oo l(s,t]Xk = l(s,t]X in [2.
T { - n - ° l n on Fa on [2 \Fa is optional for each n. It follows that R C () and hence, since R generates P, we have P C (). In the following lemma we show that certain types of stochastic intervals are predictable. 1. predictable. Stochastic intervals of the form [0, T] and (TJ, T] are Proof. Since ('I}, T] = [0, T] \ [0, TJ]' it suffices to prove that a stochastic interval of the form [0, T] is predictable. For this we use a standard approximation of T by a decreasing sequence {Tn} of countably valued optional times, defined by Tn = 2- n [2 n T + 1].
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