Anzeige
Erschienen in:
2018 | OriginalPaper | Buchkapitel
4. Other Theories of Stochastic Integration
verfasst von : Daniel W. Stroock
Erschienen in: Elements of Stochastic Calculus and Analysis
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
Doob’s presentation of Itô’s theory in his book (Doob, Stochastic Processes, 1953, [3]) indicates that he understood that one can apply Itô’s ideas to any continuous, square integrable martingale \(\bigl (M(t),\mathcal {F}_t,{\mathbb {P}}\bigr )\) for which one knows that there is a progressively measurable function \(t\rightsquigarrow A(t)\) which is continuous and non-decreasing in t and for which \(\bigl (M(t)^2-A(t),\mathcal {F}_t,{\mathbb {P}}\bigr )\) is a martingale. At the time, Doob did not know what is now called the Doob–Meyer decomposition theorem, a special case of which guarantees that such an \(A({{}\cdot {}})\) always exists. In this chapter, we will first prove this existence result and then, following Kunita and Watanabe (cf. Kunita and Watanabe, Nagoya Math J 30:209–245, 1967, [10]), develop a version of the theory that Doob had in mind.