(1)1 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

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1 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, N_t(ω) < ∞ for all ω ∈ Ω, t ≥ 0. Then

N : (Ω, F , P) → (Z^[0,∞)₊ , (2^Z⁺)^[0,∞)),

where the σ-algebra (2^Z⁺)^[0,∞) is the smallest σ-algebra that makes the projection map Πt: Z^[0,∞)₊ → Z₊

(2^Z⁺)^[0,∞)/2^Z⁺-measurable for any t ≥ 0. This is the map Πt(x) = xt, for x ∈ Z^[0,∞)₊ . I.o.w.:

Π_t: (Z^[0,∞)₊ , (2^Z⁺)^[0,∞)) → (Z₊, 2^Z⁺).

Note that 2^Z⁺ is the collection of all subsets of Z₊ (power set). Further, according to our discussion, (2^Z⁺)^[0,∞) consists of σ-cylinders.

The above implies that the map ω → Nt(ω) is F /2^Z⁺-measurable, as a composition of measurable maps.

a) Specify the simplest possible π-system for both 2^Z⁺ and the σ-algebra (2^Z⁺)^[0,∞).

b) Consider now the process N^s= (N_t)_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 Π^s_t : Z^[0,s)₊ → 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 {G_n}_n.

At time n construct inductively:

• a partition of R into a finite number of sets, denoted G_n;

• the σ-algebra G_n= σ(X⁻¹(A), A ∈ G_n);

• S_n= {x ∈ R | ∃ω ∈ Ω such that x = E(X | Gn(ω))}.

Given the partition G_n, S_n is a finite set. The construction of G_nis immediate from G_n. 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 G_n+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.

a) Perform the above construction for n = 0, 1, 2, when X= exp(1), i.o.w. P{X > x} = e^d ^−x, x ≥ 0.

b) Show that Gn+1= σ(Gn, {X > E(X | Gn)});

c) Show for A = lim sup_n{X > E(X | G_n)} that A ∈ G∞. d) Show that X^a.s.= 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 {P_t}_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 ||P_t|| ≤ e^αt for some α ≥ 0. Then the semi-group is not contracting.

i) Show that {P_te^−αt}_t is a SCCSG(S).

ii) Express the generator of {P_te^−αt}_t in terms of the generator A of {P_t}_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 Kol- mogorov forward integral equation holds:

e^dtP_tf = f + Z t

0

e^duh

P_u(Af )(x) + dP_ufi

du, x ∈ E. (0.0.1)

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 {P_t}_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

t→0lim||Ptf − f

t − Af || = 0.

v) Suppose that f satisfies the Kolmogorov Forward Integral equation Ptf (x) = f (x) +

Z t 0

Ps(Af )(x)ds, t ≥ 0, x ∈ E.

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→ P_tV (x) and t 7→ P_tV (x) continuous and

sup

x

P_tV (x)

V (x) ≤ e^αt, t ≥ 0. (0.0.2)

We now consider the transformed collections of kernels {P_t^V}_t, with P_t^V(x, B) =

R

BV (y)dPt(x, dy)

V (x) .

Consider the space C₀(E), equipped with the supremum norm || · ||.

Assume that P_t^V : C₀(E) → C₀(E), t ≥ 0.

i) Show that {P_t^V}_t is a SCSG(C0(E)) semi-group, and that ||P_t^V|| ≤ e^αt.

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 = sup_x|f (x)/V (x)|.

Show that {P_t}_t is a SCSG(C₀(E, V )) iff {P_t^V}_tis a SCSG(C₀(E)).

iii) Denote the generator of {Pt}_t and {P_t^V}_t by A and A^V respectively. Characterise generator A and domain D(A) of {P_t}_t in terms of A^V and D(A^V) of {P_t^V}_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+W_t}_twith the standard properties be given. Take V (x) = e^cx, x ∈ R, for some c > 0. We consider {P_t}_t as a SCSG(C₀(R, V )).

iv) Show that P_t^V : 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 non- explosive 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 = 2^E.

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 lim_t→∞(X_t,1+ X_t,2)? Motivate.