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Approximation du crochet de certaines semimartingales continues. (French) Zbl 0539.60045

Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 144-147 (1984).
[For the entire collection see Zbl 0527.00020.]
It is known that if X is an \(L^ 2\)-semimartingale on a complete right continuous stochastic basis (\(\Omega\),\({\mathcal F},P,({\mathcal F}_ t)_{0\leq t\leq 1})\) then \(\sum_{\sigma}E((X_{t_{i+1}}-X_{t_ i})^ 2| F_{t_ i}|)\) tends to \(<X,X>_ t\) in \(L^ 1\) if the norm of the division \(\sigma(0=t_ 0<t_ 1<...<t_ n=t)\) converges to 0. The author gives the following technical generalization of this fact:
Theorem. Suppose that \(X_ t=W_ t+\int^{t}_{0}H_ sds\) where W is a Wiener process and H an \(L^ 2\)-bounded predictable process. Suppose also that there exists another \(L^ 2\)-bounded predictable process H’ such that \(\epsilon_ s(t)=E[((W_ s-W_ t)(s-t)^{-1}(H_ s-H_ t- H'\!_ t(W_ s-W_ t))| {\mathcal F}_ t]\) uniformly tends to 0 in \(L^ 1\) when s converges to t. Then \[ L^ 1-\lim_{n}(n/u)(\sum^{n- 1}_{i=0}E[(X_{u(i+1)/n}-X_{ui/n})^ 2| {\mathcal F}_{ui/n}])=\int^{u}_{0}H^ 2_ sds+\int^{u}_{0}H'\!_ sds. \]
Reviewer: G.Zbaganu

MSC:

60G44 Martingales with continuous parameter
60G48 Generalizations of martingales

Citations:

Zbl 0527.00020
Full Text: Numdam EuDML