By A. A. Balkema (auth.), E. Çinlar, P. J. Fitzsimmons, R. J. Williams (eds.)

The 1990 Seminar on Stochastic strategies was once held on the college of British Columbia from may perhaps 10 via may perhaps 12, 1990. This used to be the 10th in a sequence of annual conferences which supply researchers with the chance to debate present paintings on stochastic techniques in an off-the-cuff and stress-free surroundings. prior seminars have been held at Northwestern college, Princeton college, the Univer sity of Florida, the collage of Virginia and the college of California, San Diego. Following the winning layout of earlier years, there have been 5 invited lectures, brought by way of M. Marcus, M. Vor, D. Nualart, M. Freidlin and L. C. G. Rogers, with the rest of the time being dedicated to casual communications and workshops on present paintings and difficulties. the passion and curiosity of the contributors created a full of life and stimulating surroundings for the seminar. A pattern of the examine mentioned there's contained during this quantity. The 1990 Seminar used to be made attainable through the help of the typical Sciences and Engin~ring examine Council of Canada, the Southwest collage arithmetic Society of British Columbia, and the college of British Columbia. to those entities and the organizers of this year's convention, Ed Perkins and John Walsh, we expand oul' thank you. ultimately, we recognize the aid and counsel of the employees at Birkhauser Boston.

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**Example text**

We have then J h1(s,T)lByd(Ix)z = J l(s,T) Hdlx E 1(S,T) 1Byd1x E L~. L~. J l(s,T)lByd(Ix )hz. L'b. is a monotone class which contains 'R, hence the above equality holds for an The class of sets B for =h which the above equality holds for alI z E B E P, and z E L'b•. Hence, for any predictable, simple process H, we have J h1(s,T)Hd(Ix)z = J l(s,T)Hd(Ix)hz. FF,G(X), Lebesgue's theorem implies that the above equality holds for H. Assume now that (h J l(s,T)H dlx J E L~. Then h J l(s,T)H dlx E L~ and 1(s,T) Hd1x,z) =/ = (/ 1(s,T) Hd1x, hz) l(s,T)Hd(Ix )hz =/ = (/ h1(s,T)HdIx,z).

Notation. Vector integration. Processes with finite variation. 2. Summable processes. Definition of summable process. Extension of Ix to stochastic intervals. Summability criteria. u-additivity and the extension of Ix. 3. The stochastic integral. Definition of the integral f H dIx. The stochastic integral. Notation and re- marks. The stochastic integral of elementary processes. Stochastic integrals and stopping times. Convergence theorems. The stochastic integral of caglad and bounded processes.

II H E L},a(X), then iar evezy t E [0,00) we have (H· X)t- E L~ and (H· X)t- = [ Hd1x. J[O,t) In particular, (H· X)oo- = (H· X)oo = The mapping t ---+ Hd1x. (H . X)t is cadlag in Lh. Proof. Let tn /' t. The l[O,tn]H n, and J J l[O,tn]H dlx ---+ l[o,t)H pointwise, 11[O,tn]HI :::; IHI for each = (H . X)t n E L~ and (H . X)t n ---+ (H . X)t-. By Stochastic Integration in Banach Spaces 51 J l[o,t)H d1x E L~ and J l[O,tn]H d1x -+ J l[o,t)H d1x Hence (H· X)t- = J l[o,t)Hd1x. 1, we have pointwise. 1. Notation and remarks If Ce FF,a(X), we denote the closure of C in FF,a(X) by FF,a(C,X).