a probabilistic approach of the reduite - Laboratoire de Probabilités ...

A PROBABILISTIC APPROACH OF THE
REDUITE
N. EL KAROUI J.-P. LEPELTIER A. MILLET
Abstract
In this work the approach of the reduite is made in a simple and unified way. More
precisely, we use the same probabilistic technique to study the optimal stopping
problem associated with the reduite, to prove the expression of the Snell envelope
in terms of the reduite under very general assumptions, and to show continuity
properties of the reduite. We finally describe the example of diffusion processes with
jumps.
Nicole EL KAROUI
Laboratoire de Probabilités
Tour 56, 3ème étage
Université Paris VI
4, place Jussieu
75230 PARIS Cedex O5
Jean-Pierre LEPELTIER
Université du Maine
Faculté des Sciences
Route de Laval
72017 LE MANS Cedex
Annie MILLET
Université Paris X
200, avenue de la République
92001 NANTERRE Cedex
and
Laboratoire de Probabilités
URA 224
1

0 Introduction
During the sixtieth and seventieth, the problem of optimal stopping of continuous
time processes was the object of many papers. Without quoting them all, let us stress
the importance of J.M. Mertens’ works [21] and [22] which deal with the problem
in its most general setting; in these articles, the Snell envelope is systematically
studied, and is characterized as the smallest super-martingale which is larger than
or equal to the process. In this general setting, the works of J.M. Bismut and B.
Skalli [8] , M.A. Mainguenau [19] , N. El Karoui [12] describe precisely an optimal
stopping time (whenever it exists) as the “beginning” of the set where the process is
equal to its Snell envelope, and more generally, give the exact default of optimality.
In the case of a Markov process (X t ) , a new question arises: if the process
one wants to stop only depends on the state of the process at time t , i.e., can be
written as g(X t ) 1 {t

elongs to M(x) , and hence
Rg(x) ≤ ˆRg(x) .
The converse inequality is much more difficult to prove; it depends on a theorem
of Rost [27] which gives a complete characterization of the set M(x) , at least in the
transient case.
The interest of this formulation is to introduce a functional analysis setting which
is well suited for solving the optimal stopping problem: under mild hypotheses, the
set M(x) as well as its graph are shown to be weakly compact.
Our point of view is quite similar to this last one: we introduce the set
A(x) = { µ ; µ = µ T for some stopping time T } ,
which we endow with the topology induced by that of J.R. Baxter and R.V. Chacon
[3] on processes; this topology is stronger than the topology of convergence in law.
More precisely, let R(x) denote the set of stopping rules starting from x , which are
measures on the set of processes ( Y t (ω) ; (ω, t) ∈ Ω × IR + ) , defined by:
R ∈ R(x) ⇐⇒ there exists a stopping time T such that R(Y ) = IE x (Y T ) ,
and let R(x) be endowed with the Baxter-Chacon topology defined as follows: R n
converges to R if and only if R n (Y ) converges to R(Y ) for every continuous process
Y . The set A(x) is convex compact in the induced topology.
It is easy to see that the graphs of the multi-valued mapping x ❀ R(x) and
x ❀ A(x) are Borel subsets of products of compact spaces. Furthermore, if the map
x ↦→ IP x is weakly continuous, then these graphs are closed.
If g is measurable, then the function sup{ µ(g) ; µ ∈ A(x) } is also measurable,
and is identified with the reduite Rg of g (in the sense of Snell’s envelope).
The same arguments show the continuity of Rg if g and X . are continuous and
if the map x ↦→ IP x is weakly continuous.
Hence the novelty of the present method lies in the fact that the restriction of a
topology, defined on a set of probabilities acting on a set of processes, to the set of
stopping measures yields a “good” convex compact topology.
The paper is organized as follows. In the first section we define the optimal
stopping problem, and show that the reduite is independent of the realization. In
the second section we define and characterize the set of stopping rules; we also prove
(0.1). Section three establishes the expression of the Snell envelope in terms of the
reduite Rg . In the fourth section we study the continuity properties of the reduite,
and the example of diffusion processes with jumps on IR d is described in the fifth
section. Finally, we prove the weak continuity of the map x ↦→ IP x for Feller processes
on a compact state space in an Appendix.
3

1 First properties of the optimal stopping problem
1.1 The Markov process
We consider a Markov process (X t ) taking on values in a metrizable Lusin space
E ; in some cases E is supposed to be LCCB. We denote its semi-group by (P t )
and its resolvent by (U α ) . We suppose that the semi-group is conservative, i.e.,
P t 1 = 1 for every t . Notice that this assumption is not restrictive. Indeed, if
the semi-group (P t ) is sub-Markovian, we extend it to a Markovian semi-group over
E ∆ = E ∪ {∆} , where ∆ is a coffin-state (see e.g. [9] or [28]). Let Π(E) denote
the set of probabilities on E .
In the paper we will make various assumptions on the semi-group (P t ) ; however
they all imply the existence of a strongly Markov realization of (P t ) (see e.g. [9] ,
[28]). There is no uniqueness of strongly Markov realizations of a semi-group (P t ) .
Thus on a given space there exists a realization which is the smallest one.
Definition 1. 1 Let X = (Ω , G t , X t , θ t , IP x ; x ∈ E) be a strongly Markov
realization of (P t ) . The canonical realization associated with X is defined on Ω
with the filtration (F t ) deduced from Ft
0 = σ(X s ; s ≤ t) by standard regularization
procedures (completeness and right-continuity), say C(X ) = (Ω , F t , X t , θ t , IP x ;
x ∈ E) .
1.2 Definition of the optimal stopping problem
Let X = (Ω , G t , X t , θ t , IP x ; x ∈ E) be a strongly Markov realization of the semigroup
(P t ) . Extend the process (X t ) to IR + = IR + ∪ {+∞} by setting X ∞ = ∆ ,
and let (F t ) denote the canonical filtration of C(X ) .
Definition 1. 2 Let T (G) denote the set of stopping times (finite or not) with
respect to the filtration (G t ) (i.e., T ∈ T (G) ⇔ ∀t , {T ≤ t} ∈ G t ) . For the
canonical filtration, simply set T = T (F) .
Let µ be a probability on E . The aim is to stop the evolution of a reward
process (Y t ) at a stopping T ∗ ∈ T (G) which maximizes the expected reward, i.e.,
such that :
IE µ ( Y T ∗) = sup{ IE µ (Y T ) ; T ∈ T (G) } .
The definition of the optimal stopping problem obviously requires integrability
conditions on Y . The simplest condition, which we will often use, is to suppose that
(Y t ) is bounded. The “good” hypothesis is to assume that (Y t ) is of class (D) with
respect to the filtration (G t ) , i.e., is uniformly integrable over the set of stopping
times, uniformly with respect to the initial condition:
[
]
lim
c→+∞
sup
µ∈Π(E)
sup
T ∈T (G)
IE µ ( |Y T | 1 {|YT |>c} )
= 0 .
A process of class (D) with respect to the canonical filtration will simply be said of
class (D) .
4

We will study thoroughly the particular case of a process (Y t ) which only
depends on the state of the process (X t ) :
∀t ∈ IR + , Y t = e −αt g(X t ) ; Y ∞ = 0 ,
where α > 0 , g : E → IR is borelian and either bounded or of class (D α ) , i.e.,
such that (Y t ) is of class (D) . We will also study the case where Y t = g(X t )
and Y ∞ = lim sup g(X t ) as in Shiryayev [29] . We refer to these situations by the
notations:
∀α > 0 , g α t = e −αt g(X t ) or g t = g(X t ) = g 0 t ,
with the convention that g∞ α = lim sup gt
α
= 1 if α = 0 ) , and g(∆) = g∞ 0 .
for α ≥ 0 , e −α∞ = 0 if α > 0 (resp.
Definition 1. 3 Let Y be a reward process of class (D) with respect to (G t ) . The
reduite of Y is the maximal payoff function
v(x, Y ) = sup{ IE x (Y T ) ; T ∈ T (G) } .
If Y = g α t (α ≥ 0) , we will denote the reduite v(x, Y ) by R α g(x) .
Remark: This last definition refers to that in potential theory where the reduite of
a function g is the smallest α-excessive function which is larger than g . One of the
aims of this paper is to show that both notions coincide by means of probabilistic
arguments, which are substantially simpler than the usual ones when g does not
have any regularity property (cf. e.g. [12]).
1.3 Randomized stopping times
A classical approach of such problems consists in introducing a convex set of randomized
stopping times containing T (G) , and extending the optimal stopping problem.
This technique is indeed natural to study the relationship between the reduite and
the realization of the semi-group.
Definition 1. 4 Let X = (Ω , G t , X t , θ t , IP x ; x ∈ E) be a strongly Markov realization
of (P t ) . A randomized stopping time is an increasing, (G t )-adapted, rightcontinuous
process (A t ) such that A ∞ = 1 IP µ a.s. for every initial probability
µ ∈ Π(E) . Let A(G) denote the set of randomized stopping times. For the canonical
filtration, simply set A = A(F) .
The set T (G) can be embedded in A(G) ; indeed, for T ∈ T (G) and t ∈ IR + , set
A t = 1 {T ≤t} . Changes of time describe the connection between stopping times and
randomized stopping times (see also [3] , [24] ). Indeed, let r t = inf{s : A s ≥ t}
denote the pseudo-inverse of (A t ) , with the convention inf ∅ = +∞ ; then each
r t is a stopping time. The following proposition shows that the maximal expected
values are the same over the sets T and A . It is essentially shown in [24] p.419.
Proposition 1. 5 Let Y be a process of class (D) with respect to the filtration (G t ) .
Then
5

{ ∫
(i) The family of random variables Y A =
[0,+∞]
Y s dA s ; A ∈ A(G)
}
is uniformly
integrable, uniformly with respect to the initial probability.
(ii) The reduite of Y is the same over the sets T (G) and A(G) . More precisely,
v(µ, Y ) = sup{ IE µ (Y T ) ; T ∈ T (G) } = sup{ IE µ (Y A ) ; A ∈ A(G) } (1.1)
for every µ ∈ Π(E) .
Finally, the following theorem shows that the reduite of an (F t ) -adapted process is
independent of the realization. It is a consequence of Property (K), which holds for
the filtrations (F t ) and (G t ) ; see [20] . We need before the following result:
Proposition 1. 6 Let X = (Ω , G t , X t , θ t , IP x ; x ∈ E) be a strongly Markov
realization of (P t ) , and let (F t ) denote its canonical filtration. Given any stopping
time T with respect to the filtration (G t ) , there exists a randomized stopping time
(A T t ) ∈ A such that
IE µ ( Z T ) = IE µ ( Z A T )
for every µ ∈ Π(E) , and every F ∞ ⊗B(IR + )-measurable process Z which is positive
(respectively bounded).
Proof : For every t ∈ [0, +∞] , let  T t = IE µ
(
1{T ≤t} | F ∞
)
be the version
independent of µ . The Markov property yields that IE µ
(
1{T ≤t} | F ∞
)
=
IE µ
(
1{T ≤t} | F t
)
IPµ a.s., and therefore that ( ÂṬ ) is IP µ a.s. (F t )-adapted,
increasing and such that ÂT ∞ = 1 IP µ a.s. Let ( A Ṭ ) denote its increasing, rightcontinuous,
(F t )-adapted regularization such that A T ∞ = 1 IP µ a.s. The definition
of A Ṭ shows that given any positive, F ∞ -measurable random variable H , and any
interval ]s, t] ,
(
IE µ H ( A
T
t − A T s ) ) ( )
= IE µ H 1{s

Proof : It is well-known (see e.g. [10] , t.1, p.38) that the uniform integrability
condition is equivalent to the existence of a convex function φ : IR + −→ IR + with
φ(t)
lim
t→∞ t
= ∞ , and such that sup
µ∈Π(E)
sup
T ∈T
IE µ [ φ( |Y T | ) ] < +∞ .
Let T ∈ T (G) and µ ∈ Π(E) ; then if A T ∈ A is the increasing process constructed
in Proposition 1.6, then:
]
IE µ [ φ( |Y T | ) ] = IE µ
[ ∫
[0,+∞]
φ( |Y s | ) dA T s
≤ sup{ IE µ [ φ( |Y T | ) ] ; T ∈ T } < +∞
by Proposition 1.5 .
To show the equality of the reduites, we use a similar argument; indeed by Proposition
1.5, we only need to compare the suprema over the sets A and T (G) . Let
T ∈ T (G) , µ ∈ Π(E) , and let A Ṭ ∈ A be the process defined as above. Then
IE µ (Y T ) = IE µ ( Y A T ) ≤ sup{ IE µ (Y A ) ; A ∈ A } .
Since F ⊂ G , the converse inequality is obvious.
✷
This last result shows that we can work with the canonical filtration, whose
topological properties we use.
2 Stopping rules and reduite
2.1 Stopping rules
Following Baxter and Chacon [3] , Bismut [6], Meyer [24], we introduce the “good”
set of parameters to solve the optimization problem, i.e., probabilities on Ω × IR +
which can be written IP µ (dω) A(ω, dt) , where the process A t (ω) = A(ω, [0, t]) is
a randomized stopping time for the canonical filtration. These are Meyer’s “temps
d’arrêt flous”.
Definition 2. 1 Let µ ∈ Π(E) ; a stopping rule starting from µ is a probability
R on Ω × IR + which can be disintegrated with respect to IP µ as
R(dω, dt) = IP µ (dω) A(ω, dt) ,
with a randomized stopping time A ∈ A .
Let R(µ) denote the set of stopping rules starting from µ . When µ is a
Dirac measure δ x , we write R(x) for simplicity. Baxter-Chacon [3] and Meyer [24]
have given characterizations of stopping rules in the case of an abstract measurable
space endowed with one probability. Our setting will be slightly different since we
are working with the canonical realization, and thus with a topological space Ω of
trajectories endowed with various probabilities IP µ .
Given an F ∞ -measurable random variable h and a Borel function f on IR + ,
set h ⊗ f(ω, t) = h(ω) f(t) .
7

Theorem 2. 2 Let µ ∈ Π(E) ; a stopping rule starting from µ is a probability on
Ω × IR + satisfying the conditions (i) et (ii):
(i) For any bounded random variable h , R(h ⊗ 1) = IE µ (h) .
(ii) For any bounded random variable h , any t in a countable dense subset D
of IR + ,
R( h ⊗ 1 [0,t] ) = R( IE µ (h|F t ) ⊗ 1 [0,t] ) .
Remark: Condition (i) ensures that the projection of R on Ω is IP µ , and
condition (ii) gives the suitable adaptation property of the disintegration of R with
respect to IP µ .
Proof : A probability R ∈ R(µ) clearly satisfies (i) and (ii). We only prove the
converse implication. Condition (i) shows that the projection of R on Ω is IP µ ,
and hence that R can be disintegrated in the form R(dω, dt) = IP µ (dω)A(ω, dt) ,
where A is a transition probability from Ω to IR + , with repartition function
A t = A(., [0, t]) . This fact has already been used in [15], p.541 and in [24], p.411;
it comes from general results concerning the disintegration of measures ([10], t.1,
p.125). Condition (ii) shows that for each a ∈ D , A a is IP µ a.s. F a -measurable.
Let à a be an F a -measurable random variable IP µ a.s. equal to A a . Set Ã∞ = 1
and let (A t ) be the increasing, right-continuous extension of (Ãa , a ∈ D) . Since
both processes A . and A . are right-continuous, they are IP µ indistinguishable, and
R(dω, dt) = IP µ (dω) A(ω, dt) . Furthermore, the right-continuity of the filtration
(F t ) shows that (A t ) is (F t )-adapted. ✷
Baxter-Chacon and Meyer have endowed the set R(µ) with a compact topology.
Since we work with a topological space Ω and several probabilities IP µ ∈ Π(Ω) ,
our definition of the topology on R(µ) will look different from theirs. For a fixed
probability IP µ , both topologies coincide.
Theorem 2. 3 Let R(µ) be endowed with the Baxter-Chacon topology, i.e., the
coarsest topology such that the maps R ∈ R(µ) ↦→ R(X) = IE µ
( ∫
[0,+∞] X s dA s
)
with R(dω, dt) = IP µ (dω)A(ω, dt) , are continuous for every bounded F ∞ ⊗ B(IR + )-
measurable process (X t (ω)) with continuous trajectories on IR + ( i.e., ∀ω , t ↦→ X t (ω)
is continuous). Then the set R(µ) is compact in the Baxter-Chacon topology.
Proof : We briefly sketch the argument, and refer to [3] and [24] for details. Theorem
2.2 and approximations of 1 [0,a] by continuous functions clearly show that R(µ) is
closed in Π(Ω × IR + ) . To show that R(µ) is relatively compact, we use a criterion
of Jacod and Memin [15] , and prove that the set of projections of R(µ) on Ω and
IR + are both weakly relatively compact. This is obvious since { R Ω ; R ∈ R(µ) } =
{ IP µ } , and { R IR +
; R ∈ R(µ) } is a set of probabilities on a compact set. ✷
2.2 Dependence on the initial condition
The characterization of stopping rules given in Theorem 2.2 allows us to precise the
dependence of the reduite of a (non necessarily adapted) process Y on the initial
condition. The proof depends on the properties of the graph of the multi-valued
8

mapping x ❀ R(x) . It is easy under additional hypothesis. We at first consider
the following particular case:
We suppose that the state space E is a Polish space. Let Ω 0 = D([0, +∞] , E)
denote the set of right-continuous and left-limited (cad-lag) maps from [0, +∞] into
E . The set Ω 0 is a Polish space (i.e., separable, complete metrizable when it is
endowed with the Skorokhod topology). Let : F 0 denote its canonical σ-algebra,
and let C(E) denote the set of continuous functions on E .
In the sequel, we will denote by H a countable family of bounded random
variables which is stable by product, and generates the σ-algebra F∞ 0 = σ(X s ;
s ∈ IR + ) . Depending on the special assumption made on the Markov process
(Feller, right, ... ) we will mainly consider several sets H , which will be the best
suited for the particular problem under study.
Let H b (I) denote the set of random variables
h =
∏
1≤i≤k
f i (X ti ) where k ≥ 1 , t 1 < t 2 < · · · < t k ∈ I , (2.1)
and the functions f i are bounded and measurable. We suppose that (t i ) belong to a
countable dense subset I ⊂ IR + , and that (f i ) belong to a countable set generating
B(E) .
If E is compact, let H c (I) denote the subset of H b (I) corresponding to functions
(f i ) that belong to a countable dense subset of C(E) . When no confusion is possible,
simply set H b and H c .
By the Markov property, for every µ ∈ Π(E) and for every h ∈ H b ,
ĥ t = IE Xt(ω)[ h(ω/t/.) ] is a version of IE µ (h|F t ) , (2.2)
where the trajectory ω/t/ω ′ is defined by (cf. e.g. [28]):
X s (ω/t/ω ′ ) =
{
Xs (ω) if s ≤ t
X s−t (ω ′ ) if s > t .
The assumptions we make in the following proposition ensure the measurability
of the map x ↦→ IP x .
Proposition 2. 4 Let (P t ) be a Borel semi-group on E . We suppose that there
exists a family (IP x ; x ∈ E) of probability on (Ω 0 , F 0 ) such that X = (Ω 0 , Ft 0 , X t ,
θ t , IP x ; x ∈ E) is a strong Markov realization of the semi-group (P t ) . Then for
each reward process Y of class (D) , the reduite
v(x, Y ) = sup{ IE x (Y T ) ; T ∈ T }
is an analytic function, and for each µ ∈ Π(E) ,
∫
v(x, Y )µ(dx) = sup{ R(Y ) ; R ∈ R(µ) } = sup{ IE µ (Y T ) ; T ∈ T } . (2.3)
Proof : We study the graph G = { (x, R) ; R ∈ R(x) } . The map x ↦→ IP x is
Borel from E into Π(Ω 0 ) , since the Borel σ-algebra on Π(Ω 0 ) is generated by the
9

maps µ ↦→ µ(Z) , where Z is F 0 -measurable. The monotone class theorem shows
that in the characterization of R(x) given in Theorem 2.2 , it suffices to consider the
maps R(h ⊗ 1) , IE x (h) , R(h ⊗ 1 [0,a] ) and R( IE x ( h | F a ) ⊗ 1 [0,a] ) = R(ĥa ⊗ 1 [0,a] )
for h ∈ H b , and a in a countable dense subset of IR + . Under our assumptions
all these functions are Borel on E ⊗ Π(Ω 0 × IR + ) . Hence the graph G is a Borel
subset of the product space E ⊗ Π(Ω 0 × IR + ) .
Suppose at first that Y is bounded. Since Y is F 0 ⊗ B(IR + )-measurable, the
map R ↦→ R(Y ) is Borel from Π(Ω 0 × IR + ) into IR , and the theory of analytic
functions ([10], t.1 p.119, Théorème 62) implies that the function
v(x, Y ) = sup{ R(Y ) ; R ∈ R(x) }
is analytic. We follow the proof of ([10], chapter X, Lemma 17). The function
v is universally measurable and consequently we can find a Borel function w(x) ,
majorized by v(x, Y ) , and µ a.s. equal to v(x, Y ) . Given ɛ > 0 , the set
G ɛ = { (x, R) ; R(Y ) + ɛ ≥ w(x) , R ∈ R(x) }
is Borel, and its section along a fixed x , say G ɛ x , is non-empty. From the section
theorem ([10], t.1, Chapter III 44-45) there exists a Borel set A carrying µ , and
a Borel section R ɛ (x, .) of G ɛ defined on A . We set R ɛ (x, .) = ɛ x on A c . Since
∫ R ɛ (x, .)µ(dx) belongs to R(µ) by Theorem 2.2 , we deduce that
∫
v(x, Y ) µ(dx) ≤ sup{ R(Y ) ; R ∈ R(µ) } .
Conversely, let R = IP µ (dω) · dA t (ω) ∈ R(µ) . Since (A t ) is a randomized stopping
time we have
∀x ∈ E , IP x (dω) · dA t (ω) ∈ R(x) .
Then
R(Y ) =
≤
∫
∫
µ(dx)
∫
µ(dx) v(x, Y ) .
IP x (dω) Y t (ω) dA t (ω)
Let Y be of class (D) ; we compare v(x, Y ) and v(x, Y c ) , where Y c is the
(bounded) truncated process (Y ∧ c) ∨ (−c) . Set
ɛ(c) = sup { IE x ( |Y T | 1 { |YT |>c } ) ; T ∈ T }
for fixed x ∈ E . Then for every T ∈ T ,
and
| IE x ( Y T ) − IE x (YT c ) | = | IE x [ (Y T − c) 1 {YT >c} + (Y T + c) 1 {YT c} ] ≤ ɛ(c) ,
| v(x, Y ) − v(x, Y c ) | ≤ sup { | IE x (Y T ) − IE x (Y c
t ) | ; T ∈ T } ≤ ɛ(c) .
Hence v(., Y ) is the pointwise limit of v(., Y c ) as c → ∞ , and the Lebesgue
theorem implies that the first equality in (2.3) holds for Y . Finally, Proposition 1.5
concludes the proof.
✷
10

2.3 Measurability of the reduite for right processes
We now consider a right semi-group (P t ) on a Lusin space E . The Ray compactification
of a right semi-group is very long to describe completely, and we refer
systematically to the notations and proofs of Getoor [14] ; see also [28]. Let E
denote the Ray-Knight compactified of E ; the extension of (P t ) to E is denoted by
(P t ) . The links between X and the Ray process associated with (P t ) are described
in the following theorem.
Theorem 2. 5 Let W be the set of applications w : IR + −→ E which are rightcontinuous
both in the initial topology and in the Ray topology, and which have left
limits in E in the Ray topology. Let (X t ) denote the coordinate process, and set
Ft 0 = σ(X s , s ≤ t) .
For each probability µ on E , IP µ is the measure constructed on (W, F 0 ) by
using (P t ) , and IP µ is the corresponding one constructed by means of (P t ) . These
probabilities IP µ and IP µ are equal, and ( X t , Ft 0 , IP µ ) is a Markov process with
semi-group (P t ) (resp. (P t ) ), if we consider E (resp. E ) as state space.
Following the remark of Getoor ([14], p.80), by changing the topology on E into
the Ray topology, the resolvent and the semi-group become Borel, and we can apply
the argument above. The kind of measurability will be the following: a function v
defined on E will be analytic if it is the restriction to E of a function ṽ which is
analytic on the Ray-Knight compactification E endowed with the metric ρ .
Theorem 2. 6 With the notations above, let C = (Ω , F 0,r
t , X t , θ t , IP x ; x ∈ E )
denote the canonical realization of a right semi-group (P t ) . For each F 0,r ⊗ B(IR + )-
measurable process Y of class (D) , the map
v(x, Y ) = sup{ IE x (Y T ) ; T ∈ T }
is universally measurable on E , and :
∫
µ(dx) v(x, Y ) = sup{IE µ (Y T ) ; T ∈ T } . (2.4)
Proof: If Y t = g(X t ) , let B denote the set of branching points, and D = B c .
If µ(B) = 0 , then (X t ) behaves like a right process taking on values in D . The
reduite v can be defined on D and extended to R by setting v = P 0 v . Given any
probability µ on E , the probability µP 0 does not charge B , and hence (see e.g.
[8])
∫
µ(dx)P 0 v(x, g(X)) = sup{ IE µP0 [(g(X T )] ; T ∈ T } = sup{ IE µ (g(X t )) ; T ∈ T } .
For a general process Y , let µ ∈ Π(E) ; Ω is not Lusin, but we can restrict ourselves
to a Borel subset Ω ′ of Ω included in W = D(IR + , E) , where E is endowed with
the Ray topology. Then replace E by a Borel subset E ′ such that µ(E ′ ) = 1 and
IP x (Ω ′ ) = 1 for x ∈ E ′ , and replace Y by a G 0 ⊗ B(IR + )-measurable process
Y ′ which is IP µ indistinguishable of Y . The proof of Proposition 2.4 shows that
v(x, Y ′ ) is ρ analytic on E ′ , and (2.4) holds by a selection theorem.
✷
11

3 Reduite and Snell’s envelope
The most important application of Theorem 2.6 is the connection between the reduite
and the Snell envelope for homogeneous processes, and more precisely, for processes
of the form gt
α = e −αt g(X t ) , t ∈ IR + (α ≥ 0) , where g is a fixed nearly Borel
function of class (D α ) . We recall the conventions we made in section 1:
X ∞ = ∆ , g α ∞ = lim sup t→∞ g α t , g(∆) = g 0 ∞ ;
set R α g(x) = v(x, g α ) for x ∈ E , and R α g(∆) = g α ∞ .
We at first apply Theorem 2.6 to prove that R α g is an α-strongly super median
function.
Proposition 3. 1 Let C = (Ω , F t , X t , θ t , IP x ; x ∈ E) be the canonical realization
of a right semi-group (P t ) . For each stopping time S ∈ T , each probability
µ ∈ Π(E) , one has that
IE µ ( e −αS R α g(X S ) ) ≤ sup { IE µ ( g α T ) ; T ≥ S , T ∈ T } (3.1)
≤ < µ, R α g > .
Proof: Fix S ∈ T , and let S α µ denote the measure on E defined by
S α µ (φ) = IE µ ( e −αS φ(X S ) 1 {S .
✷
12

Remark: The inequality (3.1) shows that the reduite R α g is the smallest α-super
median function larger than or equal to g . Its α-excessive regularization ˆR α g is
defined by:
ˆR α g = lim
t→0
e −αt P t R α g ≤ R α g on E , and ˆR α g(∆) = g α ∞ .
The converse inequality of (3.1) is established in Lemma 2.7.1 of [12] . It is clear if
g is l.s.c. on trajectories, since ˆR α g ≥ g yields that ˆR α g = R α g . Let (F 0 t ) denote
the filtration generated by (X s ; s ≤ t) .
Lemma 3. 2 Let C = (Ω , F t , X t , θ t , IP x ; x ∈ E) be the canonical realization of a
right semi-group (P t ) . Then for all stopping times T ≥ S with T ∈ F 0,+
t = ⋂ ɛ>0
F t+ɛ
and S ∈ T (F 0 t ) ,
IE µ ( g α T ) ≤ IE µ
(
e −αS R α g(X S ) ) . (3.2)
Proof : We denote by T 0 (resp. ˆT 0 ) the set of stopping times with respect to
Ft 0 (resp. F 0,+
t ) . Let S (resp. T ) be a stopping time in T 0 (resp. T ˆ 0
) , with
S ≤ T . By a slight modification of a theorem of Courrege-Priouret (cf. [10], t.1 ,
p. 237) there exists a FS 0 ⊗ F∞ 0 -measurable random variable U : Ω × Ω → [0, +∞]
such that :
(i) U(ω, w) = 0 if S(ω) = +∞ , or if S(ω) < +∞ and X 0 (w) ≠ X S (ω) .
(ii) U(ω, .) belongs to ˆT 0 .
(iii) T (ω) = S(ω) + U(ω, θ S (ω)) .
The proof is similar to theirs, using the Galmarino test for F 0,+
t stopping times
([10], t1, p.234, Théorème 101) instead of the Galmarino test for Ft 0 stopping times
([10], t.1, p.234, Théorème 100) . By the strong Markov property,
IE µ ( g α T ; S < +∞ ) = IE µ
(
e −αS IE XS (g α U(ω,.) ) ; S(ω) < +∞ ) (3.3)
≤ IE µ
(
e −αS R α g(X S ) ; S < +∞ ) .
Since gT
α and e −αS R α g(X S ) coincide on {S = +∞} , this concludes the proof of
(3.2) . ✷
Hence if g is l.s.c. on trajectories, approximating any stopping time S by a
decreasing sequence in T 0 yields that the inequality (3.2) holds for any S ∈ T .
The following lemma gives a more precise inequality for an arbitrary nearly Borel
function g .
Lemma 3. 3 Let C = (Ω , F t , X t , θ t , IP x ; x ∈ E) be the canonical realization of
a right semi-group (P t ) . Then for all stopping times T ≥ S in T ,
and R α g = g ∨ ˆR α g .
IE µ ( g α T ) ≤ IE µ
(
e −αS ( ˆR α g ∨ g)(X S ) ) . (3.4)
13

Proof : We at first extend the inequality (3.2) to stopping times S ∈ ˆT 0 . Apply
(3.2) to S = ɛ > 0 , and T = sup(V, ɛ) , where V is a strictly positive stopping
time in ˆT 0 . Then
IE x
(
g
α
sup(V,ɛ)
)
≤ IEx
(
e −αɛ R α g(X ɛ ) ) .
Since g α t
is of class (D α ) , we can let ɛ → 0 , and obtain that
IE x ( g α V ) ≤ ˆR α g(x) .
If we suppose that T > S ∈ T 0 on {T < ∞} , then the stopping time U(ω, .) is
strictly positive, and we have that:
IE µ ( g α T ; S < +∞ ) ≤ IE µ
(
e
−αS ˆRα g(X S ) ; S < +∞ ) .
Fix S ∈ ˆT 0 , T > S on {T < +∞} , and approximate it by a strictly decreasing
sequence (S n ) of F 0 t stopping times. Let (T n ) be the sequence in ˆT 0 defined by
T n = T on {T > S n } , and T n = +∞ otherwise .
The sequence (T n ) decreases to T in a stationary way (i.e., for every ω there exists
an integer N(ω) such that T n (ω) = T (ω) for all n ≥ N(ω) ) , and since T n > S n
on {T n < ∞} ,
IE µ
(
g
α
Tn
; S n < +∞ ) ≤ IE µ
(
e
−αS n ˆRα g(X Sn ) ; S n < +∞ ) .
We use the right-continuity of the process e −αt
ˆRα g(X t ) to deduce that
IE µ ( g α T ; S < +∞ ) ≤ IE µ
(
e
−αS ˆRα g(X S ) ; S < +∞ )
by letting n → ∞ .
Suppose at last that T ≥ S belongs to T , and set
ˆT = T on {S < T } , and ˆT = +∞ otherwise ,
Then
Ŝ = S on {S < T } , and
Ŝ = +∞ otherwise .
IE µ ( g α T ; {S < +∞} ) = IE µ ( g α T ; {S < T } ∩ {S < +∞} )
+ IE µ
(
e −αS g(X S ) 1 {S=T } ; S < +∞ )
= IE µ
(
g
αˆT
; Ŝ < +∞) + IE µ
(
e −αS g(X S ) 1 {S=T } ; S < +∞ )
≤ IE µ
(
e
−αŜ ˆRα g(XŜ) ; Ŝ < +∞ )
+ IE µ
(
e −αS g(X S ) 1 {S=T } ; S < +∞ )
= IE µ
(
e
−αS [ ˆRα g(X S ) 1 {S

Since ( ˆR α g ∨ g)(∆) = g∞
α on {S = +∞} , the proof of (3.4) is complete. Finally,
(3.4) applied with S = 0 yields that R α g ≤ ˆR α g ∨ g ≤ R α g . ✷
The main consequence of these two results is the description of the Snell envelope
in terms of the reduite. Let E e denote the σ-algebra generated by excessive functions.
Theorem 3. 4 Let X = (Ω , G t , X t , θ t , IP x ; x ∈ E) be a strong Markov realization
of a right semi-group (P t ) , and let g be a universally measurable function of
class (D α ) . Then if g is E e -measurable, the process
ˆR α t =: e −αt R α g(X t ) if t < ∞ , ˆRα ∞ = g α ∞
is a strong super martingale which is the Snell envelope J(g α ) of ( gt α , t ∈
[0, +∞] ) , i.e. ,
ˆR α S = ess sup {IE µ ( g α T | G S ) ; S ≤ T ∈ T (G )} .
Proof : Theorem 1.7 shows that we can use the canonical realization of (P t ) . By
Proposition 3.1 and Lemma 3.3 , we obtain that given S ∈ T and µ ∈ Π(E) ,
sup { IE µ ( g α T ) ; S ≤ T ∈ T } ≤ IE µ
(
e −αS R α g(X S ) )
≤ sup { IE µ ( g α T ); S ≤ T ∈ T }
= IE µ ( J(g α ) S ) ,
where the last equality is deduced from the property of decreasing filtrations for the
set ( IE µ (g α T ) , T ≥ S , T ∈ T ) (cf. e.g. [22] ) . Therefore,
e −αS R α g(X S ) = ess sup { IE µ (g α T | F S ) ; S ≤ T ∈ T } on {S < +∞} . (3.5)
Since ˆR α g is obviously E e -measurable, the equation R α g = ˆR α g ∨ g shows that
R α g is E e -measurable, and hence the process ˆRα g . is optional (cf. e.g. [10], t.3 or
[28] ) . Thus, equation (3.5) concludes the proof. ✷
Remark 3. 5 (1) The Snell envelope is a crucial tool in the theory of optimal stopping.
Thus it is very important to relate the reduite and the Snell envelope. For
example optimal stopping times of a process (Y t ) can be characterized in terms of
the Snell envelope J(Y ) of Y as follows (see e.g. [12]): A stopping time T ∗ is
optimal if and only if :
• Y T ∗ = J(Y ) T ∗ ;
• the process (J(Y ) t∧T ∗ , t ≥ 0) is a martingale.
The Snell envelope also gives an explicit construction of optimal stopping times
under u.s.c. assumptions on the process; cf. e.g. [12]. More precisely, let (Y t ) be an
optional process of class (D) ; then:
• If Y is u.s.c. on trajectories, then inf { t : Y t = J(Y ) t } is the smallest optimal
stopping time.
15

• If Y is u.s.c. in expectation ( i.e., EY T ≥ lim sup EY Tn for every sequence
(T n ) of stopping times converging to T ) , let J(Y ) t = M t − A t be the Doob
decomposition of the super martingale J(Y ) ; then inf { t : Y t = J(Y ) t } and
inf{ t : J(Y ) t ≠ M t } are optimal.
(2) Given an arbitrary function g , the reduite R α g we have studied is the
smallest strongly α-super median function larger than equal to g . Then:
• For fixed α > 0 , the optimal stopping time for the process g α and the
probability IP µ is the entrance time in a subset of E which does not depend
on the initial law µ .
• If g is l.s.c. , or more generally l.s.c. on trajectories (e.g. the difference of
two excessive functions), then R α g is excessive, and hence it is the “excessive”
reduite in the sense of potential theory (cf. e.g. [10] t.3).
4 Continuity properties of the reduite
In this section we prove continuity results on the reduite when both the function
f and the semi-group (P t ) have “continuity” properties, e.g., f is continuous and
(P t ) is Feller. Thus we generalize some known results about the reduite, but the
novelty lies mainly in the method which we develop. Instead of the penalization
iterative method (cf. [26], [12]) or a discretization method (cf. [18], [7], [31]), we use
the more probabilistic notion of stopping rule, which also yields functional results.
Furthermore, our method only depends on the weak continuity of the map x ↦→ IP x ,
which is well-known for diffusion Markov processes (cf. e.g. [25]). In the appendix,
we prove that this continuity property also holds for Feller processes on a compact
state space E .
Since the initial condition x ∈ E will not be fixed, we cannot use the Baxter-
Chacon topology on R = ∪R(x) ; indeed, this would imply that the laws of the
processes are strongly continuous.
In all this section we suppose that the state space is a Polish space, and sometimes
that it is compact metrizable. Let Ω = D( [0, +∞], E ) denote the canonical set of
right-continuous and left-limited maps from [0, +∞] into E , and let ω t denote the
coordinate maps. The set Ω is endowed with the Skorokhod topology, for which it
is metrizable, complete and separable (cf. e.g., [4], [16]). We recall some well-known
properties of this topology on Ω .
Proposition 4. 1 (a) For every bounded continuous ∫ function f on E , every
α > 0 and 0 ≤ t 1 < t 2 ≤ +∞ , the map ω ↦→ e −αs f(ω s ) ds is continuous.
(b) For every given probability P on (Ω, F∞ 0 ) , almost every t (with respect
to Lebesgue’s measure) , there exists a P -null set N t such that the map ω ↦→ ω t is
continuous for every ω ∉ N t .
[t 1 ,t 2 ]
In this section we will assume that the following assumptions (Ha) and (Hb) hold:
(Ha) One can define on Ω a family of probabilities (IP x ; x ∈ E) such that
16

(Ω , F t , Ω t , θ t , IP x ; x ∈ E) is a strong Markov realization of the semi-group (P t ) .
(Hb) The map x ↦→ IP x is weakly continuous.
Note that the assumption (Hb) is satisfied when E = IR d , and (IP x ) is the
solution of a “good” martingale problem. The following theorem shows that (Hb)
also holds for an arbitrary Feller process on a compact state space E (i.e., such that
P t ( C(E) ) ⊂ C(E) for each t > 0 , and P t f → f pointwise as t → 0 for each
f ∈ C(E) ), or Feller in the following weaker sense : U α ( C(E) ) ⊂ C(E) for each
α > 0 , and αU α f → f pointwise as α → +∞ for each f ∈ C(E) . Its proof is
given in the Appendix.
Theorem 4. 2 Let (Ω , F t , ω t , θ t , IP x ; x ∈ E) be the canonical realization of a
Feller semi-group (P t ) on a compact metric space E . Then the map x ↦→ IP x is
continuous when Π(Ω) is endowed with the weak topology.
We now prove regularity properties of the sets R(µ) of stopping rules when
Ω = D( [0, +∞], E ) under the assumptions (Ha) and (Hb) on the semi-group. The
set R = ⋃
R(µ) is included in Π( Ω × [0, +∞] ) , and is endowed with the
µ∈Π(E)
weak-star topology. Note that for fixed µ ∈ Π(E) , the restriction to R(µ) of
the weak-star topology is the same as the Baxter-Chacon topology on R(µ) , since
µ n → µ in the Baxter-Chacon topology if µ n (X) → µ(X) as n → +∞ for every
bounded continuous process X (see [3] or [24] p.419) . Indeed, for fixed µ choose
a countable dense subset I ⊂ IR + and a subset N ⊂ Ω such that IP µ (N) = 0 ,
and ω ↦→ ω t is continuous for each t ∈ I and ω ∉ N (cf. Proposition 4.1 ) . Then
the random variables h ∈ H c (I) and ĥt = IE Xt ( h(ω/t/.) ) are “continuous” on Ω
if t ∈ I .
Theorem 4. 3 Let Ω = D( [0, +∞], E ) , let K be a compact subset of E such
that the map x ↦→ IP x is continuous on K . Then
(a) The graph G K = { (x, R) ; x ∈ K , R ∈ R(x) } is a compact subset of
K×Π( Ω×[0, +∞] ) , where Π( Ω×[0, +∞] ) is endowed with the weak-star topology .
(b) The set R K = ∪ x∈K R(x) is a weakly compact subset of Π( Ω × [0, +∞] ) .
Proof : We at first check that G K is closed. Let (x n , R n ) be such that x n → x
and R n → R . The sequence (IP xn ) of projections of R n on Ω converges weakly to
IP x , which is hence the projection of R on Ω . To show that R ∈ R(x) , it suffices
to prove that the second condition in Theorem 2.2 is satisfied. Let I = (a i ) be a
countable dense subset of [0, +∞] such that the maps ω ↦→ X ai (ω) are continuous
except on a IP x -null set N of Ω . Let a ∈ I , h ∈ H c (I) , and φ be a continuous
function on [0, +∞] with support included in [0, a] . Then by Theorem 2.2 we have
that
R n (h ⊗ φ) = R n (ĥa ⊗ φ) , ∀n ≥ 0 .
Since the functions h and ĥa are continuous except on N × [0, +∞] , which is an
R-null set, letting n → +∞ yields that R(h ⊗ φ) = R(ĥa ⊗ φ) , and hence that G K
is closed.
17

Since G K is closed, it suffices to prove that R K is weakly compact, and hence
that the sets R Ω K and R IR +
K of projections of R K on Π(Ω) and Π( [0, +∞] ) are
tight. The continuity assumption made on the map x ↦→ IP x shows that R Ω K =
∪ x∈K IP x is a compact subset of Π(Ω) . Since [0, +∞] is compact, the tightness of
R IR +
K is obvious, and R K is compact. Since G K is a closed subset of K × R K , it
is clearly compact.
✷
The following lemma gives sufficient conditions for the upper semi-continuity of a
map defined in terms of suprema of continuous functions. The lower semi-continuity
of such maps is intuitively expected. Similar results can be found in the more general
setting of set-valued maps (cf. [2]).
Lemma 4. 4 Let X be metrizable and let R be compact metrizable. Let F : X ×
R → IR be bounded u.s.c. , and for every x ∈ X let R(x) ⊂ R be such that
G = { (x, R) ; x ∈ X , R ∈ R(x) } is closed in X × R . Then
is u.s.c.
v(x) = sup{ F (x, R) ; R ∈ R(x) }
Thus we obtain the upper semi-continuity of the reduite of an u.s.c.
which is not necessarily a function of X .
process Y
Theorem 4. 5 Let Ω = D( [0, +∞], E ) , and suppose that E is LCCB and that
the map x ↦→ IP x is continuous from E to Π(Ω) . Let Y be a F 0 ∞ ⊗ B(IR + ) -
measurable process of class (D) . If the map (ω, t) ↦→ Y t (ω) is u.s.c. on Ω×[0, +∞] ,
then the reduite v(x, Y ) = sup{ R(Y ) ; R ∈ R(x) } is u.s.c.
Proof : Fix x 0 ∈ E and let K be a compact neighborhood of x 0 .
We at first suppose that Y is bounded. Set F (x, R) = R(Y ) ; then the map R ↦→
R(Y ) is u.s.c., and hence F is u.s.c. Therefore, Lemma 4.4 applied with X = K and
R = R K (= ∪ x∈K R(x) ) shows that v is u.s.c. at x 0 .
Suppose that Y is of class (D) , and fix c > 0 . Let Y c be the truncated process
(−c) ∨ (Y ∧ c) . Then F c (R) = R(Y c ) converges uniformly to F as c → +∞ ;
indeed,
lim sup
c
sup
R∈R
| R(Y ) − R(Y c ) | ≤ lim sup R( |Y | 1 { |Y |>c } )
c
R∈R
= lim sup c
µ∈Π(E)
= 0 .
sup
T ∈T
IE µ ( |Y T | 1 |YT |>c } )
Hence the map v(., Y ) is also u.s.c.
✷
The extension of the optimal stopping problem to a larger convex compact set
has yielded the upper semi-continuity of v . As it might be expected, the lower
semi-continuity of v is easier to prove; it it obtained by restricting the study to the
smaller class T I of I-valued simple stopping times T : (i.e., taking on finitely many
values in I ⊂ IR + ).
18

Theorem 4. 6 Let Ω = D( [0, +∞] ), E ) , and let Y be a measurable (not necessarily
adapted) process of class (D) . Then
(a) If (Y t ) has lower semi-continuous trajectories, then for every µ ∈ Π(E) and
any dense subset I ⊂ IR + one has that
v(µ, Y ) := sup{ R(Y ) ; R ∈ R(µ) } = sup{ IE µ (Y T ) ; T ∈ T I } .
(b) Suppose that the map x ↦→ IP x is continuous from E in Π(Ω) , and let (Y t )
be l.s.c. on Ω × IR + ; then v(., Y ) is l.s.c.
Proof : We at first suppose that Y is bounded.
(a) Let T ∈ T , and let I be a countable dense subset of IR + . Let T n be a
decreasing sequence of I-valued simple stopping times converging to T . Then since
Y has l.c.s. trajectories,
IE µ (Y T ) ≤ lim inf IE µ ( Y Tn ) ≤ sup { IE µ (Y S ) ; S ∈ T I } ,
so that sup{ IE µ (Y T ) ; T ∈ T } ≤ sup{ IE µ (Y T ) ; T ∈ T I } ; the converse inequality
is obvious, and Theorem 1.7 concludes the proof.
(b) Let (x n ) be a sequence in E converging to x . Let I be a countable dense
subset of IR + , and N be a IP x -null subset of Ω such that for every t ∈ I , the
map ω ↦→ ω t is continuous on N c . Let T = ∑ t i 1 {T =ti } ∈ T I , and fix ɛ > 0 . For
i≤k
each i ≤ k choose an F ti -measurable random variable H i
continuous except on N , and for
such that ω ↦→ H i (ω) is
A(ω, dt) = ∑ i≤k
H i (ω)δ ti (dt) ∈ A , | IE x (Y T ) − IE x (Y A ) | ≤ ɛ .
Since the map Y is l.s.c. on Ω × IR + , the map ω ↦→ Y A (ω) is l.s.c. except on N .
Therefore, for x n → x
lim inf
n
v(x n , Y ) ≥ lim inf
n
IE xn (Y A ) ≥ IE x (Y A ) ≥ IE x (Y T ) − ɛ .
This clearly shows that lim inf v(x n , Y ) ≥ sup{ IE x (Y T ) ; T ∈ T I } = v(x, Y ) ,
and hence that the map v(., Y ) is l.s.c.
The standard truncation argument used in the proof of Theorem 4.5 yields the
lower semi-continuity of v(., Y ) for processes Y of class (D) .
✷
Theorems 4.5 and 4.6 imply the continuity of the reduite in the particular case
of continuous processes, or of continuous functions of (X t ) .
Corollary 4. 7 Let E be compact metrizable, and let Ω = D([0, +∞], E) . Suppose
that the map x ↦→ IP x is weakly continuous, and that Y is a process of class (D)
which is continuous on Ω×IR + . Then the reduite v(x, Y ) = sup{ IE x (Y T ) ; T ∈ T }
is continuous.
Thus we obtain the continuity of the reduite q α
Feller process (see e.g. [12], theorem 2.82 ).
of a continuous function of a
19

Corollary 4. 8 Let X = (Ω , G t , X t , θ t , IP x ; x ∈ E) be a strongly Markov realization
of a Feller semi-group (P t ) on a compact set E . Then for every g ∈ C(E)
and α > 0 ,
R α g(x) = sup { IE x
[
e −αT g(X T ) ] ; T ∈ T (G) }
is continuous.
Proof: We at first suppose that g is the α-potential of a continuous function f .
Then for each T ∈ T and x ∈ E , by the strong Markov property,
IE x ( e −αT U α f(X T ) ) = IE x
( ∫
[T,+∞]
The map (ω, t) ↦→ Y t (ω) =
∫
[t,+∞]
e −αs f(X s ) ds
e −αs f(ω s ) ds is continuous. Indeed, since
f is bounded, the map t ↦→ Y t (ω) is continuous uniformly in t , and for fixed t ,
Proposition 4.1 (a) implies that the map ω ↦→ Y t (ω) is continuous. Hence Corollary
4.7 shows that R α g = v(., Y ) is continuous. The uniform approximation of continuous
functions by potentials concludes the proof.
✷
)
.
Remark: Suppose that the assumptions of Corollary 4.8 are satisfied, and let g , h
be continuous functions on E . Then the map
{ [ ∫
]
}
q α (x) = sup IE x e −αT g(X T ) + e −αu h(X u ) du ; T ∈ T (G)
[0,T ]
is continuous. Indeed,
q α (x) = IE x
[ ∫
[0,+∞]
e −αu h(X u ) du
]
+ sup { IE x
[
e −αT (g − U α h)(X T ) ] ; T ∈ T }
= U α h(x) − R α (g − U α h) (x) .
Since the maps h and g are continuous and (P t ) is Feller, U α h ∈ C(E) , and
R α (g − U α h) ∈ C(E) .
✷
5 Example: optimal stopping for diffusion processes
with jumps
In this section, we prove that in the case of a “good” diffusion with jumps, the
reduite of a continuous function is also continuous. We use the continuity results
established in the previous section for potentials of continuous functions and an
exponential estimation to conclude the proof.
20

5.1 Hypothesis and notations
Let L be an integro-differential operator of the form
Lf(t, x) =
[ 1
2 a ijD 2 ijf + b j D j f + D t f
+
∫
IR d −{0}
]
(t, x)
[
f(x + u) − f(x) − 1{|u| ] S(t, x, du) ,
where we assume that the assumptions (H1) and (H2) hold (cf. [17] for details) :
(H1) ρ(t, x) = (a ij , b j )(t, x) is Borel bounded by K , ρ(t, x) is continuous for
each t , and (a ij ) is strictly positive.
(H2) S(t, x, du) is a positive kernel on IR d − {0} such that
sup{ S( t, x, |u| 2 ∧ 1 ) ; t, x } ≤ K .
Furthermore, for each bounded continuous function f and each t , S(t, ., f) is
continuous.
Then by D. Stroock [30], for each x ∈ IR d , the martingale problem corresponding
to (x, a, b, S) is “well-set”, i.e., there exists a unique probability IP x on the space
Ω = D([0, +∞] , IR d ) endowed with the canonical filtration (F t ) such that (i) and
(ii) hold:
(i) IP x (X 0 = x) = 1 ;
∫
(ii) ∀f ∈ C 1,2
b (IR + × IR d ) , f(t, X t ) − f(0, X 0 ) − Lf(s, X s )ds is a IP x -
martingale.
The assumptions made on a, b , and S imply that Lf(t, .) is continuous for
every function f ∈ C 1,2
b (IR + × IR d ) .
5.2 Continuity properties
We use the following result, which can be deduced from the exponential majorization
in the setting of differential operators ([17] , Theorem 13).
Lemma 5. 1 Given λ ∈ IR and positive numbers A, η, K , there exists a constant
k which only depends on K such that
IP x ( sup{ |X s − X 0 | > η ; 0 ≤ s ≤ t } ) ≤ 2d exp [ − λ d ( η − Kt − A )
[0,t]
+ λ2
2 kt ( 1 + e|λ| ) ] + Kt
A .
The following result generalizes a classical result for continuous diffusions; see e.g.
[25] .
Lemma 5. 2 Under the assumptions (H1) and (H2) , the map x ↦→ IP x
continuous from IR d to Π(Ω) .
is weakly
21

Sketch of proof: The technique used in [17] , Theorem 20 shows at first that the
family (IP x ; x ∈ IR d ) is weakly relatively compact. Let x n → x ; the sequence
(IP xn ) is weakly compact and let P be one of its cluster points. Given f ∈ C 1,2
b (IR + ×
IR d ) set
∫
= f(t, X t ) − f(0, X 0 ) − Lf(s, X s )ds .
H f t
The set {X 0 = x} is closed in the Skorokhod topology, and P (X 0 = x) = 1 . On
the other hand, the right-continuity of (X t ) and the definition of IP x show that, in
order to prove that P = IP x , it suffices to check that IE xn (ξH f t ) → IE x (ξH f t ) for t
in a countable dense subset of IR + such that the maps ω ↦→ X t (ω) are continuous
except on a IP x -null subset of Ω , for f ∈ C 1,2
b (IR + × IR d ) and for a bounded F s -
measurable random variable ξ . This last convergence is a direct consequence of the
continuity of H f t . ✷
[0,t]
Let f ∈ C b (IR d ) and α > 0 ; we want to prove that
R α f(x) = sup { IE x
(
e −αT f(X T ) ) ; T ∈ T }
is a bounded continuous function. We at first suppose that f = U α g with
g ∈ C b (IR d ) . Then Lemma 5.2 and Corollary 4.7 show that R α f ∈ C b (IR d ) .
In order to deduce the general result, we need the following technical lemma.
Lemma 5. 3 Let f ∈ C b (IR d ) . Then for each ɛ > 0 and each compact set K ⊂
IR d , there exists a C ∞ function gK
ɛ with compact support such that:
i) sup { | f(x) − gK(x) ɛ | ; x ∈ K} < ɛ .
ii) ||gK|| ɛ ≤ ||f|| ∞ + ɛ .
This yields the main result of this section.
Theorem 5. 4 Under the assumptions (H1) and (H2), given α > 0 and a bounded
continuous function g , the reduite R α g is continuous.
Proof : The continuity of the reduite is true for C ∞ -functions with compact support,
since they are the α-potential of some continuous functions. For every n , let K n denote
the closure of the ball B(0, n) , and let g n = g 1/n
K n
be the function constructed in
Lemma 5.3 . Given t > 0 and T ∈ T , ∣ ∣ IEx ( e −αT f(X T ) ) − IE x ( e −αT g n (X T ) ) ∣ ∣ ≤
I 1 + I 2 + I 3 , with
I 1 = ∣ [ ∣ IEx e −αT f(X T ) − e −α(T ∧t) f(X T ∧t ) ] ∣ ∣ ,
I 2 =
[ ∣ IE x e
−α(T ∧t) ( f(X T ∧t ) − g n (X T ∧t ) ) ] ∣ ∣ ,
I 3 =
[ ∣ IE x e −αT g n (X T ) − e −α(T ∧t) g n (X T ∧t ) ] ∣ ∣ .
Then I 1 ≤ 2e −αt ||f|| ∞ , and I 3 ≤ 2e −αt (||f|| ∞ + n −1 ) . Fix ɛ > 0 , and choose
t such that I 1 + I 3 ≤ ɛ . Given any n , set T n = inf{ t : X t ∉ K n } . Then,
I 2 ≤ sup { |f − g n |(x) ; x ∈ K n } + ( 2||f|| ∞ + n −1 ) IP x ( T ∧ t ≥ T n )
≤ n −1 + ( 2||f|| ∞ + n −1 ) IP x ( t ≥ T n ) .
22

Thus Lemma 5.1 shows that one can choose N such that I 2 ≤ ɛ for n > N . Hence
sup { ∣ ∣ ∣ IEx
(
e −αT [ f(X T ) − g n (X T ) ] ) ∣ ∣ ∣ ; T ∈ T , x ∈ IR
d } < 2ɛ
for each n > N , and the sequence of continuous functions R α g n converges uniformly
to R α f .
✷
6 Appendix
In this appendix we prove Theorem 4.2 , i.e. the continuity of the map x ↦→ IP x
a Feller semi-group on a compact space E .
for
(a) The proof reduces to show that (IP x ; x ∈ E) is tight. Indeed, let (IP xn )
converge to a probability P on Ω . Since E is compact, extract a subsequence (still
denoted by (x n ) ) , such that x n converges to x . By Proposition 4.1(b) choose
a countable dense I ⊂ IR + and a P -null N such that ω ↦→ ω t is continuous for
ω ∉ N and t ∈ I . Let
h =
∏
f i (X ti ) ∈ H c (I) .
1≤i≤k
The choice of I ensures that IE xn (h) → ∫ hdP . On the other hand, for every
y ∈ E ,
( ( )
IE y (h) = P t1 f1 P t2 −t 1 f2 · · · (P tk −t k−1
f k · · ·)
(y) .
Since (P t ) is Feller, the function IE . (h) is continuous, and IE xn (h) → IE x (h) . Hence
both probabilities P and IP x coincide on H c (I) , and the monotone class theorem
shows that they are equal on σ(H c (I)) = F ∞ .
(b) To establish the tightness of a sequence ( IP xn ; n ∈ IN ) of probabilities on
Ω , we use the following criteria due to Aldous [1] (see also [16]): The conditions (i)
and (ii) must be satisfied:
(i) The laws of ω 0 are tight on E .
(ii) lim lim sup sup sup IP xn ( d(ω T +s , ω T ) > ɛ ) = 0 for every ɛ > 0 , where
h→0 n T ∈T 0≤s≤h
d denotes the Skohorod metric on Ω .
Since E is compact, the first condition (i) is trivially satisfied. To prove that (ii)
holds, we apply the strong Markov property; for every y ∈ E ,
IP y ( d(ω T , ω T +s ) > ɛ ) = IE y ( IP ωT ( d(ω s , ω 0 ) > ɛ ) )
Hence the proof of (ii) reduces to show that
lim sup
h→0 x∈E
sup
0≤s≤h
≤
sup
y∈E
IP y ( d(ω s , ω 0 ) > ɛ ) .
IP x ( d(ω s , ω 0 ) > ɛ ) = 0 .
This is Dynkin’s stochastic continuity property [11], which is always satisfied by
a Feller process on a compact set. We briefly sketch the proof; set
q(u) = 1 − u/ɛ if u ≤ ɛ , and q(u) = 0 if u > ɛ .
23

The function φ x (.) = q(d(x, .)) is continuous, and ||φ x − φ y || ≤ d(x, y)/ɛ . Fix
0 < α < ɛ 2 , and let (x i ; 1 ≤ i ≤ k) be the centers of balls of radius α covering
the compact set E . Given x ∈ E , let x i be such that d(x, x i ) < α . Then for
every s ,
IP x ( d(ω 0 , ω s ) > ɛ ) = IP x ( d(x, ω s ) > ɛ )
≤ φ x (x) − IE x ( φ x (ω s ) )
≤ | φ x (x) − φ xi (x) | + | φ xi (x) − IE x (φ xi (ω s ) |
+ IE x ( |φ xi − φ x |(ω s ) )
≤ 2α/ɛ + |φ xi (x) − IE x ( φ xi (ω s ) ) | .
Since (P t ) is a Feller semi-group, and since φ xi
is continuous,
lim
h→0
sup
0≤s≤h
sup
x
| φ xi (x) − P s φ xi (x) | = 0 .
✷
Remark: Theorem 4.2 is also true for Markov processes which are Feller in
the following weaker sense: U α (C(E)) ⊂ C(E) for each α > 0 , and αU α f → f
pointwise as α → ∞ for each f ∈ C(E) . Indeed, in part (a) of the proof, replace
the class of r.v. H c (I) by the following class H u made of random variables h
h = I k (λ 1 , f 1 ; · · · ; λ k , f k ) =
∏
1≤i≤k
∫
[0,+∞]
e −λ it f i (X t ) dt
for k ≥ 1, λ i ∈ Q and f i in a dense subset of C(E) .
The characterization of potentials given in [14], p.38 shows that IE x (h) is the
λ = ∑ λ i potential of the function P 0 g , with
g(x) =
∑
1≤i≤k
f i IE x
(
Ik−1 (λ 1 , f 1 ; · · · ; ˆλ i , ˆf i ; · · · ; λ k , f k ) ) ,
in which “ˆ” indicates the quantity which has been omitted. Since U λ P 0 = U λ , an
easy induction argument together with the weaker Feller property show that IE x (h)
is the potential of a continuous function g on E . The continuity of potentials shows
that
IE xn (h) = U λ g(x n ) → U λ g(x) = IE x (h) .
The proof of (b) carries over without change. Indeed, the last convergence property
can be checked on potentials which are dense in C(E) for the uniform topology.
24

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26