Assume that the Markov process X with transition function {Pt}t, satisfies the conditions of LN Exercise 3.12

1 Homework 10 and 10,5

to be handed in: June 8, 2016

Note: the weight of this homework assignment is 1,5 times the weights of the other homework assignments.

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Assume that the Markov process X with transition function {P_t}_t, satisfies the conditions of LN Exercise 3.12. In addition, assume that E is countable, consisting only of holding points.

Assume also that E has no accumulation points.

Remind that σ_x = inf{t > 0 | X_t6= x}. Denote by α_x the parameter of the exponentially distributed sojourn time σ_x in state x. Furthermore, P is the E × E transition kernel with elements P (x, y) = P_x{X_σx = y}. Note that by this constructionP

y6=xP (x, y) = 1. Thus, X is a Markov jump process.

Define the following E × E matrix Q, called a q-matrix, with elements:

Q(x, y) =

 −α_x, y = x

α_xP (x, y), y 6= x.

Suppose that there exists a function V : E → (0, ∞), and a constant c, such that:

• V is a moment function, that is, there exists an increasing sequence of compact sets Kn∈ E, n = 1, . . ., with K_n↑ ∞, n → ∞, such that lim_n→∞infx6∈KnV (x) = ∞.

• QV (x) =P

yQ(x, y)V (y) ≤ cV (x), x ∈ E.

Then, one can show that X is non-explosive (see Chapter 4) and that P_tV (x) =X

y

P_t(x, y)V (y) ≤ e^ctV (x). (0.0.1)

Let ||f || = sup_x|f (x)|/V (x) and put

C₀(E, V ) = (

f : E → R continuous     

for each  > 0 there exists a finite set K = K(, f ), such that ^{|f (x)|}_{V (x)} ≤ , for x 6∈ K

)

C₀(E, V ) equipped with the norm || · ||_V is a Banach space.

Claim: (Pt)t is a strongly continuous semigroup on C0(E, V ).

Exercise 1 (3pts)

i) A way to prove the claim, is by first considering

P_t^V(x, y) = e^−ctP_t(x, y)V (y)

V (x) .

Show that (P_t^V)_tis a Feller-Dynkin (sub-stochastic) transition function (on C₀(E)). Use this to show that (P_t)tis a SCSG(C0(V, E)).

2

ii) Show that the generator of {P_t}_t is given by Af (x) =X

y6=x

αxP (x, y)f (y) − αxf (x), f ∈ D(A),

and that f = 1_{x}∈ D(A).

ii) Consider the example in LN Exercise 4.3. Check that there exists a moment function V for some constant c ≥ 0, such that QV (x) ≤ cV (x), x ∈ E. You can try functions of the vorm x 7→ V (x) = e^βx, β > 0, or x 7→ V (x) =Px

i=11/i. Thus, the corresponding transition function is a SCSG(C0(V, E)).

Exercise 2 (3pts) LN Exercise 4.3.

Note: in (i), the notation Q in the exercise is meant to be the q-matrix defined above, not the generator! Then conclude what the generator is.

Exercise 3 (3pts) i) Prove Theorem 5.3.4.

ii) LN Exercise 5.7. You are not allowed to use expressions of the type P_x{η_r ≥ (n + 1)t | η_r≥ nt} = .... Clearly, conditioning is on σ-algebras, not on events.

Exercise 4 (3pts) LN Exercise 5.8 (i,ii,iii).

One can e.g. solve the first part of (iii) by using that the stochastic process is the sum of two independent stochastic processes. Hence, the transition kernel corresponding to time t is a convolution of the two transition kernels corresponding to time t of the two independent processes. Of course you have to prove that the resulting generator is the sum of the respective generators. Alternatively, you can rewrite the kernel in terms of a time and space changed BM.

Exercise 5 (3pts) LN Exercise 5.11.

You may use the result of Exercise 5.10.