In this section we are concerned with multidimensional additive maps and several of its properties. Let X be a consistent progressive Markov process on a measurable space (Ω,F ) with state space S. We utilize the natural backward filtration ( ˆFt0)t∈J
of X and the σ-fields ˆFr,t = σ(Xs : s ∈ [r, t]) for all r, t ∈ J with r ≤ t. As before,
k ∈ N, and | · | is the Euclidean norm on Rk and the Frobenius norm on Rk×k.
3.15 Definition. A k-dimensional additive map (of X ) is given through a map
κ : Ω ×B(J) → [−∞, ∞]k, (ω, B) 7→ κ(B)(ω) with the following two properties:
(i) κi(·)(ω) : B(J) → [−∞, ∞], B 7→ κi(B)(ω) is a signed Borel measure for all
i ∈ {1, . . . , k} and each ω ∈ Ω.
(ii) The map κ([r, t]) : Ω → Rk, ω 7→ κ([r, t])(ω) is Fˆ
r,t-measurable for every
r, t ∈ J with r ≤ t.
If in addition κi ≥ 0 for all i ∈ {1, . . . , k}, then we call κ non-negative. Moreover, a
k-dimensional additive map κ with κ({t}) = 0 for all t ∈ J is said to be continuous.
In alignment with the literature, we call every one-dimensional additive map an
additive functional (cf. Dynkin [11, Appendix], [12, Section 1.1]). From Lemma A.19
we easily draw the conclusion that an [−∞, ∞]k-valued map κ on Ω ×B(J) is a
k-dimensional additive map if and only if all its coordinate functions κ1, . . . , κk
are additive functionals. In this case, κ is non-negative (resp. continuous) if and only if κ1, . . . , κk are. Let us prove a multidimensional analogue to statement 0.2.E
in [12, Appendix].
3.16 Lemma. Assume that κ is a non-negative k-dimensional additive map of X .
Let θ ∈ B(J × S, Rk
+) be such that the function J → R+, s 7→ θi(s, Xs(ω)) is
locally κi(·)(ω)-integrable for every i ∈ {1, . . . , k} and each ω ∈ Ω. Then the map
ν : Ω ×B(J) → [0, ∞]k given by ν(B)(ω) :=
Z
B
θ(s, Xs(ω)) κ(ds)(ω)
is another non-negative k-dimensional additive map that is continuous if κ is.
Proof. From above remarks we infer that it suffices to show the claim for k = 1.
Because X is progressive and θ is Borel measurable, the process
J × Ω → R+, (s, ω) 7→ θ(s, Xs(ω))
must be B(J) ⊗ F -measurable. Hence, the function J → R+, s 7→ θ(s, Xs(ω)) is
B(J)-measurable and, by hypothesis, locally κ(·)(ω)-integrable for each ω ∈ Ω. For this reason, ν(·)(ω) is a Borel measure on J .
It remains to verify that ν([r, t]) is ˆFr,t-measurable for all r, t ∈ J with r ≤ t.
an increasing sequence (θn)n∈N in Bb(J × S, R+) that converges pointwise to θ. By monotone convergence, lim n↑∞ Z t r θn(s, Xs(ω)) κ(ds)(ω) = Z t r θ(s, Xs(ω)) κ(ds)(ω) = ν([r, t])(ω)
for all ω ∈ Ω. Thus, as pointwise limit of a sequence of R+-valued ˆFr,t-measurable
functions on Ω, it follows that ν([r, t]) is Fˆr,t-measurable. This justifies that the
boundedness assumption on θ leads to no loss of generality. Now, the setHr,tof all
B([r, t]) ⊗ ˆFr,t-measurable bounded processes Z : [r, t] × Ω → R for which
Z t
r
Zsκ(ds) is ˆFr,t-measurable
is a monotone class on [r, t] × Ω. For each s ∈ [r, t] and all A ∈ ˆFr,t, the process
[r, t] × Ω → [0, 1], (r0, ω) 7→1[r,s]×A(r0, ω) is a member of Hr,t, since
Z t
r 1[r,s]×A
(r0, ω) κ(dr0)(ω) = κ([r, s])(ω)1A(ω)
for all ω ∈ Ω. Consequently, the Functional Monotone Class Theorem A.29 implies that Hr,t is the linear space of all real-valued B([r, t]) ⊗ ˆFr,t-measurable bounded
processes on [r, t] × Ω. As X is progressive and θ is assumed to be bounded, the process [r, t] × Ω → R+, (s, ω) 7→ θ(s, Xs(ω)) belongs toHr,t, by Lemma 3.9. Hence,
ν is a non-negative additive functional.
Lastly, from the definition of ν we conclude that ν(·)(ω) is absolutely continuous with respect to κ(·)(ω) for all ω ∈ Ω. This in turn entails that if κ is continuous, then {t} is not only a κ(·)(ω)-null set, but also a ν(·)(ω)-null set for each t ∈ J and every ω ∈ Ω. This proves the assertion.
Let temporarily µ be a signed Borel measure on J . Then the positive part µ+
and negative part µ− of µ are given by µ+(B) = sup{µ(A) | A ∈ B(J) : A ⊂ B}
and µ−(B) = sup{−µ(A) | A ∈ B(J) : A ⊂ B} for each B ∈ B(J), which are two Borel measures that satisfy the Jordan decomposition
µ = µ+− µ−.
The variation of µ is another Borel measure given by |µ| = µ++ µ− that fulfills
|µ(B)| ≤ |µ|(B) for all B ∈ B(J) (see for instance Cohn [5, Section 4.1]). In what follows, let I be a non-degenerate interval in J .
3.17 Lemma. Suppose that κ is a continuous k-dimensional additive map of X .
Let Y : I × Ω → Rk be an ( ˆF0
s)s∈I-progressively measurable process and t ∈ I such
that
Z t
r
|Y(i)
s (ω)| |κi|(ds)(ω) < ∞ (3.7)
for all i ∈ {1, . . . , k} and each (r, ω) ∈ It× Ω. Then the map Z : It× Ω → Rk
defined via
Zr(ω) :=
Z t
r
Ys(ω) κ(ds)(ω)
Proof. As Y is ( ˆFs0)s∈I-progressively measurable, its restriction to [r, t] × Ω must
be B([r, t]) ⊗ ˆFr0-measurable for all r ∈ It. Hence, from (3.7) we deduce that Z is
well-defined. Moreover, since an [−∞, ∞]k-valued map on Ω ×B(J) is a continuous
k-dimensional additive map if and only if all its coordinate functions are continuous
additive functionals, it is enough to show the assertion for k = 1.
Let us prove that Z(ω) is continuous for each ω ∈ Ω. To this end, let r ∈ It
and (rn)n∈N be a sequence in It with limn↑∞rn = r. Then it is readily checked that
limn↑∞1[rn,t](s) = 1[r,t](s) for all s ∈ It with s 6= r. Because κ({r})(ω) = 0, we
obtain from dominated convergence that lim n↑∞Zrn(ω) = limn↑∞ Z It 1[rn,t](s)Ys(ω) κ(ds)(ω) = Z It 1[r,t](s)Ys(ω) κ(ds)(ω) = Zr(ω).
In consequence, if we can verify that Z is adapted to ( ˆFs0)s∈It, then Proposition A.40
implies that Z is reconstructible, which concludes the proof.
For this purpose, we may suppose that Y is bounded. Indeed, once the claim is true in this case, then Lemma A.39 and Corollary A.24 give us a sequence (Y(n))
n∈N
of real-valued ( ˆFs0)s∈It-progressively measurable bounded processes on It× Ω with
limn↑∞Ys(n) = Ys and supn∈N|Ys(n)| ≤ |Ys| for all s ∈ It. Then
lim n↑∞ Z t r Ys(n)(ω) κ(ds)(ω) = Z t r Ys(ω) κ(ds)(ω) = Zr(ω)
for each (r, ω) ∈ It× Ω, by dominated convergence. As pointwise limit of a sequence
of real-valued ( ˆFs0)s∈It-adapted bounded processes on It× Ω, the process Z must
also be ( ˆFs0)s∈It-adapted. This explains the simplification. At last, let r ∈ It, then
the set Hr of all B([r, t]) ⊗ ˆFr0-measurable bounded processes ˆY : [r, t] × Ω → R for
which
Z t
r
ˆ
Ysκ(ds) is ˆFr0-measurable
is a monotone class in [r, t] × Ω. For every s ∈ [r, t] and each A0 ∈ ˆF0
r, the process
[r, t] × Ω → [0, 1], (r0, ω) 7→ 1[r,s]×A0(r0, ω) is a member of Hr. So, the Functional
Monotone Class Theorem A.29 entails that Hr coincides with the linear space of
all real-valued B([r, t]) ⊗ ˆFr0-measurable bounded processes on [r, t] × Ω. Because the restriction of Y to [r, t] × Ω is B([r, t]) ⊗ ˆFr0-measurable and r ∈ It has been
arbitrarily chosen, the claim follows.
Until the end of this section, we suppose that µ is a Borel measure on J with
µ({t}) = 0 for all t ∈ J and θ ∈ B(J × S, R+) is consistently bounded. Then
Lemma 3.16 guarantees that the function ν : Ω ×B(J) → [0, ∞] defined by
ν(B)(ω) :=
Z
B
θ(s, Xs(ω)) µ(ds)
is a non-negative continuous additive functional ofX . By using standard properties of conditional expectations, we show the following integral identity that is used in the upcoming section and in Chapter 4. In one dimension, this is a special case of Theorem 57 in Dellacherie and Meyer [9, Section 6.2].
3.18 Proposition. Suppose that X has Borel measurable transition probabilities.
Let b ∈ B(I × S, Rk×k) be locally µ-dominated and Y : I × Ω → Rk be reconstructible and consistently bounded. Then
Er,x " Z t r b(s, Xs)Ysν(ds) # = Er,x " Z t r b(s, Xs)Es,Xs[Ys] ν(ds) # (3.8)
for all r, t ∈ I with r ≤ t and each x ∈ S.
Proof. BecauseX is progressive, b is Borel measurable, and Y is reconstructible, it
follows from Lemma 3.9 and Proposition 3.13 that the processes
I × Ω → Rk, (s, ω) 7→ b(s, Xs(ω))Ys(ω)
and I × Ω → Rk, (s, ω) 7→ b(s, X
s(ω))Es,Xs(ω)[Ys] must be ( ˆF
0
s)s∈I-progressively
measurable. Let b ∈ B(I, R+) be a locally µ-integrable function with |b(·, y)| ≤ b
for all y ∈ S µ-a.s. on I. For each r, t ∈ I with r ≤ t, let cr,t ≥ 0 and θr,t ≥ 0 be
such that |Ys(ω)| ≤ cr,t and |θ(s, y)| ≤ θr,t for all s ∈ [r, t], each ω ∈ Ω, and every
y ∈ S. Then Z t r |b(s, Xs)|(|Ys| ∨ Es,Xs[|Ys|]) ν(ds) ≤ cr,t Z t r b(s)|θ(s, Xs)| µ(ds) ≤ cr,tθr,t Z t r b(s) µ(ds) < ∞
for each r, t ∈ I with r ≤ t. Hence, Lemma 3.17 implies that for fixed t ∈ I the processes
It× Ω → Rk, (s, ω) 7→
Z t
r
b(s, Xs(ω))Ys(ω) ν(ds)(ω)
and It × Ω → Rk, (s, ω) 7→ Rrtb(s, Xs(ω))Es,Xs(ω)[Ys] ν(ds)(ω) are reconstructible,
consistently bounded, and continuous. This also justifies that the two expectations appearing in (3.8) are well-defined for all r ∈ It and each x ∈ S.
Let us show that these two expectations coincide. We pick a µ-null set N ∈ B(J) such that |b(s, y)| ≤ b(s) for all (s, y) ∈ (Nc∩ I) × S, then from Proposition 3.7 and
the standard properties of conditional expectations we infer that
Er,x[bi,j(s, Xs)Ys(j)θ(s, Xs)] = Er,x[bi,j(s, Xs)Er,x[Ys(j)|Fs]θ(s, Xs)]
= Er,x[bi,j(s, Xs)Es,Xs[Y
(j)
s ]θ(s, Xs)]
for each s ∈ Nc∩ [r, t] and every i, j ∈ {1, . . . , k}. In combination with Fubini’s
theorem, this gives
Er,x " Z t r bi,j(s, Xs)Ys(j)ν(ds) # = Z Nc∩[r,t]Er,x[bi,j(s, Xs)Y (j) s θ(s, Xs)] µ(ds) = Z Nc∩[r,t]Er,x[bi,j(s, Xs)Es,Xs[Y (j) s ]θ(s, Xs)] µ(ds) = Er,x " Z t r bi,j(s, Xs)Es,Xs[Y (j) s ] ν(ds) #
for each i, j ∈ {1, . . . , k}. Consequently, we compute that Er,x " Z t r b(s, Xs)Ysν(ds) # i = k X j=1 Er,x " Z t r bi,j(s, Xs)Ys(j)ν(ds) # = k X j=1 Er,x " Z t r bi,j(s, Xs)Es,Xs[Y (j) s ] ν(ds) # = Er,x " Z t r b(s, Xs)Es,Xs[Ys] ν(ds) # i
for all i ∈ {1, . . . , k}. Hence, the proof is complete.
We conclude with a (right-)continuity result for consistent progressive Markov processes that are (right-hand) Feller. The idea to use dominated convergence comes from Professor Dr. Schied.
3.19 Proposition. Assume that X is (right-hand) Feller. Let ϕ ∈ B(I × S, Rk) be
locally µ-dominated and t ∈ I. If there is a µ-null set N ∈B(J) such that θ and ϕ are right-continuous at each point of (Nc∩ It) × S, then the map
ψ : It× S → Rk, ψ(r, x) := Er,x " Z t r ϕ(s, Xs) ν(ds) #
is consistently bounded and (right-)continuous.
Proof. Since ϕ is Borel measurable, the process I × Ω → Rk, (s, ω) 7→ ϕ(s, Xs(ω))
is ( ˆFs0)s∈I-progressively measurable. Let a ∈ B(I, R+) be some locally µ-integrable
function with |ϕ(·, y)| ≤ a for all y ∈ S µ-a.s. on I, and for each r ∈ Itchoose θr≥ 0
such that |θ(s, y)| ≤ θr for every (s, y) ∈ [r, t] × S. Then
Z t r |ϕ(s, Xs)| ν(ds) = Z t r |ϕ(s, Xs)||θ(s, Xs)| µ(ds) ≤ θr Z t r a(s) µ(ds) < ∞
for each r ∈ It. These considerations in combination with Lemma 3.17 imply that
the process It× Ω → Rk, (r, ω) 7→ Rrtϕ(s, Xs(ω)) ν(ds)(ω) is in fact reconstructible,
consistently bounded, and continuous. So, we have clarified that ψ is well-defined and consistently bounded.
To show that ψ is (right-)continuous, let (r, x) ∈ It× S and (rn, xn)n∈N be a
sequence in It× S (with rn ≥ r for all n ∈ N) that converges to (r, x). First, we
consider the case r = t, then ψ(t, x) = 0, since µ({t}) = 0. Let q ∈ It be such that
q ≤ rn for almost all n ∈ N. Then
|ψ(rn, xn)| ≤ θq
Z t
rn
a(s) µ(ds)
for almost each n ∈ N. By dominated convergence, limn↑∞
Rt
rna(s) µ(ds) = 0, which
Let now r < t and choose a µ-null set L ∈ B(J) such that |ϕ(s, y)| ≤ a(s) for all (s, y) ∈ (Lc∩ I) × S. For each n ∈ N we define the map φn: It→ Rk through
φn(s) := Ern,xn[ϕ(s, Xs)θ(s, Xs)], if s ∈ L
c∩ [r n, t],
and φn(s) := 0, otherwise. In a similar way, we let φ : It → Rk be defined through
φ(s) := Er,x[ϕ(s, Xs)θ(s, Xs)], if s ∈ Lc∩ [r, t], and φ(s) := 0, otherwise. Then
Fubini’s theorem entails that
ψ(rn, xn) = Z Lc∩[rn,t]Ern,xn[ϕ(s, Xs)θ(s, Xs)] µ(ds) = Z It φn(s) µ(ds), ψ(r, x) = Z Lc∩[r,t]Er,x[ϕ(s, Xs)θ(s, Xs)] µ(ds) = Z It φ(s) µ(ds)
for all n ∈ N. Let s ∈ Nc∩ Lc
∩ (r, t], then the map S → Rk, x 7→ ϕ(s, x)θ(s, x)
belongs to Cb(S, Rk), by Proposition 3.3. Hence, the (right-hand) Feller property of
X entails that the map
Is× S → Rk, (r0, x0) 7→ Er0,x0[ϕ(s, Xs)θ(s, Xs)]
is (right-)continuous. We pick n0 ∈ N such that rn < s for every n ∈ N with n > n0
and set (r(s)
n , x(s)n ) := (rn+n0, xn+n0) for all n ∈ N, then (r
(s)
n , x(s)n )n∈N is a sequence
in Is× S that converges to (r, x). Thus,
lim
n↑∞φn(s) = limn↑∞Ern(s),x
(s)
n [ϕ(s, Xs)θ(s, Xs)] = Er,x[ϕ(s, Xs)θ(s, Xs)] = φ(s).
For s ∈ I with s < r it follows that φn(s) = 0 for almost each n ∈ N. As µ({r}) = 0,
we have shown that limn↑∞φn(s) = φ(s) for µ-a.e. s ∈ It. Let us choose q ∈ I
and n0 ∈ N such that q ≤ rn for all n ∈ N with n ≥ n0. This in turn gives
supn∈N: n≥n0|φn(s)| ≤1[q,t](s)a(s)θq for µ-a.e. s ∈ It. For this reason, the Dominated
Convergence Theorem A.33 yields that lim n↑∞ψ(rn, xn) = limn↑∞ Z It φn(s) µ(ds) = Z It φ(s) µ(ds) = ψ(r, x).
This proves the proposition.