# Download e-book for iPad: Applications of Variational Inequalities in Stochastic by Alain Bensoussan

By Alain Bensoussan

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**Example text**

If t h e s e two s e t s a r e i d e n t i c a l , we p u t : lim An = l i r n i n f A, = l i m sup ,A . I n p a r t i c u l a r , i f {A 1 i s an i n c r e a s i n g ( r e s p . d e c r e a s i n g ) monotone sequence, then l i m A e x i s t s ( l i m n A n = l i m An or l i m An = l i m + An depending on whether we a r e i n m e i n c r e a s i n g o r d e c r e a s i n g c a s e ) . 3) P(limninf ),A s l i m n i d P(A ) n s limnsup P(A ) P(lim sup A ) n n . 24 (CHAP. 5) If e x i s t s , we have = P(limnAn) h P(An) < rn , .

STOCHASTIC INTEGRALS 2. 1 The Wiener process Let (B,C7,P) be a probability space. w(t ) I is a Gaussian vector with mean 0. min(t,s) V t , s Z 0 . We shall assume the existence of such a process (see, for example, MACKEAN [ll). 4) w ( t 2 ) Indeed - w(tl), w(t4) Ew(tI2 = t - w ( t 3) - are independent if t 1 5 t2 S tg 5 t,, . ’s, and are therefore independent. 5) p E[w(t) IF’] t w(s) vt 2 s S s S . 6) E[(w(t) t. We have h E [O,s] . Hence w(t) is a - w ( s ) ) 2 Ips 1 = t - s (t 2 s) . 6) characterises continuous martingales which are zero at 0 and which are Wiener processes.

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.