By Alain Bensoussan
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Extra info for Applications of Variational Inequalities in Stochastic Control
C) = 5 g (y(t),t)dt . + &(t),t)dw(t),t > 'E We say t h a t y ( t ) i s a s o l u t i o n of a s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n ( s t r o n g formulation (*)I. (*IThe weak formulation w i l l be d e f i n e d i n t h e next s e c t i o n . 2 ( C W . 6) EISl2 - g(x'tt) I + 2 IU(x,t) - U(xt,t) I - < + / o ( x , t ) ) 2 c <(l + < += . which i s a solution of ( 3 , 1 ) , ( 3 . 2 ) , ( 3 . 3 ) such that There i s uniqueness i n the following sense. 1), ( 3 . 2 ) , ( 3 . 3 ) and ( 3 .
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.
S OF ORDER 2 Additionally, from the properties of martingales. X2 ( & Xm)2 n n m (CHAP. 2) . 54) clearly holds with C1 = 3Cz. 37) is called I t o ' s formula. Y € C2"(Rn , and 3C2T2 since cy If , x ]O,T[) the formula is applicable between two arbitrary instants tl of ( 0 , t)). a + ? b dw(t) . \ We see that, in comparison with the rules of ordinary differential calculus, we now have an extra item. To express this, it is sometimes said that dw(t) is of the order of vdt (see P. LEVY [l], E. WONG [l]).