# Applications of Variational Inequalities in Stochastic by Alain Bensoussan

By Alain Bensoussan

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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. = I + If , (SEC. 2 ) 41 STOCHASTIC INTEGRALS We have t h u s shown t h a t where c ( t ) represents ( E ( t ) , t ) t o shorten t h e notation. 40) we have since N- 1 a. s . n--0 &% ( C ( n k ) J‘z’)k b2(t)dt)-jo T dt ( e ( t ) ) b 2 ( t ) d t we can s e e t h a t t o o b t a i n ( 2 . ’s OF ORDER 2 (CHAP. 48) €1 I PiIZI > PI SUP 4%T lS(t)I > M) + PIlxZl > Since, in view of the continuity of 5, PI sup IF;(t) CSST to show that, for any fixed M, we have Ef . 49) Now In fact the double products vanish as a consequence of the property E [ ( j ( nnk +l)k b(t)dw(t))2 -[21)k gk] .

Problem. However, the same numerical solution algorithms can still be applied, apart from a few very simple transformations. In the nonstationary case further difficulties arise since the regularity 20 INTRODUCTION TO STOPPING PROBLEMS ( C H A P . 1) of the function u then has Bo be investigated with respect to the t w o variables x and t. It is in this case necessary to resort to the theory of parabolic variational inequalities. Depending on whether or not the function $(x,t) is regular with respect to t, very different situations arise.