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School of Mathematics
Programmes in the School of Mathematics
Final Examination
Programmes involving Mathematics
Final Examination
4SP 06 36976 Level M
LM Stochastic Processes
Alternative Assessment
January Examinations 2021-22
Three Hours
Full marks will be obtained with complete answers to all FOUR questions. Each question carries
equal weight. You are advised to initially spend no more than 45 minutes on each question and
then to return to any incomplete questions if you have time at the end. An indication of the
number of marks allocated to parts of questions is shown in square brackets.
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1.
(a) Let {B(t) : t ≥ 0} be a standard linear Brownian motion. Let b > 0 and define
T (b) = inf{t ≥ 0 : B(t) = −b or B(t) = b}.
Show that
E[T (b)]
is independent of b.
b2
[2]
(b) Let {B(t) : t ≥ 0} be a standard Brownian motion. Define
X(t) = eB(t) for t ≥ 0.
(i) Find E[X(t)].
(ii) Find Var[X(t)].
(iii) Let 0 < s < t . Find Cov(X(s), X(t)).
[15]
(c) For a partition Π of the interval [a, b] of the form a = t0 < t1 < . . . < tn = b we denote
|Π| = max {tk − tk−1 }.
1≤k≤n
For any p > 0, we define the p-th variation of a function f over [a, b] with respect to Π by
n
(p)
VΠ ( f )[a, b] =
∑ | f (tk ) − f (tk−1)| p.
k=1
Let {B(t) : t ≥ 0} be a standard Brownian motion. Show that for all 0 < t < ∞

0 for p > 2,
(p)
lim VΠ (B)[0,t] =
+∞ for 0 < p < 2,
|Π|→0
where the convergence is in probability.
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[8]
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2.
(a) Let (Xn )≥0 be the position of particle performing an unbiased random walk on Z starting
at 0 (that is, X0 = 0). Prove that (Xn2 )n≥0 is a sub-martingale with respect to the filtration
associated with (Xn )n≥0 .
[5]
(b) Let (Xn )n≥0 be a martingale with respect to itself. Prove that (Xn − X0 )n≥0 is also a
martingale with respect to the filtration associated with (Xn )n≥0 .
[3]
(c) Let (Xn )n≥0 be a supermartingale with respect to itself. Show that (−Xn )n≥0 is a submartingale with respect to the filtration associated with (Xn )n≥0 .
[3]
(d) Let (Xk )k≥1 be a sequence of i.i.d. random variables defined on the same probability
space (Ω, F , P) such that E[|Xk |] < ∞ and E[Xk ] = 0, for all k ≥ 1. Furthermore, assume
2
that Var(Xk ) = σ 2 . For n ≥ 1, let Sn = ∑nk=1 Xk . Prove that P(max1≤k≤n |Sk | ≥ n) ≤ σn .
[6]
(e) Let (Xn )n≥0 be a martingale with respect to itself such that for all n ≥ 0 we have E[Xn2 ] <
∞.
(i) Give a proof that for any m > n ≥ 0 we have E[(Xm − Xn )Xn ] = 0.
(ii) Show that for m > n ≥ 0 we have E[Xm2 ] − E[Xn2 ] = E[(Xm − Xn )2 ].
[8]
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3. Let Bt be a standard Brownian motion and Ft be the filtration generated by Bs , 0 ≤ s ≤ t .
(a) Determine whether the process Xt
2
Xt = t Bt − 2
Z t
0
sBs ds
is a martingale with respect to {Ft }.
[3]
(b) Use the Itô formula to prove that the following stochastic process is an Ft -martingale.
1
Xt = e 2 t cos Bt .
[6]
(c) Let x > 0 be a constant and define
1
Xt = (x1/3 + Bt )3 , t ≥ 0 .
3
Show that
1 1/3
2/3
dXt = Xt dt + Xt dBt ,
3
(d) Find the adapted process f (t) such that
B3T
= E[B3T ] +
[6]
X0 = x .
Z T
0
f (t)dBt .
[4]
(e) Solve the stochastic differential equation
dYt = rdt + αYt dBt
with initial condition Y0 , where r, α are real constants. Solution should be an expression
for Yt which is in terms of r, α, B and may include a Riemann integral or an Itô integral.
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[6]
4.
(a) Let (Ω, F , P) be a probability space, and H be a sub-σ -field of F . Let also X =
(Xn )n∈N be a real-valued stochastic process with X0 a constant, and let (Fn )n∈N be its
natural filtration, i.e. Fn = σ (Xm )m≤n , for all n ∈ N.
(i) Let A be the process defined by
n
1
An =
∑ Xm,
n + 1 m=0
for all n ∈ N, and define
T = inf {n ≥ 0 | An ≤ L}
where L ∈ R is such that L < X0 .
Show that T is a (Fn )n∈N -stopping time.
(ii) Let E ∈ H . Compute P E H .
[5]
(b) Let (Ω, F , P) be a probability space and let (Un )n≥2 be random variables that are independent and identically distributed according to the U ([0, 1]) distribution (uniform on
[0, 1]). Let F0 = F1 = {0,
/ Ω} and, for n ≥ 2, Fn = σ (Um )2≤m≤n . We define a stochastic process X as follows: X0 = X1 = 0 and, for n ≥ 1,



Xn + 1 if Xn ≥ Xn−1 , Un+1 ≤
Xn+1 =


Xn − 1 if Xn ≥ Xn−1 , Un+1 >
1
or Xn < Xn−1 , Un+1 >
3
1
or Xn < Xn−1 , Un+1 ≤
3
1
3
1
.
3
(i) Is the process X adapted to (Fn )n∈N ?
(ii) Is X a Markov chain? Justify your answer informally.
(iii) Justify formally your answer to (b.ii) above.
[5]
Question 4 continued overleaf.
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Question 4 continued.
(c) An electronics store sells a particular computer, using the following business model. Each
morning, they receive a visit from their supplier, and have the opportunity to purchase one
unit of the computer. The store always does so, unless they already have 4 units, as this
is the maximum number they have decided to stock. They pay a price ∆ = £20 every day
for the “delivery” (option to purchase), and an additional π = £70 if (and only if) they do
purchase the computer. The store sells these computers to customers for σ = £100.
Then, during the day n ∈ N, they will receive a random number Bn of buyers wanting this
computer. We assume that the random variables (Bn )n∈N are independent and identically
distributed according to
1
P(Bn = 0) = ,
6
1
P(Bn = 1) = ,
2
and P(Bn = 2) =
1
.
3
We denote by Xn the number of computers in stock in the morning, before the visit of the
supplier. It is valued in S = {0, . . . , 4}. Note that if Xn = 0 and 2 buyers end up visiting
the story later on day n for this computer, only one unit will be sold.
We start on the day n = 0, and the store has X0 = 0 units of the computer.
(i) Explain informally why X = (Xn )n∈N is a Markov chain.
(ii) Identify the transition matrix P of X , and draw its transition graph.
In drawing the graph, you are requested to represent S as horizontally as possible.
(iii) What are the communication classes?
(iv) Is the Markov chain recurrent or transient?
(v) Does an invariant distribution ν exist, and is it unique? If no, explain why. If yes,
compute it.
Question 4 continued overleaf.
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Question 4 continued.
We now look at what happens after a long time, on a day n 1 (i.e. much greater than
1, and for which any limit n → +∞ applies).
(vi) What is the expected stock in the morning?
(vii) Show that the empirical frequency of days where the store starts the day at full stock,
1
and has to decline purchasing an additional unit of this computer, converges to 90
with probability 1.
(viii) What is the expected daily cost of stocking the computer? We will denote by Cn the
random cost incurred on day n.
You are requested to provide first a literal formula for Cn using ∆ and π , before giving
a fully numerical answer.
(ix) What is the expected daily profit from stocking and selling this computer? We will
denote by Sn the random sales income received and by Yn = Sn − Cn the random
profit, on day n.
You are requested to provide first a literal formula for Yn using σ , ∆ and π , before
giving a fully numerical answer.
Is the sale of these computers profitable? If not, give the minimum sale price σ that
makes it profitable.
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