New PDF release: Applications of Variational Inequalities in Stochastic

By Alain Bensoussan

ISBN-10: 0444863583

ISBN-13: 9780444863584

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A E '3 ; we have Let S,T be two stopping times such t h a t S 3s c 3* such t h a t S 2 T ; . 5 T; we have T {S < T ] , (S E T), {S > T ] E 3' and 3 S I f < , n a r e two R"-valued R . V . ' s such t h a t 5 i s 3 -measurable and n i s Let S,T be two stopping t i m e s ; aT . 3T-measurable, then {< = ql n {ST] ~ 3 'and Let T be a sequence of stopping times; Tn 4 9 , then T i s a stopping time. that f o r a l l t , i s complete ( i . e . it contains a l l t h e events of p r o b a b i l i t y z e r o ) .

Actually a function X(t;w). X(t;w) We interpret t as the time, which thus varies in a continuous fashion. We shall also consider the discrete-time case (the transpositions are straightforward). ,P',x'(t)) Suppose we have ( Q , O , P , X ( t ) ) processes. 29). (SEC. s. A process i s s a i d t o be continuous ( r i g h t continuous, Left continuous) when a . s . i t s t r a j e c t o r i e s s a t i s f y t h i s property. t Let 3 be an i n c r e a s i n g family of sub-o-algebras s s t - $ c of a, i .

3 4 ) sup Wst ]CN(t) - <(t)l I: 0 (in probability). 39) a. s . a -. a N b N -. b SUP GS41T lCN(t) in L’(0,T) in L2(0,T) - Sit) I - 0 . Using t h e p r o p e r t i e s of Y and t h e f a c t t h a t ,€ i s continuous, w e e a s i l y deduce by proceeding t o t h e l i m i t t h a t ( 2 . 3 8 ) holds w i t h a , b We can t h u s assume t h a t . (“I w ( t ) has v a l u e s i n Rm, where m can be f n , f o r g e n e r a l i t y . 40 (CHAP. 38) in the case t = T. The proof is the same for all t. s.

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Applications of Variational Inequalities in Stochastic Control by Alain Bensoussan


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