2. Basic Theory of Markov Processes
2.3. Connection to the Theory of Semigroups
where the semigroup property follows directly from the Markov property. One of the most important features of a semigroup associated with a right continuous Markov process will be the following:
(2.4) Lemma. Let X be a right continuous Markov process with semigroup (Tt, t ≥ 0).
Then the mapping R+→ R, t 7→ Ttf (x) is right continuous for every x ∈ E, f ∈ bC(E). Proof. This follows directly from the right continuity of t 7→ Xt, together with LDCT.
The following two fundamental definitions summarize all properties of definition (2.2) and implement the connection to the associated semigroup:
(2.5) Definition. Let (Tt, t ≥ 0) be a Markov semigroup on (E,E ). The tuple X =
Ω,G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E
is a right continuous simple (E -)Markov pro- cess with transition semigroup (Tt, t ≥ 0), if properties (i), (ii), (iii), (v) and (vi) of
definition (2.2) are fulfilled and if
(iv’) X has the Markov property with respect to the semigroup (Tt, t ≥ 0):
∀s, t ≥ 0, f ∈ bE : Ex f (Xs+t)
Gs
= Ttf (Xs).
(2.6) Definition. X = Ω,G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E is a right continuous simple (E -)Markov process, if it is a right continuous simple (E -)Markov process with transition semigroup (Tt, t ≥ 0) given by equation (2.3).
14 2. Basic Theory of Markov Processes
We close this introduction by recalling that, with help of the shift operators, it is possible to lift the Markov property to general bounded functions which are measurable with respect to the σ-algebra generated by the process. The following theorem is a standard result which can be proved using MCT (see, e.g., [BG69, Theorems I.1.3, I.3.6]):
(2.7) Theorem. Let X be a right continuous Markov process on (E,E ), and consider
the σ-algebraF0 := σ(Xt, t ≥ 0) generated by X. Then for all Y ∈ bF0, the mapping x 7→ Ex(Y ) is E -measurable, and for all x ∈ E, t ≥ 0,
Ex Y ◦ Θt
Gt
= EXt(Y ).
2.2. The Usual Hypotheses
We briefly summarize the “usual hypotheses”, which will be in force for the majority of our work, and the standard technique leading to them. These hypotheses will ensure the right continuity of the underlying filtration, that is
∀t ≥ 0 : Ft= ∩s>tFs=:Ft+,
as well as the measurability of most basic random times, like the first hitting times introduced in section 3. It turns out that the proper method to achieve above-mentioned features is to complete the entire σ-algebra σ(Ft, t ≥ 0), and then augment the filtration by the null sets of this completed entire σ-algebra; it does not suffice to solely complete every single σ-algebraFt of the filtration. Furthermore, as every Markov process has
a whole set of associated measures (Px, x ∈ E), completions and augmentations must
be “universal”, that is relative to this whole set. This results in the following procedure, which is completely laid out, e.g., in [BG69, Section I.5] and [Sha88, Sections 3, 6]. Before we start, we would like to remind the reader that this procedure is even necessary in the most basic cases such as in the setting of continuous stochastic processes like the Brownian motion (see, e.g., [KS91, Problem 2.7.4]).
Let E be a Radon space equipped with the Borel σ-algebraE := B(E), and define the σ-algebraEu of universally measurable subsets of E by
Eu:=\
{Eµ: µ finite measure on E},
whereEµ is the µ-completion of E (for basic results concerning universal completions, see, e.g., [Sha88, Appendices A1–A2]).
Consider an intermediate σ-algebraE ⊆ E•⊆Eu (typically, • = u or • = 0, with the
latter case beingE0 :=E ). Let X = Ω, G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E
be a right continuous simpleE•-Markov process with transition semigroup (Tt, t ≥ 0) and state space E. Define the “raw” natural filtration (Ft•, t ≥ 0) by
F•
t := σ f (Xs), s ≤ t, f ∈ bE•, t ≥ 0,
and the σ-algebra generated by the process by F•:= σ f (X
t), t ≥ 0, f ∈ bE•
2.2. The Usual Hypotheses 15
Because (Tt, t ≥ 0) is a family of kernels with Px ◦ Xt−1 = Tt(x, · ), the mapping x 7→ Px f (Xt) is E•-measurable for f ∈ bE•, t ≥ 0. Then, by MCT, the mapping x 7→ Ex(Y ) is measurable for every Y ∈ bF• (see, e.g., [BG69, Theorem I.3.6], [Sha88,
Lemma (2.6)]). Thus, we can define for every finite measure µ onE• a measure Pµby Pµ(A) :=
Z
Px(A) µ(dx), A ∈F•.
Following [Sha88, Section 3], we consider the usual augmentations:
(2.8) Definition. For every probability measure µ on (E,E•), letFµ denote the com- pletion of Fu relative to Pµ, and let Nµ denote the set of all Pµ-null sets inFµ. For any t ≥ 0, set
(i) F :=T{Fµ: µ probability measure on E},
(ii) N :=T{Nµ: µ probability measure on E},
(iii) Ftµ:=Ftu∨N µ, µ probability measure on E,
(iv) Ft:=T{Fµ
t : µ probability measure on E}.
As [Sha88] points out, this definition “is not the one most common in the literature”, which however is fixed by [Sha88, Proposition (3.8)]:
(2.9) Lemma. For every probability measure µ on E,
(i) Fµ is the Pµ-completion of F0,
(ii) for every t ≥ 0,Ftµ=Ft0∨Nµ holds true.
The following theorem [Sha88, Theorem (3.9)] simplifies the work with the augmented filtration (Ft, t ≥ 0):
(2.10) Theorem. For every t ≥ 0, Ft=Ft0∨F0u∨N = Ftu∨N holds true. That is,
Ft is generated by random variables of the form
f (X0) f1(Xt1) · · · fn(Xtn) + H,
with 0 < t1 < · · · < tn, f ∈ bEu, f1, . . . , fn∈ bE and H ∈ bF with {H 6= 0} ∈ N .
We define the augmentation of the filtration (Gt, t ≥ 0) analogously to definition (2.8): (2.11) Definition. For every x ∈ E, let Nx(G ) denote the set of all Px-null sets in the completion Gx ofG relative to Px. For any t ≥ 0, set
(i) G :=T
x∈EGx,
(ii) N (G ) :=T
16 2. Basic Theory of Markov Processes
(iii) Gtx :=Gt∨N x(G ), x ∈ E,
(iv) Gt:=T
x∈EGtx.
The basic principle is that, “roughly speaking, one can replace the σ-algebras Gt and F0
t [in the definitions and results of subsection 2.1] by Gt and Ft, provided one replaces
E by Eu” (cf. [BG69, p. 28]). We summarize [BG69, Propositions I.5.8–I.5.12]:
(2.12) Theorem. For all F ∈ bF , s, t ≥ 0, the mapping x 7→ Ex(F ) isEu-measurable, Xt isFt/Eu-measurable, Θt isFs+t/Fs-measurable, and for any x ∈ E,
Ex(F ◦ Θt|Gt) = Ex(F ).
In many cases, right continuity of the augmented natural filtration (Ft, t ≥ 0) is ensured by the following result (cf. [BG69, Proposition I.8.12]):
(2.13) Lemma. If X admits the Markov property relative to the filtration (Ft+0 , t ≥ 0), thenFt=Ft+ holds for all t ≥ 0.
Therefore, we usually can (and will) assume the filtrations (Ft, t ≥ 0) and (Gt, t ≥ 0)
of a Markov process to be augmented, and the natural filtration (Ft, t ≥ 0) to be right
continuous. These conditions are called the usual hypotheses.
We end this section by citing Blumenthal’s zero–one law [BG69, Proposition I.5.17], which really gains its power through the augmentation (the same result forF00 instead ofF0 would be trivial due to the normality of the process):
(2.14) Corollary. Let X be a Markov process. Then for all x ∈ E, A ∈F0, Px(A) ∈ {0, 1}.
2.3. Connection to the Theory of Semigroups
Let X = Ω,G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈Ebe a right continuous simple Markov
process with transition semigroup (Tt, t ≥ 0) and state space E.
(2.15) Definition. The resolvent of X is the family of linear operators (Uα, α ≥ 0) on
(E,E ), defined for all α ≥ 0, f ∈ pEu or α > 0, f ∈ bEu by ∀x ∈ E : UαXf (x) = Ex Z ∞ 0 e−αtf (Xt) dt .
An interchange of the order of integration (justified by [Sha88, Proposition 4.3]) gives
(2.16) Theorem. For α ≥ 0, f ∈ pEu or α > 0, f ∈ bEu, UαXf (x) =
Z ∞
0
2.3. Connection to the Theory of Semigroups 17
Thus, the resolvent (UX, α > 0) of the Markov process X coincides with the resolvent (Uα, α > 0) of the corresponding semigroup (Tt, t ≥ 0) (on their shared domain), and we
will omit the superscript X of UX. All properties of the resolvent U thus hold for UX as well. Especially, as the semigroup is uniquely characterized by its restriction to bC(E) and t 7→ Ttf (x) = Ex f (Xt) is right continuous for every f ∈ bC(E) by lemma (2.4),
theorem (1.11) immediately yields:
(2.17) Corollary. The resolvent of the semigroup (Tt, t ≥ 0) of a right continuous Markov
process completely determines (Tt, t ≥ 0).
As the semigroup property of (Tt, t ≥ 0) is the reflection of the Markov property of the underlying process X, it is not surprising that the Markov property can be equivalently characterized by a condition on the resolvent:
(2.18) Theorem. Let X = Ω,G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E
satisfy the prop- erties (i), (ii), (iii), (v) and (vi) of definition (2.2) with respect to E = Eu. Set Ttf (x) := Ex f (Xt) for all t ≥ 0, f ∈ bEu, x ∈ E. Then X is a right continuous
simple Eu-Markov process with transition semigroup (Tt, t ≥ 0) on the state space E, if and only if
(iv”) for all α > 0, s ≥ 0, f ∈ bC(E), J ∈ bGs,
Ex Z ∞ 0 e−αtf (Xs+t) dt · J = Ex Uαf (Xs) · J.
Proof. We need to show the equivalence of (iv”) and (iv’). We note that, by the MCT, (iv’) is equivalent to its restriction on f ∈ bC(E), as
H :=
f ∈ bEu : Ex f (Xs+t) Gs= Ttf (Xs)
is a MVS and the Borel σ-algebra E is generated by bC(E); so if bC(E) ⊆ H , then bE ⊆ H , and, by using sandwiching, H = bEu holds true. As Ttf (Xs) isGs-measurable,
condition (iv’) holds, if and only if for all s, t ≥ 0, f ∈ bC(E), J ∈ bGs:
Ex f (Xs+t) J= Ex Ttf (Xs) J.
(2.19)
But then (iv”) is just the Laplace transform of (iv’): Both sides of above equation (2.19) are right continuous in t (see lemma (2.4) and its proof), so it is equivalent to its Laplace- transformed version. That is, (iv’) holds, if and only if we have for all α > 0, s ≥ 0, f ∈ bC(E), J ∈ bGs: Z ∞ 0 e−αtEx f (Xs+t) J dt = Z ∞ 0 e−αtEx Ttf (Xs) J dt,
which, after an interchange of the order of integration with Fubini–Tonelli’s theorem (see [Sha88, Proposition 4.3]), is just condition (iv”).