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 a continuous increasing process. It is named for J. 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 Z is of Class D if Z₀ = 0 and the collection

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

is uniformly integrable.^[5]

The theorem

Let Z be a cadlag supermartingale of class D with Z₀ = 0. Then there exists a unique, increasing, predictable process A with A₀ = 0 such that M_t = Z_t + A_t is a uniformly integrable martingale.^[6]

See also

Doob decomposition theorem

Notes

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

References

• Doob, J.L. (1953). Stochastic Processes. Wiley. 
• Meyer, Paul (1962). "A Decomposition theorem for supermartingales". Illinois Journal of Mathematics 6: 193–205. 
• Meyer, Paul (1963). "Decomposition of supermartingales: the uniqueness theorem". Illinois Journal of Mathematics 7: 1–17. 
• Protter, Philip (2005). Stochastic Integration and Differential Equations. Springer-Verlag. pp. 107–113. ISBN 3-540-00313-4. 

External links

• The Doob–Meyer decomposition theorem with proof, by Jan van Neerven