Calcul Stochastique et Problèmes de Martingales by Jean Jacod (auth.)

By Jean Jacod (auth.)

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Extra resources for Calcul Stochastique et Problèmes de Martingales

Sample text

2) ~q : {MEMloc: ~IMllHq~co} et, si l'on veut sp@cifier par rapport A quelle probabilit& ou A quelle filtration on se place, on @crit Hq(P) ou Hq(FP,P) . llHq. 19)). 4) Si M ~ ~loc et si le processus H~=loc (remarquons que si ~M AM est localement born@, on a est localement born@, la variable Me AM 0 =M 0 est born@e). 5) IrMcollco IJM~II ~< q _ ~ q = IIMCOIICO ~IMcollq si qE]1,co[ si q : CO (la premi@re in@galit@, et l'@galit@ corresponda~,t au cas q=co, sont triviales; la seconde in@galit@ est l'in@galit@ de Doob: voir Doob [~] ou Meyer [I] V T 22).

7) que l'ensemble o@ A([)C [T~oo} . _ E ( ( M o o - M T _ ) 2 1 F T ) >~ (c + £ ) 2 la d 6 f i n i t i o n E((Moo -MTA(£ )_)21__FT)>. (c + £ ) 2 p ( T A ( ~ ) < o o ) A([)c [T1 , . Alors de Banach) est d@coule de Si Hq dense que Supposons D'apr@s toute q =I dense . 35) que H= ~ est pour montre bien qu'on H= I .

De plus IAYI<. < n + a). Par suite Tn= Y e =p S . fi- 31 Y =M +A Soit comme hag { = AA. Mais la d @ c o m p o s i t i o n et IAYI{a, AM = AY- AA v@rifie Cette D@mo~stration. 15) moins importantes, a =I/2 . on a M' = M a ( v ,= e-A (Tn) sur la c a r a c t @ r e d ' a b o r d que ces e s p a c e s de L= , V= et ~p= (~p)loc" une suite que (no~ n @ c e s s a i r e m e n t sUP(n) X 6(Sp)lo c et Tn ~ ~ (T n) . La d @ c o m p o s i t i o n et Ae ~N~loc sont ~loc " X e S= . n X =M + A canonique (An+l) Tn = A n ; tels que crols- et que pour tout XI[o, Tn [ : Y ( n ) I [ O , T n [ .

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