By Ash C.

ISBN-10: 0780310519

ISBN-13: 9780780310513

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44 Stochastic processes As a function of two variables (t, ω), a stochastic process should be measurable with respect to both variables to allow a minimum of “good behavior”. 9 A stochastic process {X t } with t ∈ [0, ∞) on a probability space { , F, P} is measurable if, for all Borel sets B in the Borel σ -field B(IRd ), {(ω, t) : X t (ω) ∈ B} ∈ F ⊗ B([0, ∞)). If the probability space { , F, P} is equipped with a filtration {Ft } then a much stronger statement of measurability which relates measurability in t and ω with the filtration {Ft } is progressive measurability.

N ≥ 0}. 2 Let following filtration: 47 = {ωi ; i = 1, . . , 8} and the time index T = {1, 2, 3}. Consider the F1 = σ {{ω1 , ω2 , ω3 , ω4 , ω5 , ω6 }, {ω7 , ω8 }}, F2 = σ {{ω1 , ω2 }, {ω3 , ω4 }, {ω5 , ω6 }, {ω7 , ω8 }}, F3 = σ {{ω1 }, {ω2 }, {ω3 }, {ω4 }, {ω5 }, {ω6 }, {ω7 }, {ω8 }}. Now define the random variable α(ω1 ) = α(ω2 ) = α(ω5 ) = α(ω6 ) = 2, α(ω3 ) = α(ω4 ) = α(ω7 ) = α(ω8 ) = 3, so that {α = 0} = ∅, {α = 1} = ∅, {α = 2} = {ω1 , ω2 , ω5 , ω6 }, {α = 3} = {ω3 , ω4 , ω7 , ω8 }, and α is a stopping time.

6. 7. 8. 9. 10. 11. 12. 13. 35 1 1 1 5 , P(ω2 ) = , P(ω3 ) = , and P(ω4 ) = . 12 6 3 12 {ω1 , ω3 } if n is odd, {ω2 , ω4 } if n is even. Find P(lim sup An ), P(lim inf An ), lim sup P(An ), and lim inf P(An ) and compare. 36. Show that a σ -field is either finite or uncountably infinite. Show that if X is a random variable, then σ {|X |} ⊆ σ {X }. Show that the set B0 of countable unions of open intervals in IR is not closed under complementation and hence is not a σ -field. (Hint: enumerate the rational numbers and choose, for each one of them, an open interval containing it.

### The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists by Ash C.

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