## Caractérisation des semimartingales.(English)Zbl 0539.60046

Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 148-153 (1984).
[For the entire collection see Zbl 0527.00020.]
Let ($$\Omega$$,$${\mathcal F},P,({\mathcal F}_ t|_{0\leq t\leq 1})$$ be a complete right-continuous stochastic basis and X be an adapted process. If X can be written in the form $$X=M+A$$, where M is a martingale and A has finite variation, then X is called a semimartingale. If $$M_ 1$$ and Var(A) belong to $$L^ 2$$, then X is named an $$L^ 2$$-semimartingale.
The paper solves the following problem: for a given X, when can one exchange the probability P with an equivalent one, say Q, such that X is an $$L^ 2$$-semimartingale? Let $$H=\{\sum^{n}_{i=1}a_ i(X_{t_{i+1}}-X_{t_ i});n\geq 1,0=t_ 0<t_ 1<...<t_ n=1,a_ i\in L^{\infty}({\mathcal F}_{t_ i}),-1\leq a_ i\leq 1\}.$$ The main result is the following: The answer to the question is ”yes” iff H is bounded in $$L^ 0({\mathcal F})$$.
Reviewer: G.Zbaganu

### MSC:

 60G44 Martingales with continuous parameter 60G48 Generalizations of martingales 60H05 Stochastic integrals

### Keywords:

predictable processes; semimartingale

Zbl 0527.00020
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