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By Alain Bensoussan

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E . 3t. a family such t h a t t . We say t h a t t h e process X ( t ) i s adapted t o t h e family 3 i f t Vt, X(t) i s 3 -measurable. V. 34) (*) at, if . 35) Vt. The following p r o p e r t i e s of stopping times w i l l be very u s e f u l l a t e r ( t h e family 3t i s f i x e d ) : Let S , T be two stopping t i m e s ; ing times. Let T be a stopping time; then t h e R . V . ' s SAT, SVT, S+T a r e stopp- T then T i s 3 -measurable. V. then S i s a stopping time. Let S,T be two stopping times, T ~n f 3 .

This will be done systematically in what follows, without particular mention of the fact being made. ’s OF ORDER 2 32 (CHAP. 56) is called Doob’s optional sampling theorem; it generalises the submartingale inequalities to stopping times. In practice this will generally have to be applied over a finite interval C0,al and hence S S T S a. 54) will still hold. 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 .

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. 30) applied to (*) 'p n+l -% we have We have not needed to use this property until now.

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