Comparison of Processes

Thus, we have found a canonical process Y , equivalent to the original process X, whose new sample space is the path space restricted to paths admitting the property Π, and which can be “pulled back” to the original measures by equation (7.2). These results can be further extended to fit into the context of right processes, see, e.g., [Sha88, Proposition 19.6 and Theorem 19.7].

This construction of switching to the (restricted) path space is strictly easier than other procedures presented in the literature, such as in [BB96, Section 38], which are used to restrict the original sample space and the underlying σ-algebra to a (not necessarily measurable) subset having full outer measure. We can resort to the easier method above, as the σ-algebraFY = σ(Yt, t ≥ 0) generated by the coordinate process will turn out

sufficient for our applications and all paths—in contrast to a subset of paths with full (outer) measure—will feature the desired property Π.

7.2. Comparison of Processes

The procedure above can be used to establish a common basis on which we are then able to compare two equivalent processes which are defined on different probability spaces: Let X, Z be two right continuous, (E,E )-valued stochastic processes on (ΩX,FX, PX), (ΩZ,FZ_{, P}Z_{) respectively, having the same finite dimensional distributions, that is,}

satisfying for all n ∈ N, f1, . . . , fn∈ bE , t1, . . . , tn∈ R+: EX f1(Xt1) · · · fn(Xtn)

= EZ f1(Zt1) · · · fn(Ztn)

8. Stopping 51

Define Ω := {ω : R+→ E | ω is right continuous}, together with the path mappings ΦX: ΩX → Ω, ωX 7→ ΦX(ωX) with ΦX(ωX)(t) := X_t(ωX), t ≥ 0,

ΦZ: ΩZ → Ω, ωZ 7→ ΦZ(ωZ) with ΦZ(ωZ)(t) := Z_t(ωZ), t ≥ 0,

and consider the right continuous canonical coordinate process Y on Ω with its generated σ-algebra FY as above. Then, as seen in equation (7.1), we have

Yt◦ ΦX = Xt on ΩX, Yt◦ ΦZ= Zt on ΩZ,

so the equivalence of the processes X and Z yields that for all n ∈ N, f1, . . . , fn ∈ bE , t1, . . . , tn∈ R+, EX f1(Yt1 ◦ ΦX) · · · fn(Ytn◦ Φ X₎
= EZ f1(Yt1◦ ΦZ) · · · fn(Ytn◦ Φ Z₎
. As f1(Yt1) · · · fn(Ytn); n ∈ N, f1, . . . , fn∈ bE , t1, . . . , tn∈ R+ is an ∩-stable generator of FY = σ(Yt, t ≥ 0), the MCT then concludes that

∀G ∈ bFY _:

EX(G ◦ ΦX) = EZ(G ◦ ΦZ).

8. Stopping

We are going to consider the following question: Let X be a (strong) Markov process and τ be a random time. Is the stopped process X· ∧τ still a (strong) Markov process? It seems that this problem is not commonly treated in the literature, the only source known to us is [Dyn65, Section X.2]. One may suspect that the random time τ must be a stopping time which admits the “memoryless” property, encoded in the concept of terminal times, in order to prevent a “memory structure” to be introduced by the transformation of stopping. Furthermore, if the transformed process possesses the Markov property, it must stop immediately again when restarted at the stopping point, that is, the stopping time τ should “trigger instantly” when the original process (re-)starts at X_τ. Indeed, a rigorous refinement of these heuristic conditions ensures the stopped process to be (strongly) Markovian: Let X = Ω,G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E
be a right

continuous (strong) Markov process on (E,E ), and assume that there exists a constant path in x for every x ∈ E, that is,

∃ω_x∈ Ω : ∀t ≥ 0 : Xt(ωx) = x.

(8.1)

Furthermore, let τ be a terminal time for X, which satisfies for all x ∈ E the condition PXτ(τ = 0) = 1 Px-a.s. .

(8.2)

Define the processX resulting from stopping X at τ by the processe Xe_t= X_t∧τ, t ≥ 0,

52 9. Time Change shift operators e Θt(ω) := ( Θt(ω), t < τ (ω), ωXτ(ω), t ≥ τ (ω), ω ∈ Ω, t ≥ 0.

Then the stopped processX is (strongly) Markovian:e

(8.3) Theorem. Let X = Ω,G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E

be a right continuous Markov process on (E,E ) and τ be a terminal time for X, which satisfy conditions (8.1) and (8.2). Then the stopped process X = Ω,e G , (Ff_t)_t≥0, (Xe_t)_t≥0, (Θe_t)_t≥0, (P_x)_x∈E

is a right continuous Markov process on (E,E ). If X is a strong Markov process, thenX is ae

strong Markov process as well.

This theorem is a slight generalization of [Dyn65, Theorem 10.2], its proof proceeds completely analogously and can be found in [Wer10]. It has also been shown in [Wer10] that the conditions of the theorem above cannot be weakened in general.

The conditions on τ are always satisfied by the first hitting time H_A = inf{t ≥ 0 : Xt∈ A} of any closed set A: It is a terminal time by theorem (3.8), right continuity of X implies that XHA ∈ ¯A = A holds, and normality then ensures (8.2). The existence of

constant paths (8.1) is not very restrictive and usually can be achieved by adjoining a null set of the needed points to the sample space Ω, see, e.g., [Dyn65, footnote on p. 79].

9. Time Change

We briefly remind the technique of time-changing a Markov process. This technique seems to date back to [Boc55] and [Vol58], and found extensive applications in potential theory, see, e.g., [BG69, Chapter V] or [Sha88, Chapter IV]. Nowadays, results on this topic can fill up libraries alone (cf. [CF11] for a modern treatment). We only give one result concerning the time change of a right process with respect to a perfect continuous additive functional, which we will employ later.

We try to motivate this technique by giving the following basic idea, fitting to our context: If the time scale of a stochastic process t 7→ Xtis changed by some increasing

function τ tending to infinity, then the time changed process t 7→ X_{τ (t)} will assume the same hitting distributions as the original process, only the hitting times of any given set will be adjusted. Thus, in regard to Dynkin’s formula for the generator (3.18), it is natural to expect that the generator of a time scaled Markov process equals the original generator, rescaled at every point subject to the time changing function τ (for a rigorous result, see [Dyn65, Theorem 10.12]). On the other hand, under some regularity conditions, it can be shown that any two Markov processes with the same hitting distributions are equivalent up to a time change (see [BG69, Theorem 5.1]).

9.1. Additive Functionals

There is a finely tuned classification for additive functionals, see [Sha88, Section IV.35, Chapter VIII]. We only consider a special case, following [RW00a, Definition III.16.3]:

9.2. Basic Result 53

(9.1) Definition. A perfect continuous additive functional with respect to some Markov

process X = Ω,G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E
is an (Ft, t ≥ 0)-adapted process A = (At, t ≥ 0) with values in R+, which satisfies the following properties on some set Ω₀ ∈G with P_x(Ω₀) = 1 for all x ∈ E:

(i) A0 = 0;

(ii) t 7→ At is monotone increasing and continuous;

(iii) As+t = As+ At◦ Θs holds for all s, t ≥ 0.

Let R_A:= inf{t ≥ 0 : At> 0}. The fine support of A is given by

supp(A) := reg(RA) =x ∈ E : Px(RA= 0) = 1 .

(9.2) Example. Let B be the standard Brownian motion on R and L = (Lt, t ≥ 0) be

its local time at the origin (cf. section 15). Then L is a perfect continuous additive functional with respect to B with fine support supp(L) = {0}. Furthermore,

At:= t + c Lt, t ≥ 0,

is a perfect continuous additive functional for any c ≥ 0, with supp (At, t ≥ 0)
= R. 

9.2. Basic Result

We will only need one well-known result, which can be found, e.g., in [Sha88, Theorem 65.9] or [CF11, Theorem A.3.11]:

(9.3) Theorem. Let X = Ω,G , (Gt)t≥0, (Xt)t≥0, (Θt)t≥0, (Px)x∈E

be a right process and A = (A_t, t ≥ 0) be a perfect continuous additive functional with respect to X. Define the right continuous “pseudoinverse process” (τt, t ≥ 0) of A by

τt:= inf{s ≥ 0 : As> t}, t ≥ 0,

and the time changed process Y with its shift operators by Yt:= Xτ (t), Θˆt:= Θτ (t), t ≥ 0.

Then Y = Ω,G , (G_{τ (t)})t≥0, (Yt)t≥0, ( ˆΘt)t≥0, (Px)x∈supp(A)

is a right process on supp(A).

10. Killing

Construction of subprocesses by curtailing the lifetime of a Markov process is mainly done by killing with respect to multiplicative functionals. This is a classic, well-understood field, which has deep applications in potential theory (see [BG69, Chapter III]), and which is also applicable to right processes, see [Sha88, Chapter VII]. However, we will not need these results in their full generality, so we restrict our attention to two easier methods:

54 10. Killing