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Continuity of the Îto stochastic integral in Hilbert spaces. (English) Zbl 0868.60047

The article is devoted to the convergence of the stochastic integrals \(\int K^n_-dX^n\), where for each \(n\), \(X^n\) is a Hilbert-valued semimartingale with respect to the stochastic basis \[ (\Omega^n,F^n,(F^n_t)_{t\in[0;1]},\;P^n) \] and \(K^n\) is adapted to the corresponding stochastic flow random process, also Hilbert-valued. “Dot” means scalar or tensor product, or an action of Hilbert-Schmidt operator. The author supposes that \(K^n\) and \(X^n\) converge in the Skorokhod topology to \(K\), \(X\), resp. Sufficient conditions for \(X\) to be a semimartingale and for the convergence of stochastic integrals to \(\int K_-\cdot dX\) are given. The main role is played by the so-called UT condition, which in one-dimensional case means that the family of stochastic integrals \(\int H^n_-\cdot dX^n\) with the step integrands (with the random moments of jumps) bounded by 1 is uniformly tight. In the infinite-dimensional case it is enough to suppose that UT condition holds only for the scalar product integrals. A new infinite-dimensional problem, which is solved by the author is a checking of the “flat concentration” in this situation. As a consequence of the results a statement about the convergence of quadratic variations is obtained.

MSC:

60H05 Stochastic integrals
60F17 Functional limit theorems; invariance principles
60B11 Probability theory on linear topological spaces
60G48 Generalizations of martingales
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