales

Citations:

Zbl 0447.00009