Caractérisation des semimartingales. (English) Zbl 0539.60046

Stricker, C.

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

Citations:

Zbl 0527.00020