By Yoshikazu Sawaragi
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Extra resources for Modeling, Estimation, and Their Applications for Distributed Parameter Systems
2. Stochastic d i f f e r e n t i a l equation in H i l b e r t spaces. We are now prepared to discuss stochastic d i f f e r e n t i a l H i l b e r t space K. E [ [[ UOI[ 2 ] equations in the So, i f we suppose that U0 is an K-valued random variable with < ~ , that A(t) is a regulated mapping of [ O , T ] into and that C(t) is an element of L2( O,T; ~ ( H , ~(K, K), K) ), then we can consider the following stochastic integral equation: U ( t , m ) : Uo(m) + fO t A(s) U ( s , m ) d s + fOt C(s) dW(s,m).
Hence, by proving the following lemma the proof of the theorem was completed. D. 15 ] Let Let T1 be a compact interval and let X be a Banach space. f ( t ) map T1 into ~ ( X , X) and let g(t) map T1 into ~ ( X , X ) ( o r I f one of the maps f,g is in L2( T1 ; -~j( X, X ) ) ( or,in the case of g, LP( T1 ~j( X, X)) or LP( TI , X ) ) and the other is regulated, then f ( t ) g(t) element of LP( TI , [ PROOF ] LP( TI; X). is an ;~( X, X )) ( or LP( TI , X ) ~ . For example, suppose that f ( t ) is regulated and that g(t) is in ~( X, X )).
COROLLARY2 . 33). 37) 1 (H, K),and A(s,m) is a K-valued stochastic II A(s,m) Ilds < for all t in [ O , T ] . A(t,m) dt + C ( t , ~) dW(t). 3. I t o ' s lemma in Hilbert spaces. Let us now state and prove I t o ' s lemma in H i l b e r t spaces. with the f o l l o w i n g l e m m a w h i c h i s a n i m p o r t a n t t o o l F i r s t of a l l , we assume in t h i s section that We begin in the proof of the main theorem. the covariance operator Q(t) is independent of the time t since t h i s assumption enables us to make the d e r i v a t i o n of I t o ' s lemma very easy.