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Multiple stochastic integrals – a counter example. (English) Zbl 0564.60053

Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 258-262 (1985).
[For the entire collection see Zbl 0549.00007.]
The note gives an example of a continuous square integrable martingale M such that \(d<M,M>_ t<<dt\) but for which the multiple stochastic integral \(\iint_{\{0<s<t<\infty \}}f_{st}dM_ sdM_ t\) does not exist as an \(L^ 0\)-integrator on the space of bounded predictable integrands. The example shows, in fact, that the condition \(<M,M>_ t- <M,M>_ s\leq m(t)-m(s)\) for some deterministic m, is needed to consider multiple integrals as iterated integrals and to extend the multiple integral as \(L^ 0\)-integrator to bounded predictable processes.
Reviewer: Y.S.Mishura

MSC:

60H05 Stochastic integrals

Citations:

Zbl 0549.00007
Full Text: Numdam EuDML