Martingales And Stochastic Analysis (Series on Multivariate by J Yeh

By J Yeh

This e-book is an intensive and self-contained treatise of martingales as a device in stochastic research, stochastic integrals and stochastic differential equations. The booklet is obviously written and information of proofs are labored out.

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Np, so is X ( n ) . Thus we have a sequence {XM : n G N} of left-continuous {&}-adapted processes on the filtered space such that lim X{n) = X on R+ x Q. ■ 71—fOO §3 Stopping Times [I] Stopping Times for Stochastic Processes with Continuous Time Stopping times are a method of truncating the sample functions of a stochastic process. Let X = {Xt : t G R+} be an adapted process on a filtered space (£2,5, {&}, P). Suppose for instance that the sample functions X(-,ui), w G £2, are to be stopped from varying with t G R+ after some time point and suppose whether to stop a sample function or not at any given time depends on the behavior of the sample function up to the time point.

Since A\ g &; and A2 € &; and since fo | as t -> oo we have A. f~l A2 g &>. Thus the intersection is of the type (a) if it is not 0. 18. For the predictable a-algebra 6 and the w-class 1H of predictable rectan­ gles in a filtered space (£2, J , {&}, P), we have 6 = ^(OT) = d(V\). §2. STOCHASTIC 21 PROCESSES Proof. 1) Let R G £ft and consider the mapping X of R+ x £1 into R defined by X = 1R. lfR = (t',t"] x A where t',t" e R+, t' < t", and 4 G &,, then X is a left-continuous {&}-adapted process since X(t,u) = lWtnxA(t,D) for (t,u) 6 R+ x Q.

Next, let us consider the measurability of XT as a function on the probability space (£2, g, P)- The next lemma shows that the measurability of XT is unrelated to the filtration in terms of which the stopping time T is defined. 26. Let X = {Xt : t G R+} be a stochastic process, S be an ^-valued ran­ dom variable, and X^ be an extended real valued random variable on a probability space (£2, g, P)- With X = {Xt : t € R+}, let Xs be an extended real valued function on £2 de­ fined by Xs(w) = X(S(LL>),ui)fortD G £2.

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