MATH 5050/6815: Homework 2
Due on Tuesday, Feb. 21, by the end of lecture.
Problem 1: Further properties of conditioning The conditional variance of a random
variable X with E(X 2 ) < ∞, with respect to a σ− algebra F is a random variable defined
by
V ar(X|F) = E(X 2 |F) − (E(X|F))2 .
Show that
(1) V ar(X) = E(V ar(X|F)) + V ar(E(X|F)).
(2) Let Y1 , · · · Yn , · · · ... a sequence of i.i.d. random variables with mean µ and variance
N
X
σ 2 , and N an independent Poisson(λ) random variable. Let X =
Yi , (i.e. N
i=1
specifies how many Y terms you are summing up). Find E(X) and show that
V ar(X) = σ 2 E(N ) + µ2 V ar(N ).
(3) Show that if X, Y are random variables with E(Y |G) = X and E(Y 2 ) = E(X 2 ) < ∞
then it must be that X = Y almost everywhere. Hint: A random variable Z is
constant almost everywhere if and only if V ar(Z) = 0. Also, don’t forget that
E(Y |G) is G− measurable.
Problem 2 Suppose Xn is a martingale with respect to its own filtration such that E(Xn2 ) <
∞ and n > m.
2.
(1) Show that E((Xn − Xm )2 |Fm ) = E(Xn2 |Fm ) − Xm
(2) Assume further that for all m, n with n > m, the increments Xn − Xm are independent of Fm . Show that Mn = Xn2 − V ar(Xn ) is a martingale with respect to Fn .
You should use Problem 1.1.
Problem 3 Do problem 5.10 page 127 in your book.
Problem 4 Let Xn , n ≥ 0 a sequence of i.i.d. Bernoulli(p),
Let Sn =
n
X
i=1
P{Xi = 1} = p = 1 − P{Xi = −1}.
1−p x
.
Xi and φ(x) =
p
(1) Show that Ln = Sn − (2p − 1)n is an Fn - martingale.
(2) Let Mn = φ(Sn ) and Fn = σ{X1 , · · · , Xn }. Show that Mn is an Fn − martingale.
(3) Let S0 = a, 0 < a < N and now assume that p < 1/2. Let T = inf{n : Sn =
0 or Sn = N }. Use optional sampling on Mn (by carefully checking the hypotheses)
to compute P{ST = 0}. Check your answer with the one from Gambler’s ruin in
chapter 1.
1