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Justify the preceding assertion. – Optional and predictable projections can be deﬁned for every nonnegative measurable process X as the limit in n of the corresponding projections of the processes X ∧ n. In this case, projections may take value +∞. 7] are still valid. – Let X, Y and Z be bounded measurable stochastic processes. Moreover, assume that Y is optional and Z is predictable. Prove that Π(OX) = ΠX, O(XY ) = Y (OX), Π(XZ) = Z(ΠX). 8] The following states that the difference between the progressively measurable and the optional σ-algebras is rather small.

If T ⊆ A, then T is a predictable stopping time. 6, T is a stopping time. 5. 6. They are given as exercises. – Let X be a right-continuous predictable stochastic process with values in [0, ∞], whose all trajectories are nondecreasing, a ∈ [0, ∞]. Prove that T := inf {t : Xt a} is a predictable stopping time. We should warn the reader against a possible mistake in a situation which is quite typical in stochastic calculus (though it does not appear in this book). For example, let a = (at ) be a nonnegative progressively measurable (or predictable) stochastic t a ds, where the integral is taken pathwise.

4, we get EXT {T <∞} = Eξ {T = EMT − s} {T = EM∞ s} {T = EZT s} = EE(M∞ |FT − ) {T s} {T <∞} for every predictable stopping time T . Hence, Z = ΠX. 14 at the beginning of the proof, and we should take Y := M [0,s] at the end. – Let X be a bounded measurable process. Show that there exist a sequence {Tn } of stopping times and an evanescent set N such that {OX = ΠX} ⊆ Tn ∪ N. 3 on monotone classes. The following theorem establishes a connection between projections and conditional expectations. – Let X be a bounded measurable process.