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Additional Exercises
Exercises on Chapter 1
Excercise E.1.1 Consider the construction of the process N = (Nt )t≥0 on p.4 of the LN. So,
Nt (ω) < ∞ for all ω ∈ Ω, t ≥ 0. Then
[0,∞)
N : (Ω, F, P) → (Z+
, (2Z+ )[0,∞) ),
where the σ-algebra (2Z+ )[0,∞) is the smallest σ-algebra that makes the projection map
[0,∞)
Πt : Z+
→ Z+
[0,∞)
(2Z+ )[0,∞) /2Z+ -measurable for any t ≥ 0. This is the map Πt (x) = xt , for x ∈ Z+
[0,∞)
Πt : (Z+
. I.o.w.:
, (2Z+ )[0,∞) ) → (Z+ , 2Z+ ).
Note that 2Z+ is the collection of all subsets of Z+ (power set). Further, according to our
discussion, (2Z+ )[0,∞) consists of σ-cylinders.
The above implies that the map ω → Nt (ω) is F/2Z+ -measurable, as a composition of
measurable maps.
a) Specify the simplest possible π-system for both 2Z+ and the σ-algebra (2Z+ )[0,∞) .
b) Consider now the process N s = (Nt )t≤s , for some s ≥ 0. This is the process N restricted
to index set [0, s]. Specify the smallest σ-algebra that makes the projection map Πst :
[0,s)
Z+ → Z+ measurable for all t ∈ [0, s]. Give the simplest possible π-system for this
σ-algebra.
c) Determine a π-system (in F) for σ(Nt , t ≤ s) and motivate.
Exercises on Chapter 2
Exercise E.2.1
Consider the integrable random variable X : (Ω, F, P) → (R, B), with EX = m0 . We are
going to associate a Doob martingale M = (Mn )n=0,1,... with X by a recursive construction
of a filtration {Gn }n .
At time n construct inductively:
• a partition of R into a finite number of sets, denoted Gn ;
• the σ-algebra Gn = σ(X −1 (A), A ∈ Gn );
• Sn = {x ∈ R | ∃ω ∈ Ω such that x = E(X | Gn (ω))}.
Given the partition Gn , Sn is a finite set. The construction of Gn is immediate from Gn . We
therefore have to prescribe the initial values, and the iterative construction of the partition.
For n = 0: G0 = {R}, G0 = {∅, Ω}, S0 = {m0 }. The partition Gn+1 has the following
properties. For all A ∈ Gn , let xA = Sn ∩ A.
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• If P{X −1 ({xA })} = P{X −1 (A)} then A ∈ Gn+1 ;
• if P{X −1 ({xA })} < P{X −1 (A)} then A ∩ (−∞, xA ], A ∩ (xA , ∞) ∈ Gn+1 .
d
a) Perform the above construction for n = 0, 1, 2, when X = exp(1), i.o.w. P{X > x} = e−x ,
x ≥ 0.
b) Show that Gn+1 = σ(Gn , {X > E(X | Gn )});
c) Show for A = lim supn {X > E(X | Gn )} that A ∈ G∞ .
a.s.
d) Show that X = E(X | G∞ ).
The martingale is therefore a martingale that ‘recovers’ the random variable X through
conditional means. It shows that any random variable can be approximated arbitrarily close
(in L1 -sense) by a discrete random variable with the same mean.
Exercises on Chapters 4-5
Exercise E 5.1 Let {Pt }t≥0 be a SCSG(S) on the space (E, E), where E is a Polish space,
equipped with the Borel-σ-algebra E, and S is the Banach space of functions f : E → R,
equipped with norm || · ||. Denote A the generator and by D(A) the associated domain.
Suppose that ||Pt || ≤ eαt for some α ≥ 0. Then the semi-group is not contracting.
i) Show that {Pt e−αt }t is a SCCSG(S).
ii) Express the generator of {Pt e−αt }t in terms of the generator A of {Pt }t and show that it
has domain D(A).
iii) Show that for any d ∈ R, t ≥ 0 and f ∈ D(A) the following time-version of the Kolmogorov forward integral equation holds:
Z t
h
i
dt
e Pt f = f +
edu Pu (Af )(x) + dPu f du, x ∈ E.
(0.0.1)
0
iv) Show that (0.0.1) implies (as an alternative route) that there is no function f ∈ D(A), such
that Af = λf for some λ > α. (D(A) does not contain eigenfunctions to non-negative
eigenvalues λ > α).
Next restrict to S = C0 (E), so that {Pt }t is a Feller-Dynkin semigroup. Suppose that
f ∈ C0 (E), with Af ∈ C0 (E), where by Af we simply understand the image of f under the
operator A. The question is whether indeed Af satisfies
lim ||
t→0
Pt f − f
− Af || = 0.
t
v) Suppose that f satisfies the Kolmogorov Forward Integral equation
Z t
Pt f (x) = f (x) +
Ps (Af )(x)ds, t ≥ 0, x ∈ E.
0
Show that f ∈ D(A).
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Exercise E 5.2 (continuation of E 5.1) Clearly, for practical purposes, C0 (E) is not a very
desirable Banach space of functions to consider. We would like to enlarge the space, but still
profit from the Feller-Dynkin process advantages.
Suppose that there exists a strictly positive continuous function V : E → R+ and a
constant α > 0 such that x 7→ Pt V (x) and t 7→ Pt V (x) continuous and
sup
x
Pt V (x)
≤ eαt ,
V (x)
t ≥ 0.
(0.0.2)
We now consider the transformed collections of kernels {PtV }t , with
R
V (y)dPt (x, dy)
V
.
Pt (x, B) = B
V (x)
Consider the space C0 (E), equipped with the supremum norm || · ||.
Assume that PtV : C0 (E) → C0 (E), t ≥ 0.
i) Show that {PtV }t is a SCSG(C0 (E)) semi-group, and that ||PtV || ≤ eαt .
ii) Let C0 (E, V ) = {f : E → R | f continuous, with x → f (x)/V (x) is a C0 (E) function}.
Equip it with the weighted supremum norm || · ||V given by ||f ||V = supx |f (x)/V (x)|.
Show that {Pt }t is a SCSG(C0 (E, V )) iff {PtV }t is a SCSG(C0 (E)).
iii) Denote the generator of {Pt }t and {PtV }t by A and AV respectively. Characterise generator A and domain D(A) of {Pt }t in terms of AV and D(AV ) of {PtV }t . You may
assume the necessary properties to hold for V in order that the relation that you derive
be valid. Explain why you need these, if any extra conditions are needed.
iv) Is Dynkin’s formula applicable to functions in the domain? This amounts to going through
the proof.
Let now Brownian motion {X0 +Wt }t with the standard properties be given. Take V (x) = ecx ,
x ∈ R, for some c > 0. We consider {Pt }t as a SCSG(C0 (R, V )).
iv) Show that PtV : C0 (R) → C0 (R). Determine the minimum value of the coefficient α in
(0.0.2). Calculate domain and generator.
Exercise E 5.3
i) Consider the branching process from LN 4.3. Show that the branching process is nonexplosive by constructing a suitable moment function.
ii) Consider the following server farm model. Customers arrive at a server farm according to
a Poisson process at rate λ, each requesting a machine for an amount of time that has
an exponential distribution with mean 1/µ, independently of the other customers and
of the arrival process.
The server farm possesses an unlimited amount of machines. Each machine can be busy
(serving a customer), it can be idle (i.e. it is in the ‘on’ state, but not serving), or it
can be ‘off’ (switched off). If an idle machine is available, then it is assigned to the new
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customer. If there is no idle machine, an ‘off’ machine is switched on instanteneously
and assigned to the new customer. Idling costs energy, but switching on an ‘off’ machine
costs time.
The question is when and whether a machine should be switched off at the moment it
becomes available. However, here we assume that a machine is never switched off, but
left in the idle state (consuming energy), when it becomes available. The state space
is E = Z2+ , where state (x, y) denotes that x machines are busy and y are idle. It is
equipped with the σ-algebra E = 2E .
Consider the stochastic process X, with Xt = (Xt,1 = x, Xt,2 = y) denoting the number
x of busy machines at time t and y the number of idle machines. By describing a
suitable construction, motivate that X is a Markov process (you do not need to prove
the details) and specify the generator. Show by constructing a moment function that
X is non-explosive. What is limt→∞ (Xt,1 + Xt,2 )? Motivate.