By Gordon Wassermann

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5in 14 RozVol N. V. Krylov and B. L. Rozovskii for all i, j and any basis {hi }, where dP × d x ∞ µ(A) = E is the differential of the measure t χA (t, ω)d x t , 0 defined on the product of F and the Borel σ-algebra on [0, ∞). We call the process Qx the correlation operator of x. ) for any t ≥ 0, then there exists a square-integrable martingale y(t) in E which is strongly continuous in t and such that, for every orthonormal basis {hi } and every v ∈ E, T ≥ 0, n lim E sup v y(t) − n→∞ t≤T 2 t vB(s)hi d(hi x(s)) = 0.

There are two groups of these “selected” questions: a) construction of a stochastic integral over a square-integrable martingale with values in a Hilbert space; b) derivation of Itˆ o’s formula for the square of the norm of a semimartinagale is a rigged Hilbert space. 2 we consider questions of group a) in a very simple situation — the integration with respect to a continuous martingale. This section is entirely of survey character and there are practically no proofs given. It is based on Refs.

Stochastic integrals for functions in LQx (H, E) have previously been defined in Ref. [32], but the construction presented here is somewhat different. 1). ) t Ay(t) = AB(s)dx(s). 3) 0 We choose an element e ∈ E and by means of it define an operator e ∈ L(E, R) by the formula ey = ey, where ey is the scalar product in E. 3) we then have t ey(t) = eB(s)dx(s). 0 1/2 We observe that the operator eB(s) acts on Qx H by the formula h → eB(s)h, while the latter is equal to (B ∗ e, h) if B ∈ L(H, E). ), then we write t t h(s)dx(s) = 0 h(s)dx(s).