Applications of Variational Inequalities in Stochastic by Alain Bensoussan

By Alain Bensoussan

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22) = Erp(S)W(d Vrp,'? measurable 2 0 . - We s h a l l n b e making use of t h e following r e s u l t : suppose we have x C2 R measurable w i t h r e s p e c t t o t h e product o-algebra b ( R n ) x a 5 ( g e n e r a t e d by events of t h e form B x A where B E R(R ) a n d A E a ) . Let B be a sub-o-algebra o f 4; l e t 5 be a ,&-measurable R . V . w i t h v a l u e s i n Rn. 3 Distribution function; Let X be a R . V . 26) with v a l u e s i n R n . ,x x c h a r a c t e r i s t i c function i E R We p u t n = P{X, I x l , , X.

In this connection, we can prove the principle of separation (see Chapter 4); this signifies that the optimal decision (control and stopping time) can be obtsined at each instant as a function of the best estimate of the state of the system at that instant. 7. 9)). In the case where the data are not regular, other characterisations are possible. First, let us consider the stationary case. For simplicity, we take f = 0, h = 0. 9), when this is applicable, is written in the form We then consider the process (SEC.

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 . 3 0 ) , view of the continuity ( * ) of w(t)). and for arbitrary q, in 0, we can find a modification of the process I(t) which is continuous.

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