Doob–Meyer decomposition theorem

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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 submartingale $Z$ is of Class D if $Z_0=0$ and the collection

$\{Z_T \mid \text{T a finite valued stopping time} \}$

The theorem

Let $Z$ be a cadlag submartingale of class D with $Z_0 =0$. Then there exists a unique, increasing, predictable process $A$ with $A_0 =0$ such that $M_t = Z_t - A_t$ is a uniformly integrable martingale.[5]

Notes

1. ^ Doob 1953
2. ^ Meyer 1952
3. ^ Meyer 1963
4. ^ Protter 2005
5. ^ 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.