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

**Example text**

Dxn where A is a symmetric positive-definite matrix. The characteristic function of the normal law is . [p(ul,. 29) (Abu) -2 . We have m E A = E(X-m)(X-m)* EX , and A is termed the covariance matrix ( * ) . To $onclyde, we give the following useful result: g : R +. 4 then for a t 0 we have suppose we have we obtain the BienaymQ-Tchebichev inequality. General discussion on stochastic processes Let ( Q , a , P ) be a probability space. A mapping t +. 's with values in Rn is termed a stochastic process with values in Rn.

The space L (P,d,P;R ) denotes t h e space of t h e (equiva ence c l a s s e s ) of R . V . ' s whose e s s e n t i a l upper bound i s f i n i t e ( i f n = 1 we w r i t e L ( Q , Q , P ) ) . 7) (*) f k -. f a . s . U o s t surely. (*I if{w:fk-. f j I (SEC. 1) PROBABILITY AND STOCHASTIC PROCESSES 25 N a t u r a l l y , we a l s o have a l l t h e c o n c e p t s of convergence i n t h e s p a c e s LP, l s p s m ( i n t h e s t r o n g , weak, s t a r s e n s e ) a l t h o u g h p r o b a b i l i s t s do n o t employ any s p e c i a l terminology i n t h i s c a s e ( e x c e p t p e r h a p s f o r s t r o n g convergence i n L2, which is sometimes termed convergence i n t h e q u a d r a t i c mean).

V . (random v a r i a b l e ) i f we have f-'(B) E0 , VB E B(E) . Let f . , i E I , be a family o f mappings from 62 -+ E. We denote by d f i , 1 E I) t h e s m a l i e s t o-algebra of p a r t i t i o n s of 0 , f o r which all t h e mappings f . a r e measu r a b l e , We c a l l 3(fi, i E I) t h e o-algebra generated by t h e f u n c t i o n s l f i . 's such t h a t fk(W) f(w) v i then f ( w ) i s a R . V . 2 Let f c o n d i t i o n a l expectation be an Rn-valued R . V . which i s integrable re1 t i v e t o t h e me s u r e P.