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.
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 [math]\displaystyle{ Z }[/math] is of Class D if [math]\displaystyle{ Z_0=0 }[/math] and the collection
- [math]\displaystyle{ \{Z_T \mid T \text{ a finite-valued stopping time} \} }[/math]
The theorem
Let [math]\displaystyle{ Z }[/math] be a cadlag supermartingale of class D. Then there exists a unique, non-decreasing, predictable process [math]\displaystyle{ A }[/math] with [math]\displaystyle{ A_0 =0 }[/math] such that [math]\displaystyle{ M_t = Z_t + A_t }[/math] is a uniformly integrable martingale.[5]
See also
Notes
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. doi:10.1215/ijm/1255632318.
- Meyer, Paul-André (1963). "Decomposition of Supermartingales: the Uniqueness Theorem". Illinois Journal of Mathematics 7 (1): 1–17. doi:10.1215/ijm/1255637477.
- Protter, Philip (2005). Stochastic Integration and Differential Equations. Springer-Verlag. pp. 107–113. ISBN 3-540-00313-4. https://archive.org/details/stochasticintegr00prot_960.
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