Doob–Meyer decomposition theorem

Doob–Meyer decomposition theorem The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. It is named for Joseph L. Doob and Paul-André Meyer.

Contents 1 History 2 Class D supermartingales 3 The theorem 4 See also 5 Notes 6 References History In 1953, Doob published the Doob decomposition theorem which gives a unique decomposition for certain discrete time martingales.[1] He conjectured a continuous time version of the theorem and in two publications in 1962 and 1963 Paul-André Meyer proved such a theorem, which became known as the Doob-Meyer decomposition.[2][3] In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.[4] Class D supermartingales A càdlàg supermartingale {displaystyle Z} is of Class D if {displaystyle Z_{0}=0} and the collection {displaystyle {Z_{T}mid T{text{ a finite-valued stopping time}}}} is uniformly integrable.[5] The theorem Let {displaystyle Z} be a cadlag supermartingale of class D. Then there exists a unique, increasing, predictable process {displaystyle A} with {displaystyle A_{0}=0} such that {displaystyle M_{t}=Z_{t}+A_{t}} is a uniformly integrable martingale.[5] See also Doob decomposition theorem Notes ^ Doob 1953 ^ Meyer 1952 ^ Meyer 1963 ^ Protter 2005 ^ Jump up to: a b Protter (2005) References Doob, J. L. (1953). Stochastic Processes. Wiley. Meyer, Paul-André (1962). "A Decomposition theorem for supermartingales". Illinois Journal of Mathematics. 6 (2): 193–205. Meyer, Paul-André (1963). "Decomposition of Supermartingales: the Uniqueness Theorem". Illinois Journal of Mathematics. 7 (1): 1–17. Protter, Philip (2005). Stochastic Integration and Differential Equations. Springer-Verlag. pp. 107–113. ISBN 3-540-00313-4. Categories: Martingale theoryTheorems in statisticsProbability theorems

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