MOCK EXAM – ELE3917 STOCHASTIC PROCESSES FOR FINANCE AND
ECONOMICS
APRIL 25, 2023– 09:00 - 12:00
Note: Remember to explain your answer properly. Feel free to explain with words where it is
challenging to convey your argument with mathematical expressions. You are only allowed to use a
English dictionary and a calculator as support material.
Problem 1
(1) What is a random variable?
(2) If X and Y are two random variables, what does it mean that X and Y are independent?
Write down the mathematical implication of this assumption.
(3) Show that for three subsets A, B, C ⊂ Ω which are all disjoint then
P(A ∪ B ∪ C) = P(A) + P(B) + P(C).
(4) Let X : Ω → {1, 2, 3, 4} is a uniformly distributed random variable. Compute the probabilities
P(X = 1) =?,
P(X = 4 or X = 2) =?
(5) Let X and Y be two independent normally distributed random variables such that X ∼
2 ) and Y ∼ N (µ , σ 2 ). Show that
N (µX , σX
Y
Y
2
+ σY2 .
V ar(X + Y ) = σX
Problem 2
(1) A stochastic process {X(t)|t ∈ {1, 2, ..., N }} is a collection of random variables labeled by a
time index. Assume that E[|X(t)|] = 1. Show that
1
P(X(t) ≥ n) ≤ .
n
Hint Use Markov’s inequality.
(2) Consider the stochastic process {X(t)|t ∈ {1, 2, ..., N }} given by
X(t) =
t
X
Yi ,
i=1
where {Yi } is a collection of i.i.d. standard normally distributed random variables with Yi ∼
N (0, 1) for all i. Show that
E[X(t)] = 0,
and V ar(X(t)) = t.
(3) Construct a new stochastic process Z(t) = exp(X(t)). Show that
t
E[Z(t)] = exp( ).
2
Hint: Here you may use that for a standard normal random variable U ∼ N (0, 1) then
1
E[exp(U )] = exp( ).
2
Problem 3
(1) What is a martingale? List and explain the defining properties.
(2) Let X be a random variable with finite second moment, and let {Ft |t ∈ [0, T ]} be a filtration,
such that X is measurable with respect to FT (this means that E[X|FT ] = X). Define the
stochastic process Z(t) = E[X|Ft ]. Show that Z(t) is a martingale. Hint: You may use
Hölders inequality telling us that for any p ≥ 1
E[|X|] ≤ E[|X|p ]1/p .
1
MOCK EXAM – ELE3917 STOCHASTIC PROCESSES FOR FINANCE AND ECONOMICS
Problem 4
(1) What is a Brownian motion?
(2) Let B(t) be a Brownian motion. Show that it is a martingale.
(3) Construct a new process for σ > 0 by
X(t) = −σ 2 t/2 + σB(t).
And define another process Z(t) = exp(X(t)). Show that Z(t) is a martingale.
2