Stochastic Integrals and Quadratic Variation

  • Olav Kallenberg
Part of the Probability and Its Applications book series (PIA)


Continuous local martingales and semimartingales; quadratic variation and covariation; existence and basic properties of the integral; integration by parts and Itô’s formula; Fisk-Stratonovich integral; approximation and uniqueness; random time-change; dependence on parameter


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  1. The first stochastic integral with a random integrand was defined by Ito (1942a, 1944), who used Brownian motion as the integrator and assumed the integrand to be product measurable and adapted. Doob (1953) noted the connection with martingale theory. A first version of the fundamental substitution rule was proved by Itô (1951a). The result was later extended by many authors. The compensated integral in Corollary 17.21 was introduced by Fisk, and independently by Stratonovich (1966).Google Scholar
  2. The existence of the quadratic variation process was originally deduced from the Doob-Meyer decomposition. Fisk (1966) showed how the quadratic variation can also be obtained directly from the process, as in Proposition 17.17. The present construction was inspired by Rogers and Williams (2000b). The Bdg inequalities were originally proved for p > 1 and discrete time by Burkholder (1966). Millar (1968) noted the extension to continuous martingales, in which context the further extension to arbitrary p 0 was obtained independently by Burkholder and Gundy (1970) and Novikov (1971). Kunita and Watanabe (1967) introduced the covariation of two martingales and proved the associated characterization of the integral. They further established some general inequalities related to Proposition 17.9.Google Scholar
  3. The Ito integral was extended to square-integrable martingales by Courrège (1962–63) and Kunita and Watanabe (1967) and to continuous semimartingales by Doléans-Dade and Meyer (1970). The idea of localization is due to Itô and Watanabe (1965). Theorem 17.24 was obtained by Kazamaki (1972) as part of a general theory of random time change. Stochastic integrals depending on a parameter were studied by Doléans (1967b) and Stricker and Yor (1978), and the functional representation of Proposition 17.26 first appeared in Kallenberg (1996a).Google Scholar
  4. Elementary introductions to Ito integration appear in many textbooks, such as Chung and Williams (1983) and 0ksendal (1998). For more advanced accounts and for further information, see Ikeda and Watanabe (1989), Rogers and Williams (2000b), Karatzas and Shreve (1991), and Revuz and Yor (1999).Google Scholar

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© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Olav Kallenberg
    • 1
  1. 1.Department of MathematicsAuburn UniversityAuburnUSA

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