By Alain Bensoussan

ISBN-10: 0444863583

ISBN-13: 9780444863584

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**Extra resources for Applications of Variational Inequalities in Stochastic Control**

**Example text**

2 I n t r o d u c t i o n of s t o c h a s t i c i n t e g r a l s t t , t E [OPT] and w ( t ) a continuous 3 martingale We t a k e an i n c r e a s i n g family a with v a l u e s i n Rn, zero at 0 and s a t i s f y i n g ( 2 . 9 ) ( * ) . 1 0) Lz(O,T) = (q(t;U) a . s . 1 1 ) <(O,T) i s measurable, adapted, scalar-valued, and < = ( q E L:(O,T) a] ,p2 s 1 / E :llp(t)lPdt - < + -1 We s i m i l a r l y d e f i n e L$(O,T;Rn) and #(0,T;Rn) f o r processes with values i n Rn. The space M $ ( o , T ; R ~ ) i s a Banach &space of Lp((O,T) x Q,dt Q dP;Rn).

11 vtl . e. V. i s n e c e s s a r i l y constant on B and on This i s t h u s expressed as follows CB. We then s e e t h a t ( i f P ( B ) lO,l[) E j J X dP B = xB(")P(B) + B' X dP CB l-P(B) /X dP The nwnber B is realised. I6) E@ i s 1 a l i n e a r operator from L (p,o,P) and c o n t r a c t i n g such t h a t #I = 1 . q E Il,m[ 1 1 , +-= P 9 1. 18) (JenSen's Inequality): a real-valued R . V . s. (SEC. 20) L1 ; t h e n . 20)extend t o R . V . ' s R1, b2 of We say t h a t two sub-a-algebras E aare i n Rn.

It would be p r e f e r a b l e t o w r i t e flt(O,T) 3 , but t h i s would be t o o cumbersome. 3L (CHAP. 12) 'p E = 'Pk(t) The sequence 'pk E 0 . that , we 0 I 1,; associate 'pk defined by ~ j ( ds)ds,nks ~ - ~ t) < ~ < t k . 2 'p in L (0,T;L2(Q,g,P;Rn)) Furthermore, it is clear Thus 'pk -. 'p in 9. 13) E 0 is a dense subspace in 0 . * < t * = 'p 6 there thus exists Q ; T v ( t ) = 'pn in [t,, tn+,[, n = 0 , t E 0 , 'p" must be 3 n measurable. N-1 . n denotes the inner product of two vectors E,n in R".

### Applications of Variational Inequalities in Stochastic Control by Alain Bensoussan

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