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The resulting stochastic integrals are of two types: (i) deterministic integrands with a stochastic integrator (or random measure), and (ii) a general integration with both a stochastic integral and a stochastic integrator. This leads to the Wiener and Itˆ o type integrals respectively. Thus Young develops an extension of the classical Riemann-Stieltjes type integral relative to a ‘measure’ that has finite quadratic variation, extending the well-known case of finite (Vitali) variation. Then his second extension is designed to include the martingale integrators which do not necessarily have finite variations but have increments that are ‘nearly orthogonal’ and are termed near-martingales or nigh-martingales (and informally called by him also as “nightingales”).

Be measurable scalar functions and β : B1 × B2 → C be a bimeasure of finite Fr´echet variation (which is automatic if the Bi , i = 1, 2 are σ-algebras). Let h1 , h2 be β-integrable in the strict MT-sense, and let |fin | ≤ |hi |, i = 1, 2. If fin → fi as n → ∞ then the fin as well as the limits fi , i = 1, 2 are β-integrable in the same sense, and moreover for each A ∈ B1 , and B ∈ B2 , lim m,n→∞ (f1m (s1 ), f2m (s2 ))β(ds1 , ds2 ) = A B (f1 (s1 ), f2 (s2 ))β(ds1 , ds2 ) A B (19) holds, the order of the limits being immaterial.

If this sequence is Cauchy in X for each A ∈ A with limit αA (∈ X), then we set αA = f dZ = lim A n→∞ A fn dZ, A ∈ A. (2) 26 2 Second Order Random Measures and Representations This integral is called the Dunford-Schwartz (or D–S) integral and is uniquely defined in that it does not depend on the sequence {fn , n ≥ 1} approximating f. 4. 10). It is seen that f → A f dZ is additive for each A ∈ A and µf : A → A f dZ is a vector measure for each f for which the integral is defined. In particular, the set of Z-integrable functions forms a vector space over R or C, and contains all bounded measurable functions.

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