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Affine Markov processes on a general state space
Veerman, E.
Publication date
2011
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Citation for published version (APA):
Veerman, E. (2011). Affine Markov processes on a general state space. Uitgeverij BOXPress.
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Chapter
3
Markov processes
In this chapter we consider Markov processes and Feller processes. We develop some general theory for Markov processes where the state space assumes the par-ticular form X × Rp−m, as given in (3.2). This will be applied in Chapter 4 to
obtain the desired characterization for affine processes living on such a state space. After defining Markov transition functions, Markov semigroups and stochastic con-tinuity in Section 3.1, we derive some geometric properties of the state space in Section 3.2. Next we define in Section 3.3 the concept of regularity of a Markov process, similar to the definition of regularity for affine processes in [17, 38, 10], and we translate results of [38] from the affine setting to the more general setting. For this we make use of the symbol of a regular Markov process, a concept that can be found in [32]. We show in Proposition 3.14 that the symbol has a L´ evy-Khintchine representation under regularity conditions, which is used in Section 3.4 to characterize Feller processes as solutions of the martingale problem, see Theo-rem 3.20. This connects Chapter 2 with Chapter 3. For the proof of the theoTheo-rem, Proposition 2.5 will be crucial.
3.1
Definitions
As in the previous chapter, E ⊂ Rp is a closed set and E
∆ denotes the one-point
compactification.
Definition 3.1. A function pt(x, dz) on [0, ∞) × E × B(E) is called a
time-homogeneous Markov transition function on E if it satisfies the following prop-erties:
• pt(x, ·) is a substochastic measure on E (i.e. a measure with pt(x, E) ≤ 1),
for all t ≥ 0, x ∈ E;
• p0(x, ·) = δx, for x ∈ E;
• x 7→ pt(x, B) is Borel-measurable for B ∈ B(E), t ≥ 0;
• the Chapman-Kolmogorov relation holds, i.e. pt+s(x, B) =
Z
ps(y, B)pt(x, dy), t, s ≥ 0, x ∈ E, B ∈ B(E).
Likewise we define a Markov transition function on E∆, with the additional
re-quirement that pt(x, ·) is a probability measure on E∆.
A Markov transition function pt(x, dz) on E can always be extended to a
Markov transition function on E∆ by putting pt(∆, {∆}) = 1 and pt(x, ∆) =
1 − pt(x, E) for x ∈ E. One verifies that the above properties remain satisfied in
this extended setting. Conversely, if pt(x, dz) is a Markov transition function on
E∆such that pt(∆, {∆}) = 1 and pt(x, ∆) = 1 − pt(x, E) for x ∈ E, then pt(x, dz)
restricted to [0, ∞) × E × B(E) is a Markov transition function on E. With this in mind, we give the following definition of a time-homogeneous Markov process.
Definition 3.2. On some filtered measurable space (Ω, F , (Ft)t≥0), let (Px)x∈Ebe
a family of probability measures and let X be an E∆-valued adapted stochastic
pro-cess with Px(X0= x) = 1 for all x ∈ E. We call (X, (Px)x∈E) a time-homogeneous
Markov process with Markov transition function pt(x, dz) on E if
Px(Xt+s∈ B|FtX) = ps(Xt, B), Px-a.s., for x ∈ E, B ∈ B(E∆), (3.1)
where we put ps(∆, {∆}) = 1 and pt(x, ∆) = 1 − pt(x, E) for x ∈ E.
A Markov transition function pt(x, dz) on E induces a semi-group (Pt)t≥0
act-ing on functions f ∈ B(E) by
Ptf (x) =
Z
f (z)pt(x, dz), x ∈ E.
Indeed, the Chapman-Kolmogorov relation yields the semi-group property Pt+s =
PtPs. Note that we can write Ptf (x) = Exf (Xt) for f ∈ B(E), x ∈ E, in view
of (3.1), where (X, (Px)x∈E) is the corresponding Markov process. Here we put
f (∆) = 0 by convention. Hence if we let 1 denote the function in B(E) with constant value 1, then we can write
3.1. Definitions 53
If Pt1 = 1 for all t ≥ 0, then we call the Markov process (X, (Px)x∈E) and its
associated semi-group (Pt)t≥0 conservative.
It is worth noting that from Proposition 2.8 we infer that (X, (Px)x∈E) is
a Markov process when Px is the unique solution of the martingale problem for
(A, δx) in Ω given by (2.8) and X is the coordinate process. Indeed, define pt(x, dz)
by
pt(x, B) = Px(Xt∈ B), for t ≥ 0, x ∈ E∆, B ∈ B(E∆).
Then Proposition 2.8 yields measurability of x 7→ pt(x, B) and the Markov
prop-erty (3.1), which on its turn implies the Chapman-Kolmogorov relation. Since pt(∆, {∆}) = P∆(Xt= ∆) = 1, the restriction of pt(x, dz) to [0, ∞) × E × B(E) is
a Markov transition function on E, see the remark preceding Definition 3.2. Hence (X, (Px)x∈E) is a Markov process with transition function pt(x, dz) = Px(Xt∈ dz).
In this chapter we are interested in the converse situation, that is, we provide con-ditions such that a Markov process can be characterized as the solution of a certain martingale problem. This requires certain continuity properties of the process, to begin with stochastic continuity.
Definition 3.3. A Markov process (X, (Px)x∈E) is called stochastically continuous
if ps(x, ·) w
→ pt(x, ·) on E, for s → t, for all t ≥ 0, x ∈ E, that is, if
Z
f (z)ps(x, dz) →
Z
f (z)pt(x, dz),
as s → t, for all f ∈ Cb(E).
Remark 3.4. It is important to notice that there exist Markov processes with continuous sample paths in E∆, that are not stochastically continuous in the sense
of Definition 3.3. For example, consider the ODE
∂tf (t, x) = f (t, x)2, f (0, x) = x.
with x ∈ E := R+. Its unique solution is given by
f (t, x) = 0 for x = 0, t ≥ 0, (x−1− t)−1, for x > 0, 0 ≤ t < x−1, ∆ for x > 0, t ≥ x−1.
Take Ω = R+, F = B(R+) and define the filtration (Ft)t≥0 by Ft = B(R+) for
all t ≥ 0. Define probability measures (Px)x∈E on (Ω, F , (Ft)t≥0) by Px({x}) = 1
Px(Xt = f (t, x)) = 1 for all x ∈ E. Then (X, (Px)x∈E) is a Markov process on
E = R+ with Markov transition function pt(x, dz) on E given by
pt(x, B) = 1B(0) for x = 0, 1B((x−1− t)−1), for x > 0, 0 ≤ t < x−1, 0 for for x > 0, t ≥ x−1,
for B ∈ B(R+). The sample paths of X are continuous in E∆, Px-a.s. for all
x ∈ E. However, stochastic continuity does not hold. For example take x > 0, t = 1/x. Then R 1E(z)ps(x, dz) = 1 for all s < t, which does not converge to
R 1E(z)pt(x, dz) = 0 for s ↑ t.
3.2
Properties of the state space
Henceforth and throughout we assume that the state space E ⊂ Rp takes the
special form
E = X × Rp−m, (3.2) for some 1 ≤ m ≤ p, where X ⊂ Rmis a closed, convex set with non-empty interior
and such that
U := {u ∈ Cm: sup
x∈X
Re u>x < ∞} (3.3)
has non-empty interior. We decompose {1, . . . , p} = I ∪ J with I = {1, . . . , m}, J = {m+1, . . . , p} and accordingly a point x ∈ Cpis decomposed as x = (xI, xJ) ∈
Cm× Cp−m. For u ∈ Cp we let fu ∈ C(E) denote the function fu(x) = eu
>x
. Similarly we define fuI ∈ C(X ) and fuJ ∈ C(R
p−m), so that we have the
decom-position fuI,uJ(x) := fu(x) = fuI(xI)fuJ(xJ). Note that u ∈ U × iR
p−m if and
only if fu ∈ Cb(E). In particular it holds that iRm ⊂ U . On the other hand,
iRm∩ U◦ = ∅. This is a consequence of the next lemma, which characterizes the
interior of U .
Lemma 3.5. It holds that fu∈ C0(X ) if and only if u ∈ U◦.
Proof. First we prove that fu∈ C0(X ) for u ∈ Re U◦. Fix u ∈ Re U◦. There exists
ε > 0 such that u ± εei∈ Re U◦ for all i. Therefore,
sup
x∈X
3.2. Properties of the state space 55
Let (xn) be a sequence in E such that |xn| → ∞. Arguing by contradiction,
suppose lim supn→∞u>xn> −∞. Then there exists a subsequence, also denoted
by (xn), such that lim infn→∞u>xn > −∞, and in addition xn,i → ∞ or xn,i→
−∞. Then (3.4) yields u>x
n → −∞, a contradiction. Hence u>xn → −∞
whenever u ∈ Re U◦ and |x
n| → ∞ with xn∈ E, in other words, fu ∈ C0(X ) for
all u ∈ Re U◦. For u ∈ U◦ we have |fu(x)| ≤ fRe u(x), so by the previous we have
fu∈ C0(X ) for all u ∈ U◦.
Now suppose u ∈ U \U◦. Translating X does not affect U , so we may assume without loss of generality that 0 ∈ X . There exists v ∈ B(0, 1) such that u + εv 6∈ U for all ε > 0. This yields the existence of a sequence xn ∈ X such that
(u + v/n)>xn= n. If u>xn6→ −∞, then we are done. If u>xn→ −∞, then there
exists N > 0 such that u>xn< −1 for all n ≥ N . Henceexn = −xn/(u
>x n) ∈ X ,
for n ≥ N , by convexity of X and the fact that 0 ∈ X and xn ∈ X . We have
|exn||v| ≥ v>xen= −n
2/(u>x
n) + n ≥ n,
soxen→ ∆. On the other hand, u>xen= −1, which yields that fu6∈ C0(X ), as we needed to show.
Corollary 3.6. If u ∈ U◦ and v ∈ U , then v + λu ∈ U◦ for all λ ∈ (0, 1].
Proof. Translating X does not affect U , so we may assume without loss of gener-ality that 0 ∈ X . Let |xn| → ∞, with xn ∈ X and λ ∈ (0, 1] be arbitrary. Since
0 ∈ X , it holds that yn:= λxn∈ X and |yn| → ∞. Hence the result follows from
Lemma 3.5, as fv+λu(xn) = fv(xn)fu(yn) → 0 for n → ∞, since fv ∈ Cb(X ) and
fu∈ C0(X ).
As we saw in Lemma 3.5, there is equivalence between u ∈ U◦and fu∈ C0(X ).
The following lemma shows that a similar equivalence holds between convergence in U◦ and convergence in C0(X ).
Lemma 3.7. It holds that un → u in U◦ if and only if fun → fu in C0(X ) (i.e.
kfun− fuk∞→ 0).
Proof. The “if”-part is easy. We show the “only if”-part. First assume un and
u are real-valued. Fix ε > 0 arbitrarily. We first show that there exists M > 0 and N > 0 such that for all x ∈ X with |x| ≥ M it holds that |fu(x)| < ε and
|fun(x)| < ε for all n ≥ N . Let δ > 0 be such that vi := u + δei ∈ U
◦ and
wi:= u − δei ∈ U◦, for all i = 1, . . . , p. By Lemma 3.5 we have fvi, fwi ∈ C0(X )
such that for all x ∈ X with |x| > M it holds that |fu(x)| < ε, |fvi(x)| < ε and
|fwi(x)| < ε, for all i, i.e.
u>x ≤ log ε, v>i x ≤ log ε, w>i x ≤ log ε, for all x ∈ X . (3.5)
For all n ∈ N we can write
un= u + p X i=1 λniei− p X i=1 µniei,
with λni, µni ≥ 0. Since un → u, there exists N > 0 such that both P p i=1λ n i and Pp i=1µ n
i are bounded by δ/2 for n ≥ N . Fix n ≥ N and define [0, 1]-valued
numbers ti= λni/δ, si= µni/δ, t0= 1 − p X i=1 (ti+ si).
Then we can write
un= t0u + p X i=1 tivi+ p X i=1 siwi,
so that it follows that for all x ∈ X with |x| > M we have u>
nx ≤ log ε in view of
(3.5). Hence |fun(x)| ≤ ε for |x| > M and n ≥ N .
Next we write
fun(x) − fu(x) = (e
(un−u)>x− 1)eu>x.
For x ∈ X with |x| > M , the left-hand side is bounded by 2ε for n ≥ N . For |x| ≤ M we can bound the right-hand side by
exp(|(un− u)>x|) exp(|u>x|) ≤ exp(M |un− u|) exp(M |u|),
which can be made arbitrarily small by choosing n large enough. It follows that fun(x) → fu(x) uniformly in x.
Now assume un and u are complex-valued. Let un= an+ ibn, u = a + ib, with
an→ a in Re U◦and bn→ b in Rp. We can write
fun(x) − fu(x) = (e
a>nx− ea>x)eib>nx+ e(a+ib)>x(ei(bn−b)>x− 1),
and bound it by
|ea>nx− ea>x| + C|b n− b|,
with C = exp(supx∈Xa>x) < ∞. Both terms tend to zero uniformly in x, which
3.2. Properties of the state space 57
Corollary 3.8. For all compact K ⊂ U◦ it holds that supu∈Kkfuk∞< ∞.
Proof. By Lemma 3.7 it holds that u 7→ kfuk∞ is continuous on U◦. This
imme-diately gives the result.
We now investigate continuity properties of u 7→ R fu(x)µ(dx), where µ is
some finite measure on E. An immediate consequence of Lemma 3.7 is that u 7→ R fu(x)µ(dx) is continuous on U◦× iRp−m. However, continuity on U × iRp−m
fails in general, as is demonstrated in the next example.
Example 3.9. Consider E = X = {x ∈ R2 : x
2 ≥ x21}. Then one verifies that
Re U = (R × (−∞, 0)) ∪ {(0, 0)}. Define the finite measure µ on E with support {(k, k2
) : k ∈ N} by
µ({(k, k2)}) = 1/k2, for k ∈ N.
Take un= (n−1, −n−4) ∈ Re U , which converges to u0= 0 ∈ Re U . Then it holds
that Z fu0(x)µ(dx) = ∞ X k=1 1 k2 = π2 6 , while Z fun(x)µ(dx) = ∞ X k=1 1 k2exp(k/n − k 2 /n4) ≥ 1 n4exp(n − 1) → ∞,
for n → ∞. HenceR fun(x)µ(dx) does not converge toR fu0(x)µ(dx).
Although continuity fails on the whole of U × iRp−m, it does hold on certain
compacta of U × iRp−m, as we show in the following lemma.
Lemma 3.10. Let u ∈ U , K ⊂ U◦ be compact and define the compact set C = {λu + (1 − λ)v : λ ∈ [0, 1], v ∈ K} ⊂ U . Then for all finite measures µ on E it holds that z 7→R fz(x)µ(dx) is continuous on C × iRp−m.
Proof. Let zn → z with zn, z ∈ C × iRp−m. Then zn,I = λnu + (1 − λn)vn, for
some λn∈ [0, 1], vn∈ K. It follows that
kfznk∞≤ kfuk∞kfvnk∞,
which is uniformly bounded in n ∈ N, in view of Corollary 3.8. Dominated con-vergence yields
Z
fzn(x)µ(dx) →
Z
fz(x)µ(dx),
The next continuity property will be needed later on.
Lemma 3.11. Let (X, (Px)x∈E) be a stochastically continuous Markov process
with state space E and let the compact set C ⊂ Rm be given as in Lemma 3.10.
Then for fixed x ∈ E, it holds that (t, u) 7→ Ptfu(x) is jointly continuous on
R+× C × iRp−m.
Proof. Let (tn, un) → (t, u) in R+× C × Rp−m. As in the proof of Lemma 3.10 it
holds that kfunk∞is uniformly bounded in n. We have to show that Ptnfun(x) →
Ptfu(x). By stochastic continuity we have ptn(x, dz)
w
→ pt(x, dz). Note that if
pt(x, E) = 0, then
|Ptnfun(x)| ≤ kfunk∞ptn(x, E) → 0 = Ptfu(x).
which yields the result. Therefore, suppose pt(x, E) > 0. Then we have ptn(x, E) >
0 for n large enough. Define probability measures on E by
e pn(dz) = ptn(x, dz) ptn(x, E) , ep(dz) =pt(x, dz) pt(x, E) . Then pen w
→p. By Skorohod’s Representation Theorem [34, Theorem 4.30] theree exist E-valued random variables Yn, Y defined on a common probability space
(Ω, F , P ) such that P ◦ Yn−1=pen, P ◦ Y−1=p and Ye n→ Y , P -a.s. By dominated convergence we have Z fun(z)epn(dz) = Z fun(Yn(ω))P (dω) → Z fu(Y (ω))P (dω) = Z fu(z)p(dz).e
This gives the result.
3.3
The symbol of a regular Markov process
In this section we consider regular Markov processes. Intuitively speaking, such a process behaves locally like a L´evy process. Recall that if (X, (Px)x∈E) is a L´evy
process with L´evy triplet (b, c, K) with respect to a truncation function χ, then its Fourier-Laplace transform can be written as
Ptfu(x) = fu(x) exp(tΨ(u)),
for u ∈ U × iRp−m, where Ψ is of the L´evy-Khintchine form
Ψ(u) = u>b +12u>cu + Z
3.3. The symbol of a regular Markov process 59
Hence, we call a Markov processes regular if this holds “locally”, that is, if
Ptfu(x) ≈ fu(x) exp(tΨ(u, x)),
for some Ψ(u, x), t ≥ 0 small enough. We make this precise in the next definition.
Definition 3.12. We call (X, (Px)x∈E) regular if the right-hand derivative
∂+t
t=0Ptfu(x)
exists for all x ∈ E, u ∈ U◦× iRp−m, and is continuous in u for all x ∈ E. In that
case we define the symbol of the process X by
Ψ(u, x) = ∂ + t t=0Ptfu(x) fu(x) , (3.6)
for u ∈ U◦× iRp−m, x ∈ E. We call X strongly regular if it is regular and in
addition if for all x ∈ E, y ∈ U◦, compact K ⊂ Rp−m, there exists T > 0 such that sup z∈K,t∈[0,T ] 1 t(Ptfy,iz(x) − fy,iz(x)) < ∞.
Note that our definition of regularity slightly differs from the definition in [17], as the parameter u is restricted to U◦× iRp−m. Moreover, one usually defines the
symbol with a minus-sign on the right-hand side of (3.6), see [32]. Similar as for L´evy processes, the symbol Ψ of a regular process is also of the L´evy-Khintchine form. This is proved in Proposition 3.14 below, for which we need the following lemma. The proofs are based on the proof of [38, Theorem 4.3]. Recall that µn
converges vaguely to µ, denoted by µn v
→ µ, if Z
f (z)µn(dz) →
Z
f (z)µ(dz), for all f ∈ C0(E).
Lemma 3.13. Let µn be a sequence of finite measures on E and suppose
Z
fu(z)µn(dz) → Φ(u),
for all u ∈ U◦× iRp−m, with Φ : U◦× iRp−m → C a continuous function. Then
there exists a finite measure µ such that
(i) fθ,0(z)µn(dz) w → fθ,0(z)µ(dz) for all θ ∈ Re U◦, (ii) µn v → µ,
(iv) If in addition µn(E) → µ(E), then µn w
→ µ.
Proof. Let θ ∈ Re U◦ be arbitrary. By assumption, the characteristic functions of the exponentially tilted measures fθ,0(z)µn(dz) converge to u 7→ Φ(θ + uI, uJ),
which is continuous in 0 as u 7→ Φ(u) is continuous on U◦× iRp−m. By L´evy’s
continuity theorem we have
fθ,0(z)µn(dz) w
→ µθ(dz) (3.7)
for some measure µθ(dz) with characteristic function u 7→ Φ(θ + uI, uJ). Let µ be
a vague accumulation point of µn, which exists by Helly’s selection theorem (see
[4, Corollary 5.7.6]). Then it holds that
fθ,0(z)µn(dz) v
→ fθ,0(z)µ(dz), (3.8)
along a subsequence. Since weak convergence implies vague convergence, we infer from the uniqueness of the vague limit that
µθ(dz) = fθ,0(z)µ(dz), (3.9)
by combining (3.7) and (3.8). This yields the first assertion. Moreover, we conclude that µnhas only one vague accumulation point µ, which yields the second assertion.
Since µθ(dz) has characteristic function u 7→ Φ(θ + uI, uJ), it follows from (3.9)
that Φ(θ + uI, uJ) = Z fu(z)µθ(dz) = Z fθ+uI,uJ(z)µ(dz),
for all u ∈ iRp. This gives the third assertion, as θ ∈ Re U◦was chosen arbitrarily. The last assertion is a consequence of [41, II.6.8] (but notice the difference in terminology).
Proposition 3.14. Let (X, (Px)x∈E) be a regular Markov process with symbol Ψ.
Then there exist functions b : E → Rp, c : E → Sp+, γ : E → R+, a transition
kernel K from E to Rp\{0} with supp K(x, dz) ⊂ E − x and
Z (|z|2∧ 1)K(x, dz) < ∞, (3.10) such that Ψ(u, x) = u>b(x) +12u>c(x)u − γ(x) + Z (eu>z− 1 − u>χ(z))K(x, dz), (3.11) for u ∈ U◦× iRp−m, x ∈ E, where χ denotes a continuous truncation function.
3.3. The symbol of a regular Markov process 61
Proof. Fix x ∈ E and let tn ↓ 0. We define measures Kn(x, dz) on Rp\{0} with
supp Kn(x, dz) ⊂ E − x, by Kn(x, B) = 1 tn Z 1B(z − x)ptn(x, dz), for B ∈ B(R p\{0}).
Note that (3.6) can be rewritten as
Ψ(u, x) = lim n→∞ Z (eu>z− 1)Kn(x, dz)) + 1 tn (ptn(x, E) − 1) , for u ∈ U◦× iRp−m. Let e
µn(x, dz) be the compound Poisson distribution with
L´evy measure Kn(x, dz) and define the infinitely divisible substochastic measure
µn(x, dz) by µn(x, dz) = exp( 1 tn (ptn(x, E) − 1))µen(x, dz) Then we have exp(Ψ(u, x)) = lim n→∞exp Z (eu>z− 1)Kn(x, dz) exp 1 tn (ptn(x, E) − 1) = lim n→∞ Z fu(z)µn(x, dz),
for u ∈ U◦× iRp−m. By Lemma 3.13, there exists a finite measure µ(x, dz) that
is the vague limit of µn(x, dz). Moreover, for all θ ∈ Re U◦ we have
fθ,0(z)µn(x, dz) w
→ fθ,0(z)µ(x, dz).
Since the class of infinitely divisible measures is closed under exponentially tilting and weak convergence, it follows that fθ,0(z)µ(x, dz) is infinitely divisible and
hence the same holds for µ(x, dz). In addition, we have by Lemma 3.13 that
exp(Ψ(u, x)) = Z
fu(z)µ(x, dz), (3.12)
for u ∈ U◦× iRp−m. In particular µ(x, E) > 0. Since µ(x, E) is the vague limit
of a sequence of substochastic measures, we also have µ(x, E) ≤ 1. Therefore, γ(x) := − log µ(x, E) ≥ 0. By the L´evy-Khintchine formula, there exist b(x) ∈ Rp, c(x) ∈ S+p and a measure K(x, dz) on B(Rp\{0}) satisfying (3.10) such that
Z fu(z)µ(x, dz) = exp − γ(x) + u>b(x) +12u>c(x)u + Z (eu>z− 1 − u>χ(z))K(x, dz) ,
for u ∈ iRp
and even for u ∈ U × iRp−m in view of [48, Theorem 27.17]. Here χ
denotes a truncation function, which we choose to be continuous. Combining the above display with (3.12) gives
Ψ(u, x) + k(x, u)πi = γ(x) + u>b(x) +1 2u
>c(x)u +Z (eu>z
− 1 − u>χ(z))K(x, dz), for some k(x, u) ∈ 2Z, possibly depending on x and u. Since Ψ(u, x) as well as the right-hand side of the above display are continuous in u on U◦× iRp−m by
Lemma 3.7, it holds that k(x, u) does not depend on u. In addition, choosing u ∈ Re U◦× {0} gives k(x, u) = 0, as all other terms are real-valued. Thus we have derived that
Ψ(u, x) = γ(x) + u>b(x) +1 2u
>c(x)u +Z (eu>z
− 1 − u>χ(z))K(x, dz). Since χ is chosen to be continuous, [48, Theorem 8.7] yields
Z
f (z)Kn(x, dz) →
Z
f (z)K(x, dz), as n → ∞,
for f ∈ Cb(Rp) vanishing on a neighborhood of 0. Note that if f (z) = 0 for z ∈ E −
x, then the left-hand side of the above display is zero, since supp Kn(x, dz) ⊂ E −x.
This yields that the support of K(x, dz) is contained in E − x and we have proved the result.
3.4
Feller processes
In this section we recall the definition of a Feller process and provide conditions under which a (regular) Feller process solves a certain martingale problem.
Definition 3.15. A Markov process (X, (Px)x∈E) is called a Feller process and
the corresponding semigroup (Pt)t≥0is called a Feller semi-group if
(i) PtC0(E) ⊂ C0(E) for all t ≥ 0;
(ii) Ptf (x) → f (x) as t ↓ 0, uniformly in x ∈ E, for f ∈ C0(E).
With every Feller semi-group one can associate its infinitesimal generator , which is the linear operator A on C0(E) (i.e. linear operator from a subspace D(A) ⊂ C0(E)
to C0(E)) defined by
Af = lim
t↓0
Ptf − f
3.4. Feller processes 63
where the limit is taken with respect to the supremum norm. The domain D(A) ⊂ C0(E) of A is the subspace of those f ∈ C0(E) for which this limit exists.
Note that for a Feller process (X, (Px)x∈E) we may take Ω to be of the form
(2.8) and X the coordinate process, in view of [34, Theorem 19.15] and the ensuing paragraph. In that case we call (X, (Px)x∈E) a canonical Feller process. For a
stochastically continuous Markov process to be Feller, it suffices to have Ptf ∈
C0(E) for f in a certain subclass of C0(E). This is the content of Proposition 3.17
below, the proof of which uses the following lemma, taken from [38, Theorem 3.5].
Lemma 3.16. Define the class of functions H by
H = {x 7→ fy(xI)
Z
fiz(xJ)g(z)dz : y ∈ U◦, g ∈ Cc∞(R
p−m)}. (3.14)
It holds that the linear span of H is dense in C0(E) with respect to the supremum
norm.
Proof. Lemma 3.5 yields that xI 7→ fy(xI) ∈ C0(X ) for all u ∈ U◦, while xJ 7→
R fiz(xJ)g(z)dz ∈ C0(Rp−m) for all g ∈ Cc∞(Rp−m) by the Riemann-Lebesgue
Lemma. Hence H ⊂ C0(E). Note that the functions in H can be written as
x 7→ fy(xI)bg(xJ), where bg denotes the Fourier transform of g. Therefore, the product of two elements
x 7→ fy1(xI)gb1(xJ) and x 7→ fy2(xI)gb2(xJ) can be written as
x 7→ fy1+y2(xI) \g1∗ g2(xJ),
where g1 ∗ g2 denotes the convolution. Since U◦ is closed under addition and
Cc∞(Rp−m) is closed under convolution, it follows that H is closed under multipli-cation, i.e. H is a subalgebra of C0(E). In addition, H vanishes nowhere (i.e. there
is no x0 ∈ E such that h(x0) = 0 for all h ∈ H) and H separates points (i.e. for
all x, y ∈ E with x 6= y there exists h ∈ H with h(x) 6= h(y)). Since E is closed, it is locally compact, whence the result follows from a locally compact version of the Stone-Weierstrass Theorem (see [49, Corollary 7.3.9]).
Proposition 3.17. Let (X, (Px)x∈E) be a Markov process and let H be given by
(3.14). Then X is a Feller process if
(i) Pt(H) ⊂ C0(E) for all t ≥ 0;
Proof. We check the properties stated in Definition 3.15. Let f ∈ C0(E).
Accord-ing to Lemma 3.16 there exists a sequence fn in the linear span of H such that
kfn− f k∞ → 0, as n → ∞. By the contraction property of the semigroup Pt
we have kPtfn − Ptf k∞ ≤ kfn− f k → 0. Since Ptfn ∈ C0(E) and C0(E) is a
complete space with respect to the supremum norm, it follows that Ptf ∈ C0(E),
which gives the first property. Stochastic continuity immediately yields
Ptf (x) =
Z
f (z)pt(x, dz) →
Z
f (z)p0(x, dz) = f (x), for f ∈ Cb(E),
so in particular for f ∈ C0(E). The uniform convergence follows from [34,
Theo-rem 19.6], whence we have shown the second property.
In Theorem 3.20 below we provide conditions such that the generator of a regular Feller process (X, (Px)x∈E) assumes the form (2.2) for f ∈ Cc2(E) and we
show that Px is a solution of the corresponding martingale problem. The special
form of the generator is a consequence of the L´evy-Khintchine form of the symbol Ψ, while we use Proposition 2.5 to deduce that Px solves the martingale problem.
Lemma 3.18. Let f ∈ Ck,0
(Rp×Rq
) and define g : Rp→ C by g(x) =R
Cf (x, y)dy,
for some compact set C ⊂ Rq. Then it holds that g ∈ Ck
(Rp) and
∂xg(x) =
Z
C
∂xf (x, y)dy.
Proof. It is sufficient to prove this for k = p = 1. By Fubini we can write
g(x) = g(0) + Z C Z x 0 ∂tf (t, y)dtdy = g(0) + Z x 0 Z C ∂tf (t, y)dydt,
since (t, y) 7→ ∂tf (t, y) is continuous, whence bounded on [0, x] × C. Dominated
convergence yields that t 7→RC∂tf (t, y)dy is continuous, so that we infer from the
fundamental theorem of calculus that g is continuously differentiable with
∂xg(x) =
Z
C
∂xf (x, y)dy.
Lemma 3.19. Let K be a transition kernel from E to Rp\{0} with supp K(x, dz) ⊂
E − x, and satisfying (3.10). Let f ∈ Cb2,0(E × Rq
) and define g : E → C by g(x) =R
Cf (x, y)dy, for some compact set C ⊂ R
q. Then it holds that g ∈ C2(E)
and Z (g(x + z) − g(x) − ∇g(x)>χ(z))K(x, dz) = Z C Z (f (x + z, y) − f (x, y) − ∇f (x, y)>χ(z))K(x, dz)dy.
3.4. Feller processes 65
Proof. By Lemma 3.18 we have g ∈ C2(E) and ∇g(x) =R
C∇f (x, y)dy. Therefore, Z (g(x + z) − g(x) − ∇g(x)>χ(z))K(x, dz) = Z Z C (f (x + z, y) − f (x, y) − ∇f (x, y)>χ(z))dyK(x, dz).
It remains to show that the order of integration can be interchanged by Fubini’s theorem. Fix x0∈ E. SinceR (|z|2∧1)K(x0, dz) < ∞ and supp K(x0, dz) ⊂ E−x0,
it suffices to show that
|f (x0+ z, y) − f (x0, y) − ∇f (x0, y)>χ(z)| ≤ M (|z|2∧ 1), (3.15)
for some M > 0, for all z ∈ E − x0, y ∈ C. Recall that χ(z) = z in a neighborhood
of 0, so there exists ε > 0 such that we can write
f (x0+ z, y) − f (x0, y) − ∇f (x0, y)>χ(z) = X i,j Z 1 0 Z 1 0 fij(x0+ stz, y)tzizjds dt,
for all |z| ≤ ε, where fij is short-hand notation for ∂xi∂xjf . By assumption
(x, y) 7→ fij(x, y) is continuous. In particular it is bounded on B(x0, ε) × C.
It follows that
|f (x0+ z, y) − f (x0, y) − ∇f (x0, y)>χ(z)| ≤ M1|z|2,
for some M1 > 0, for all |z| ≤ ε, y ∈ C. Since f and χ are bounded and since
y 7→ ∇f (x0, y) is bounded on C by continuity, we have
|f (x0+ z, y) − f (x0, y) − ∇f (x0, y)>χ(z)| ≤ M2,
for some M2 > 0, for all |z| > ε, y ∈ C. This yields (3.15) and the lemma is
proved.
Theorem 3.20. Let (X, (Px)x∈E) be a strongly regular canonical Feller process,
with infinitesimal generator A and symbol Ψ given by (3.11), for measurable func-tions b : E → Rp, c : E → Sp
+, γ : E → R+, a transition kernel K from E
to Rp\{0} with supp K(x, dz) ⊂ E − x, and satisfying (2.1). In addition, let A]: C2
b(E) → M (E) be the linear operator given by
A]f (x) = ∇f (x)>b(x) +1 2tr (∇ 2f (x)c(x)) − γ(x)f (x) + Z (f (x + z) − f (x) − ∇f (x)>χ(z))K(x, dz).
Assume A](Cc2(E)) ⊂ C0(E) as well as A](H) ⊂ C0(E), where H is given by
(3.14). Then C2
c(E) ⊂ D(A) and A|C2
c(E) = A
]| C2
c(E). In particular, Px is a
Proof. Throughout we make use of the equality Z A]f y,iz(x)g(z)dz = A] Z fy,iz(x)g(z)dz , for y ∈ U◦, g ∈ Cc∞(Rp−m), (3.16)
which is a consequence of Lemmas 3.18 and 3.19. We first show that H ⊂ D(A) and that
Ah = A]h, for all h ∈ H. (3.17)
Let y ∈ U◦, g ∈ Cc(Rp−m) and take h(x) =R fy,iz(x)g(z)dz. Then h ∈ C0(E) by
Lemma 3.16. Note that for u ∈ U × iRp−m we can write A]fu(x) = fu(x) u>b(x) +1 2u >c(x)u − γ(x) + Z (eu>z− 1 − u>χ(z))K(x, dz) . (3.18)
Fubini and strong regularity of X yields 1
t(Pth(x) − h(x)) = Z 1
t(Ptfy,iz(x) − fy,iz(x))g(z)dz → Z
Ψ(y, iz, x)fy,iz(x)g(z)dz,
as t ↓ 0, for all x ∈ E. From (3.11) we infer that Ψ(u, x)fu(x) = A]fu(x) for
u ∈ U◦× iRp−m, x ∈ E. Together with (3.16) this gives
1
t(Pth(x) − h(x)) → A
]h(x),
as t ↓ 0, for all x ∈ E. Since (Pt)t≥0 is a Feller semi-group with generator A
and A]h ∈ C0(E) by assumption, the pointwise convergence in the above display
suffices for uniform convergence, see [48, Lemma 31.7]. Hence h ∈ D(A) and (3.17) holds.
Let Tn be the stopping time given by (2.7), n ∈ N. By Dynkin’s formula [34,
Lemma 19.21], it holds for all h ∈ H that
h(XTn
t ) − h(X0) −
Z t∧Tn
0
Ah(Xs)ds
is a Px-martingale for all x ∈ E. Let (y, iz0) ∈ U◦× iRp−m be arbitrary. Take
a sequence of non-negative functions (gk) in Cc∞(Rp−m) with support contained
in the unit ball B(0, 1) in Rp−m, withR gk(z)dz = 1 for all k ∈ N and such that
gk(z)dz converges weakly to δz0(dz). We define a sequence (hk) in H by
hk(x) =
Z
3.4. Feller processes 67 We show that hk(XtTn) − Z t∧Tn 0 A]h k(Xs)ds L1 → fy,iz0(X Tn t ) − Z t∧Tn 0 A]f y,iz0(Xs)ds, (3.19)
as k → ∞, so that the right-hand side, being an L1
-limit of a sequence of Px
-martingales, is a Px-martingale itself for all x ∈ E. Since hk(x) → fy,iz0(x) and
khkk∞= kfyk∞< ∞, we have hk(XtTn) L1 → fy,iz0(X Tn t ) as k → ∞. By (3.16) it holds that Z t∧Tn 0 A]h k(Xs)ds = Z t∧Tn 0 Z A]f y,iz(Xs)gk(z)dzds.
From (3.18) together with (2.1) it follows that z 7→ A]fy,iz is continuous, whence
bounded on B(0, 1). Hence
Z
A]fy,iz(x)gk(z)dz → A]fy,iz0(x),
as gk(z)dz w
→ δz0(dz). It also follows from (3.18) together with (2.1) that
sup |x|≤n,|z|≤1 |A]f y,iz(x)| := C < ∞. This gives Z A]f y,iz(Xs)gk(z)dz ≤ C Z gk(z)dz = C,
for s < Tn. From dominated convergence we deduce that
Z t∧Tn 0 Z A]f y,iz(Xs)gk(z)dzds L1 → Z t∧Tn 0 A]f y,iz0(Xs)ds,
whence we have that (3.19) holds. It remains to show that fu(XtTn) −
Rt∧Tn
0 A
]f
u(Xs)ds is a Px-martingale for
all x ∈ E, u ∈ iRp, n ∈ N. Indeed, in that case, Proposition 2.5 yields that Px is
a solution of the martingale problem for (A], δ
x) for all x ∈ E, while Lemma 2.11
gives A]f (x) = ∂t+ t=0Exf (Xt) = ∂ + t t=0Ptf (x), for f ∈ C 2 c(E).
Since (Pt)t≥0 is a Feller semi-group with generator A and A](C0(E)) ⊂ C0(E)
uniform convergence, see [48, Lemma 31.7], whence f ∈ D(A) and A]f = Af for
f ∈ C2 c(E).
Let u0∈ iRpbe arbitrary and take a sequence (uk) in U◦×iRp−mthat converges
to u0 and with the property that
sup
k∈N
kfukk∞< ∞, (3.20)
which is possible in view of the proof of Lemma 3.10. Since χ(z) = z in a neigh-borhood of 0, there exists ε > 0 such that for |z| ≤ ε it holds that
|eu>kz− 1 − u> kχ(z)| ≤ |uk|2|z|2 Z 1 0 Z 1 0 exp(u>kzst)t ds dt ≤ C|z|2,
for some constant C > 0, uniformly in k. Also, since χ is bounded and since we have (3.20), there exists C > 0 such that for all x ∈ E , |z| > ε with z ∈ E − x, k ∈ N, we have that |fuk(x)(e u>kz− 1 − u> kχ(z))| = |fuk(x + z) − fuk(x)(1 + u > kχ(z))| ≤ C.
By assumption supp K(x, dz) ⊂ E − x, whence by the previous we have
|fuk(x) Z (eu>kz− 1 − u> kχ(z))K(x, dz)| ≤ C Z (|z|2∧ 1)K(x, dz), for some C > 0, for all x ∈ E. Hence (3.18) together with (2.1) gives
sup
|x|≤n,k∈N
A]f
uk(x) < ∞,
while dominated convergence gives
Z (eu>kz− 1 − u> kχ(z))K(x, dz) → Z (eu>0z− 1 − u> 0χ(z))K(x, dz), as k → ∞. Hence A]fuk(x) → A ]f
u0(x) for all x ∈ E, as k → ∞, in view of
(3.18). Using dominated convergence again, we obtain
fuk(X Tn t ) − Z t∧Tn 0 A]f uk(Xs)ds L1 → fu(XtTn) − Z t∧Tn 0 A]f u(Xs)ds,
whence the right-hand side, being an L1
-limit of a sequence of Px-martingales, is
