By Alain Bensoussan

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61) E[Y(w(t)) - Y(w(8)) I $1 = - $lu12 E[f: Y(w(h))dhl$] , (SEC. V. E y ( w ( t ) ) q = E 's(w(s))tl It follows from ( 2 . 6 1 ) t h a t - 4u ' j t , . ~(i(w(h))tldh If we put f ( t ) = E Y(w(t))tl we have f o r t t s f(t) = f(s) -$lU/'j: f(A)dA , and hence We have t h u s proved t h a t 2 which proves t h a t w ( t ) - w ( s ) i s independent of $and Gaussian law with mean 0 and with v a r i a n c e t - s . t h a t w ( t ) - w ( s ) follows a Before g i v i n g a second example of a p p l i c a t i o n o f I t o ' s formula, we s h a l l f i r s t consider a s t o c h a s t i c i n t e g r a l with a stopping time a s an upper bound.

C(t)dt 11p ' 119 V cp,+ f Q . 19): Remark 2 . dw(t) where 9 . is the ith row of cp(t)dw(t) 0 'p. (z) converges on I o if z > N . s. V. 's OF ORDER 2 (CHAP. 22). dw(t) in probability. ; , . 26) also holds with 5 instead of - c2 I2dt Icp(t)I2dt . (use 5= (Y-O2 and the linearity). dw(t) laa] = 0 (SEC. 29) . dw(s) We are thereby defining a stochastic process. 44)). 27). Ej:l'p(s) . I2ds We thus have Furthermore, if is piecewise constant, then I(t) is a continuous process (in We shall now show that by virtue of ( 2 .

Some of these will be treated in Volume 2: diffusions with reflection at the boundary of the domain, processes of the Poisson or semi-Markov type, degenerate processes, etc ... e. the evolution of y). In this case, the form of the optimal decisions mentioned above has one very attractive aspect: all the information necessary for making the optimal decision at each instant is concentrated in the state of the system at that instant. Information regarding past history is superfluous. Putting it yet another way, the optimal decisions are Markov.