Optimization of stochastic systems by Aoki M.

By Aoki M.

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W e SI. 14. If Y is a a(X)-measurable random variable then there exists a measurable function g :ffi—> M. s. 6 Martingales and Stopping Times We compile a list of results from martingale theory and refer for proofs to standard references including H. M. Dudley [90], St. Ethier and Th. Kurtz [98], J. Neveu [276], Ph. Protter [292], D. Revuz and M. Yor [299], Chr. Rogers and D. Williams [301] and D. Williams [371]. ) Martingales had been introduced by J. L. 4. Still a good reference work for martingale theory is of course the monograph [88] of J.

Kurtz [98] or in D. Revuz and M. Yor [299]. But it seems worth to make a few remarks to the proofs. Suppose that (Xt)t>o is a sub-martingale. s. 3 on cadlag-functions. Once the existence of these limits are proved we can construct the cadlag-modification using these limits. 12. Let Z be an integrable random variable and let (Tt)t>o be any filtration. 126) as s —> t+. e. Ft0)-martingale. Clearly, enlarging the filtration may destroy the martingale property. However we have, compare D. Revuz and M.

Let (XJ)J€I, Xj : Cl —> flj, be a family of random variables on (Ct,A,P), where (Clj,Aj) is a measurable space. Further let for eachj € / a measurable mapping Yj : Clj —> Cl'j be given where (QpA'A is a further measur able space. If the family (XJ)J^I is independent then the family (Yj o Xj) j e / is independent. , m let Xj : Q —> fij, (Clj, Aj) being a measurable space, be a random variable. , Px1®-0Xm on ®™=1Aj. 8. 74) Pxr<»-

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