Homework 1
(1) Show that if X = Y on B ∈ G , then E[X |G ] = E[Y |G ] a.s. on B .
(2) Suppose X ≥ 0 and E[X] = ∞. Show that there is a unique Y ∈ G with 0 ≤ Y ≤ ∞ so that
ˆ
ˆ
XdP =
A
Y dP for all A ∈ G.
A
(Hint: Consider Xn = X ∧ n.)
(3) Prove the following conditional limit theorems.
(a) Fatou: If X1 , X2 , ... are nonnegative, then E [lim inf n Xn |G ] ≤ lim inf n E [Xn |G ] a.s.
(b) DCT: If Xn → X a.s. and there is an integrable Z with |Xn | ≤ |Z|, then E[Xn |G ] → E[X |G ]
a.s.
(4) Give an example on Ω = {a, b, c} in which E [E[X |F1 ] |F2 ] 6= E [E[X |F2 ] |F1 ].
(5) Suppose that G1 ⊆ G2 and E[X 2 ] < ∞. Show that
i
h
i
h
2
2
E (X − E[X |G2 ]) ≤ E (X − E[X |G1 ]) .
(6) Suppose that E[X 2 ] < ∞, and dene Var (X |G ) = E[X 2 |G ] − E[X |G ]2 . Show that
Var(X) = E [Var (X |G )] + Var (E[X |G ]) .
(7) Show that if E[X |G ] = Y and E[X 2 ] = E[Y 2 ] < ∞, then X = Y a.s.
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(8) Suppose that X and Y have joint density f (x, y) > 0. Let
´
f (x, y)dx
µ(y, A) = ´A
.
f (x, y)dx
Show that µ(Y (ω), A) is a r.c.d. for X given σ(Y ).
(9) Prove that a random variable N : Ω → N ∪ {∞} on a ltered probability space (Ω, F, {Fn }∞
n=1 , P )
is a stopping time if and only if {N ≤ n} ∈ Fn for all n ∈ N.
(10) Suppose that S and T are stopping times.
(a) Show that S ∨ T and S ∧ T are stopping times.
(Since T ≡ n is a stopping time, this shows that S ∨ n and S ∧ n are stopping times for any n.)
(b) Is S + T a stopping time? If S < T , is T − S a stopping time? Prove or give a counterexample.
(11) Suppose that M and N are stopping times with M ≤ N .
(a) Show that FM ⊆ FN .
(
(b) Show that if A ∈ FM , then L(ω) =
M (ω),
ω∈A
N (ω),
ω ∈ AC
is a stopping time.
(12) Show that if Yn ∈ Fn and N is a stopping time, then YN ∈ FN . Conclude that if f : S → R is
P
measurable, Tn = ni=1 f (Xi ), and Mn = maxm≤n Tm , then TN , MN ∈ FN .
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