By Alain Bensoussan
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Extra info for Stochastic Control by Functional Analysis Methods
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S. we deduce ( 3 . 2 2 ) . 0 0 The estimate ( 3 . 2 2 ) is due to Burkholder [ I ] . 4. STOCHASTIC DIFFERENTIAL EQUATIONS Let Zt be an increasing family of sub-a-algebras of (G&,P) Let w(t) be a standard m dimensional Wiener process with values in Rm, which is an Let T zt Wiener. be an measurable. V. which is zT We set We define the following problem. in Rn a,where is a probability space. 1) (1 (4* 2) y(t) x,(t))y(t) is adapted to 3 t is continuous 33 STOCHASTIC CALCULUS Clearly 7. 4) dy = Y (T) = 5 .
Let 6 be Zt measurable and bounded. ads - the conditional expectation entering in the last equality. We can write But result ( 5 . 7 ) a+i$ with $ applies (as easily checked) when a is replaced by deterministic. 15). ,An) = wk (t) - Jt ak(s)ds is a vector of Rn Rn. 15) we get 0 which proves the desired result. Corollary 5 . 1 . Let be adapted processes, a. (t), b.. (t) 13 which are a . s . 16) y(t) Then on (n,a,F), z3e = y(0) + Jt Let [O,T]. a(s)ds + ft b(s)dw(s) bk i,j = 1,. s. s. (cf. 1 . ) For step functions one easily checks i t bk(s)dw(s) = {t bk(s)dG(s) + J NOW for a subsequence (1) bk + b 2 .
W, $ we obtain We can then go from any Borel positive bounded function to any Borel positive function by an approximation argument. 2 . Let y(t) be a stochastic process with values in t Rs continuous with left links. Let open bounded subset of We write R". = u(y(X),s