Unformatted text preview: uence of σalgebras, ℱ ⊂ ℱ
a. (5 points)
Let be a random variable whose values belong to
. Prove that
is a sequence of integrable random variables, i.e.
ℱ
∞
and 
b. (7 points)
Let τ be a stopping time with respect to ℱ . Prove that the following
holds:
,
)
ℱ(
ℱ
c. (6 points)
Assume that
ℱ
is adapted. Let
be a
monotone increasing sequence of stoppingtimes with respect to ℱ so
that
∞, a.s. Use part b to show that if
ℱ
is a
supermartingale for each n, then
ℱ
is a supermartingale as
well....
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 Spring '12
 STAFF
 Probability, Probability theory

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