Stochastics II
Stochastic Processes
Winterterm 2016/17
Prof. Dr. U. Rösler
S. Hallmann
Sheet 10
Exercise 1
Let X, Y be integrable random variables. Is
E(X|Y ) = E(X)
a.s.
sufficient for X and Y to be stochastically independent?
Exercise 2
Let Xi , i ≥ 0 be independent random variables, where
EXi2 = c < ∞ for i = 1, 2, . . .. Show that


!2
n
X

Xi − nc
i=1
n∈
EXi = 0 and
N
is a martingale (w.r.t. the filtration generated by the Xi , i.e. An := σ(X1 , . . . , Xn )).
Exercise 3
Let Xi , i ≥ 0 be integrable random variables, and An := σ(X1 , . . . , Xn ).
Assume for n ≥ 1,
E(Xn+1|An) = aXn + bXn−1,
where a ∈ (0, 1) and a+b = 1. For what value(s) of α does Sn := αXn +Xn−1
define a (An )-martingale?
Exercise 4
Let XiP
, i ≥ 0 be square-integrable, An := σ(X1 , . . . , Xn ), and assume that
Sn := ni=1 Xi defines a (An )-martingale.
Show that E(Xi Xj ) = 0 for any i 6= j.
Exercise 5
Consider a (fair) game of heads and tails (in which a winning bet of ke is
rewarded 2ke). You adopt the following strategy. You bet 1e at the first
hand. If you lose your n first bets, you bet 2n+1 e at the (n + 1)-th hand.
Let Xi , i ≥ 0 be independent random variables where P(Xi = 1) = 1/2 =
P(Xi = −1). The outcome of the n-th hand is given by Xn, where Xn = 1, if
you win the n-th hand and Xn = −1, if you lose the n-th hand. Moreover, as
soon as you win one bet, you stop playing, i.e. at time τ := inf {n : XP
n = 1}.
n−1
The payoff of the n-th hand is given by Yn := 2 Xn and Mn := ni=1 Yi
denotes your net profit just after the n-th hand has been played.
The filtration is given by An := σ(X1 , . . . , Xn ).
Show that
(a) (Mn )n∈N is a (An )-martingale.
(b) Mτ = 1. Why is this no contradiction to the Optional-Sampling-Theorem?
(c) Consider now the case P(Xi = 1) = p = 1 − P(Xi = −1), where p 6= 1/2.
Determine EMτ ∧n .
Hand in until friday, 20.01.2017, 12:00