STOCHASTIC ORDER RELATIONS AND LATTICES OF
PROBABILITY MEASURES∗
ALFRED MÜLLER† AND MARCO SCARSINI‡
Abstract. We study various partially ordered spaces of probability measures and we determine which of them are lattices. This has important consequences for optimization problems with
stochastic dominance constraints. In particular we show that the space of probability measures on
R is a lattice under most of the known partial orders, whereas the space of probability measures on
Rd typically is not. Nevertheless, some subsets of this space, defined by imposing strong conditions
on the dependence structure of the measures, are lattices.
Key words. Copula, zonoid, lattice, univariate stochastic orders, multivariate stochastic orders,
optimization with stochastic dominance constraints.
AMS subject classifications. 60E15, 06B23
1. Introduction. A partially ordered set is called a lattice if every pair of elements has a supremum and an infimum. A great deal of literature has appeared in
the recent decades about ordered sets of probability measures (see for instance [34]
and [25] for the state of the art), but surprisingly little attention has been given to
the lattice structure of these sets. To the best of our knowledge the only exceptions
are [19], [17], [7], [11], and [20, 21].
Lattice structures of ordered sets of probability measures have important implications for optimization problems where ordered sets of probability measures occur as
constraint sets. As a concrete example of an application where a lattice structure is
very helpful we consider optimization problems with stochastic dominance constraints
as considered e.g. in [5]. There optimization problems of the form
max f (X)
subject to X ≤∗ Yi ,
X∈C
i = 1, ..., n,
(1.1)
(1.2)
(1.3)
are considered, where f is some real valued functional, C is a set of random variables,
and ≤∗ is some stochastic order relation. If the stochastic order relation ≤∗ leads to a
lattice, then the problem with multiple stochastic dominance constraints is equivalent
to the problem
max f (X)
subject to X ≤∗ Y1 ∧∗ ... ∧∗ Yn ,
X∈C
with only one constraint, which is much more easy to solve. Similar optimization
problems can be found in [8] and [16].
In several other fields of research it is also often useful to resort to classes of
distributions rather than one single distribution. This happens for instance in robust
∗ Partially
supported by the Vigoni Program and MIUR-COFIN
für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Geb. 20.21,
D-76128 Karlsruhe, Germany, [email protected]
‡ Dipartimento di Statistica e Matematica Applicata, Università di Torino, Piazza Arbarello 8,
I–10122 Torino, Italy, [email protected]
† Institut
1
2
A. MÜLLER AND M.SCARSINI
Bayesian statistics, where one considers families of prior distributions, for instance
the so called ε-contamination classes (see e.g. [2], [3], and [28]).
Whereas in decision theory à la Savage, the decision maker maximizes her expected utility with respect to her subjective probability measure, more recent developments in the field have led to paradigms of choice that involve a whole class of
probability measures rather than a single probability. This allows to incorporate in
the model the idea of ambiguity (see e.g. [12] and the subsequent literature).
An interesting area where the concepts of robustness and ambiguity coexist is
robust control under economic model uncertainty (see for instance [14] and [13]).
In mathematical finance, classes of equivalent martingale measures occur when
dealing with incomplete markets (see e.g. [18]).
In all these situations it may be useful to compute bounds with respect to some
order. That is, given a class C of distributions, it may be interesting to find, among
all the distributions that are larger than all distributions in C, the smallest one, where
of course larger and smaller refer to some pre-specified partial order. For the above
problem to be well defined it is necessary to have a lattice structure on the space of
distributions.
In this paper we will try to study this issue in a more systematic way. It will
turn out that the space of probability measures on R (or some suitable subsets of
it) is a lattice when endowed with most of the well known stochastic orders like
usual stochastic order, convex order, dispersion order, hazard rate order etc., whereas
the space of probability measures on Rd is in general not a lattice. In order to
obtain a lattice structure for sets of probability measures on Rd we need to put severe
restrictions on their dependence structure.
The paper is organized as follows. Section 2 contains some preliminary definition
and results, Section 3 is devoted to the space of probability measures on R, and
Section 4 deals with the space of probability measures on Rd . Section 5 studies in
details the properties of a special order.
2. Preliminaries. In this section we will introduce notation and all the definitions used in the rest of the paper.
2.1. Orders and lattices. We first recall the basic definitions of the theory of
lattices.
Definition 2.1. Let (X , ≤∗ ) be an ordered set. For x, y ∈ X let U (x, y) = {z ∈
X : x ≤∗ z, y ≤∗ z}. If U (x, y) has a smallest element z̃ such that z̃ ≤∗ z for all z ∈
U (x, y) then z̃ is called the supremum of x and y, denoted by z̃ = x ∨∗ y = sup{x, y}.
Similarly, if there is a unique largest element z 0 smaller than x and y, then this is
called the infimum, denoted by z 0 = x ∧∗ y = inf{x, y}.
If x ∨∗ y and x ∧∗ y exist for all x, y ∈ X , then (X , ≤∗ ) is called a lattice.
A subset Z ⊂ X of a lattice is called a sublattice, if x, y ∈ Z implies x ∨∗ y ∈ Z
and x ∧∗ y ∈ Z. Notice that (Z, ≤∗ ) can be a lattice in its own right without being a
sublattice.
For properties of lattices the reader is referred to [4] and [1].
The following lattice and some of its sublattices will be used quite frequently in
the following. Let S be an arbitrary set and let X be the set of all functions f : S → R,
endowed with the pointwise order
f ≤g
if f (s) ≤ g(s)
for all s ∈ S.
STOCHASTIC ORDER RELATIONS AND LATTICES
3
This is obviously a lattice with
f ∨ g(·) = max{f (·), g(·)}
and
f ∧ g(·) = min{f (·), g(·)}.
(2.1)
Remark 1. Consider the lattices induced by the two orders ¹1 , ¹2 on the same
space X . Let ¹1 be stronger than ¹2 , namely, for all x, y ∈ X , let x ¹1 y imply
x ¹2 y. Then x ∧1 y ¹2 x ∧2 y, and x ∨2 y ¹2 x ∨1 y. Therefore comparability of the
orders induces comparability of the suprema and infima (with respect to the weaker
order).
The following order ≤↑ will be of interest in the sequel. Let S = [a, b] ⊂ R be a
finite interval and let BV(S) be the set of functions F : S → R, which are of bounded
variation. Endow BV(S) with the following relation ≤↑ .
For F, G ∈ BV(S),
F ≤↑ G if s 7→ G(s) − F (s)
is increasing.
Properties of ≤↑ will be studied in Section 5.
2.2. Probability. Next, we will collect some basic notations from probability
that are needed in the sequel.
Definition 2.2. Given a topological space Y, B(Y) will indicate its Borel σfield, M(Y) will be the set of σ-additive probability measures on (Y, B(Y)), and for
some measure µ on (Y, B(Y)), Mµ (Y) ⊂ M(Y) will be the set of probability measures
dominated by the measure µ.
When Y is a linear space, then M∗ (Y) ⊂ M(Y) will be the set of probability
measures with finite expectation, and Ma (Y) ⊂ M∗ (Y) will be the set of probability
measures with finite expectation equal to a.
We denote by δx the degenerate probability measure at x.
Definition 2.3. Given a probability measure P ∈ M(R), define FP the associated distribution function, and F P the associated survival function, i. e.
FP (x) = P ((−∞, x]),
F P (x) = P ((x, ∞)) = 1 − FP (x).
Define the quantile function
FP−1 (u) = sup{x : FP (x) ≤ u},
0 < u < 1.
When P ∈ Mµ (R), define fP the associated density function, and rP the associated hazard rate function, i. e.
dP
(x), µ-a.s.,
dµ
fP (x)
rP (x) =
, µ-a.s..
F P (x)
fP (x) =
(2)
Given P ∈ M∗ (R), let F P be the associated integrated survival function,
Z ∞
(2)
F P (x) =
F P (t) dt.
x
For P ∈ M∗ (R+ ) we call mP the associated mean residual life function, i.e.
(2)
mP (x) =
F P (x)
.
F P (x)
The name is due to the fact that, if the nonnegative random variable X has law P ,
then mP (x) = E[X − x|X > x].
4
A. MÜLLER AND M.SCARSINI
2.3. Univariate stochastic orders. The following definitions of stochastic orders can be found e.g. in [34] or [25].
Definition 2.4. Given P, Q ∈ M(R) we define
Z
Z
P ≤st Q if
φ dP ≤ φ dQ for all increasing φ,
P ≤disp Q
P ≤hr Q
if
if
FP−1 (t) − FP−1 (s) ≤ FQ−1 (t) − FQ−1 (s)
t 7→
F Q (t)
F P (t)
for all 0 < s < t < 1,
is increasing.
Given P, Q ∈ M∗ (R) we define
Z
Z
P ≤cx Q if
φ dP ≤ φ dQ for all convex φ,
Z
Z
P ≤icx Q if
φ dP ≤ φ dQ for all increasing convex φ.
Given P, Q ∈ M∗ (R+ ) we define
P ≤mrl Q
if
mP (t) ≤ mQ (t)
for all t ∈ R+ .
Given P, Q ∈ Mµ (R) we define
P ≤lr Q
if
fP (t)fQ (s) ≤ fP (s)fQ (t)
for all s ≤ t.
Remark 2. In all the integral orders (≤st , ≤cx , ≤icx ) the defining inequality is
assumed to hold whenever the expectations exist.
Remark 3. If P, Q ∈ Mµ (R), then P ≤hr Q iff rP (x) ≥ rQ (x) for all x ∈ R. If
furthermore fP > 0, then P ≤lr Q iff fQ /fP is increasing.
Remark 4. Given a random variable X with law P , and a number a ∈ R, we
denote by Pa the law of X + a. Then we have
P ≤disp Q
iff Pa ≤disp Q for all
a ∈ R.
Therefore the relation ≤disp is not antisymmetric, which implies that it is not a partial
order on M(R). It is a partial order on the quotient space (M(R)/∼ ) with respect to
the relation
P ∼ Q iff
Q = Pa
for some
a ∈ R.
2.4. Multivariate stochastic orders. Definition 2.5. Any function C :
[0, 1]d → [0, 1] which is (the restriction of ) a d-variate distribution function with
uniform marginals on [0, 1] is called a copula.
Lemma 2.6 ([35]). Let P ∈ M(Rd ). For i ∈ {1, . . . , d} let Pi ∈ M(R) be the i-th
unidimensional marginal of P (i.e. Pi (A) = P (R × · · · × R × A × R · · · × R)).
Then there exists a copula CP such that
P (×di=1 (−∞, xi ]) = CP (P1 ((−∞, x1 ]), . . . , Pd ((−∞, xd ])).
For the properties of copulae the reader is referred to [32], [15], or [26]. For the
properties of copulae of probability measures on more general product spaces, see
[31].
STOCHASTIC ORDER RELATIONS AND LATTICES
5
Definition 2.7. Let M(C) (Rd ) be the set of probability measures with a common
copula C, and let Ma (Rd ) be the set of probability measure with expectation equal to
a.
Definition 2.8. A random variable X is stochastically increasing in the random
vector Y (denoted by X ↑st Y) if for all s ≤ t we have L(X|Y = s) ≤st L(X|Y = t),
where L(X|A) is the conditional law of X given A.
Definition 2.9 ([24]). A random vector X = (X1 , . . . , Xd ) is said to be conditionally increasing (CI), if
Xi ↑st (Xj ,
j ∈ J)
for all J ⊂ {1, . . . , d} and i 6∈ J.
A copula is called conditionally increasing if it is the copula of the distribution of a
CI random vector.
Definition 2.10. A function φ : Rd → R is called supermodular if
φ(x) + φ(y) ≤ φ(x ∨ y) + φ(x ∧ y)
for all x, y,
where Rd is endowed with the usual componentwise order and the corresponding lattice
structure.
A function φ : Rd → R is called directionally convex if for all xi ∈ Rd , i =
1, 2, 3, 4, such that x1 ≤ x2 ≤ x4 , x1 ≤ x3 ≤ x4 and x1 + x4 = x2 + x3 ,
φ(x2 ) + φ(x3 ) ≤ φ(x1 ) + φ(x4 ).
Notice that a function is directionally convex, if it is supermodular and convex in
each variable, when the others are held fixed.
For the following definitions of stochastic orders the reader is referred again to
[34] or [25].
Definition 2.11. Given P, Q ∈ M(Rd ) we define
Z
Z
P ≤st Q if
φ dP ≤ φ dQ for all increasing φ,
Z
Z
P ≤cx Q if
φ dP ≤ φ dQ for all convex φ,
Z
Z
P ≤sm Q if
φ dP ≤ φ dQ for all supermodular φ,
Z
Z
P ≤dcx Q if
φ dP ≤ φ dQ for all directionally convex φ,
Z
Z
P ≤lcx Q if
φ dP ≤ φ dQ, for all φ such that
φ(x) = ψ(`(x)) with ψ : R → R convex and ` : Rd → R linear,
Z
Z
P ≤plcx Q if
φ dP ≤ φ dQ, for all φ such that
φ(x) = ψ(`(x)) with ψ : R → R convex and ` : Rd → R linear and increasing,
¡
¢
¡
¢
P ≤lo Q if P ×di=1 (−∞, xi ] ≤ Q ×di=1 (−∞, xi ]
for all (x1 , . . . , xd ) ∈ Rd ,
¡ d
¢
¡ d
¢
P ≤uo Q if P ×i=1 (xi , ∞) ≤ Q ×i=1 (xi , ∞)
for all (x1 , . . . , xd ) ∈ Rd .
6
A. MÜLLER AND M.SCARSINI
3. Lattices of measures on R. Now we will investigate whether the orders
defined in Subsection 2.3 lead to a lattice structure. In case of ≤st this is easy. The
well known fact that P ≤st Q if and only if F P (x) ≤ F Q (x) for all x ∈ R immediately
implies the following result.
Theorem 3.1. The ordered set (M(R), ≤st ) is a lattice with
F P ∧st Q = min{F P , F Q }
and
F P ∨st Q = max{F P , F Q }.
In the next result we use the notation
vex(f )(x) = sup{g(x) : g is convex and g(y) ≤ f (y) for all y ∈ R}.
for the convex hull operator, yielding the largest convex function smaller than a given
one.
Theorem 3.2. The ordered set (M∗ (R), ≤icx ) is a lattice with
(2)
(2)
(2)
F P ∧icx Q = vex(min{F P (x), F Q (x)}),
(2)
(2)
(2)
F P ∨icx Q = max{F P (x), F Q (x)}.
Proof. It is well known that increasing convex order can be characterized by
pointwise comparison of the integrated survival functions, i.e. P ≤icx Q holds if and
only if
Z ∞
Z ∞
(2)
(2)
F P (x) =
F P (t) dt ≤
F Q (t) dt = F Q (x)
x
x
for all real x. Denote by F (2) (R) the class of all integrated survival functions.
F (2) (R) contains all functions f that are continuous, decreasing, convex, and satisfy limx→−∞ f (x) − x = a for some a ∈ R, and limx→∞ f (x) = 0, see e.g. Theorem
1.5.10 in [25]. Therefore the pointwise maximum of two such functions f and g again
is such a function, and clearly the smallest integrated survival function larger than f
and g. The pointwise minimum of two such functions is not necessarily convex, but
there is always a largest integrated survival function h ∈ F (2) (R) smaller than f and
g, namely h = vex(min{f, g}).
The ordered set (M∗ (R), ≤cx ) is not a lattice, as P ≤cx Q can only hold for
distributions with the same mean. Therefore only the set (Ma (R), ≤cx ) containing
all distributions with some fixed mean a ∈ R can be a lattice. Since in this case
(2)
limx→−∞ F P (x) − x = a, the following result can be proved exactly as 3.2.
Theorem 3.3. For all a ∈ R the ordered set (Ma (R), ≤cx ) is a lattice with
(2)
(2)
(2)
F P ∧cx Q = vex(min{F P (x), F Q (x)}),
(2)
(2)
(2)
F P ∨cx Q = max{F P (x), F Q (x)}.
The problem examined in Theorem 3.2 and 3.3 has been studied extensively by
[20, 21]. The reader is also referred to [19], [7], and [11].
In the next result we investigate the lattice structure of the mean residual life
order ≤mrl for distributions on R+ with a finite mean.
STOCHASTIC ORDER RELATIONS AND LATTICES
7
Theorem 3.4. The ordered set (M∗ (R+ ), ≤mrl ) is a lattice with
µ Z x
¶
1
min{mP (0), mQ (0)}
F P ∧mrl Q (x) = exp −
dt ·
,
min{m
(t),
m
(t)}
min{m
P
Q
P (x), mQ (x)}
0
µ Z x
¶
max{mP (0), mQ (0)}
1
dt ·
.
F P ∨mrl Q (x) = exp −
max{mP (x), mQ (x)}
0 max{mP (t), mQ (t)}
Proof. A function m : R+ → R is a mean residual life function of some probability
measure P on R+ , if and only if it has the following properties: It is non-negative,
right-continuous, and such that t 7→ m(t) + t is increasing, and if there exists t0 such
that m(t0 ) = 0, then m(t) = 0 for all t > t0 . If such a t0 does not exist, then
Z ∞
1
dt = ∞
m(t)
0
(see [36], [33, 34]). The class of these functions is closed under pointwise minimum
and maximum. As the survival function F P is uniquely determined by the mean
residual life function mP via
µ Z x
¶
1
mP (0)
F P (x) = exp −
dt ·
,
mP (x)
0 mP (t)
the given representation for the survival functions of P ∨mrl Q and P ∧mrl Q follows.
The following theorems will make use of properties of the order ≤↑ that are proved
is Section 5.
Theorem 3.5. The ordered set ((M(R)/∼ ), ≤disp ) is a lattice with
³
´−1
FP ∧disp Q = FP−1 ∧↑ FQ−1
,
³
´−1
FP ∨disp Q = FP−1 ∨↑ FQ−1
.
Proof. Notice that P ≤disp Q, if and only if FP−1 ≤↑ FQ−1 . Therefore the assertion
follows from Lemma 5.2.
Remark 5. The set (M(R), ≤hr ) is not a lattice. Notice that P ≤hr Q holds if and
only if log(F̄P ) ≤↑ log(F̄Q ). Therefore (M(R), ≤hr ) would be a lattice, if the set of
logarithms of survival functions, endowed with ≤↑ were a lattice. However, whereas
log(F̄P ) ∧↑ log(F̄Q ) is always a logarithm of a survival function, this in not necessarily
the case for log(F̄P ) ∨↑ log(F̄Q ). In this case it may happen that the limit for x → ∞
is finite. If for example P has as support all even numbers and Q has as support the
odd numbers, then log(F̄P ) ∨↑ log(F̄Q ) ≡ 0, and this obviously is not a logarithm of
a survival function of a distribution on R.
However, the order relation ≤hr defines a lattice for distributions on the extended
real line R ∪ {+∞}, allowing explicit mass on {+∞}. Thus the logarithm of the
survival function is allowed to have a finite limit as x → ∞. The function f ≡ 0, for
instance, then is the logarithm of the distribution with P ({+∞}) = 1.
Theorem 3.6. The set (M(R ∪ {+∞}), ≤hr ) is a lattice with
F P ∧hr Q = exp(log(F P ) ∧↑ log(F Q )),
F P ∨hr Q = exp(log(F P ) ∨↑ log(F Q )).
8
A. MÜLLER AND M.SCARSINI
Proof. As P ≤hr Q holds if and only if log(F̄P ) ≤↑ log(F̄Q ), this is an immediate
consequence of Lemma 5.1.
Example 1. We will illustrate how the suprema and infima vary with respect to
the various orders by comparing two simple discrete distributions, which have the
same mean and variance, and therefore are not comparable with respect to any of the
mentioned orders.
Let
1
1
1
3
1
P = δ1 + δ3 ,
Q = δ0 + δ2 + δ4 .
2
2
8
4
8
Then
3
3
1
1
P ∧st Q = δ0 + δ1 + δ2 + δ3 ,
8
8
8
8
1
3
3
1
P ∨st Q = δ1 + δ2 + δ3 + δ4 ;
8
8
8
8
1
1
1
P ∧cx Q = P ∧icx Q = δ1 + δ2 + δ3 ,
4
2
4
1
3
3
1
P ∨cx Q = P ∨icx Q = δ0 + δ4/3 + δ8/3 + δ4 ;
8
8
8
8
P ∧disp Q = δ1 ,
1
3
3
1
P ∨disp Q = δ1 + δ3 + δ5 + δ7 ;
8
8
8
8
7
3
1
1
P ∧hr Q = δ0 + δ1 + δ2 + δ3 ,
8
16
8
16
P ∨hr Q = δ4 ;
5
2
2
P ∧mrl Q = δ1 + δ2 + δ3 ,
9
9
9
1
10
9
9
P ∨mrl Q = δ0 + δ1 + δ2 + δ4 .
8
32
32
32
Notice that in most cases the support is contained in the union of the supports of P
and Q, whereas
supp(P ∨cx Q) 6⊆ supp(P ) ∪ supp(Q) =: S = {0, 1, 2, 3, 4}.
Therefore (M(S), ≤cx ) is not a sublattice of (M(R), ≤cx ). Notice, however, that
(M(S), ≤cx ) is still a lattice in its own right. In the lattice (M(S), ≤cx ) the supremum
of P and Q from the above example is given by
P ∨cx Q =
1
1
1
1
1
δ0 + δ1 + δ2 + δ3 + δ4 .
8
4
4
4
8
Theorem 3.7.
(a) For any P, Q ∈ M(R) we have
supp(P ∨st Q) ⊆ supp(P ) ∪ supp(Q),
supp(P ∧st Q) ⊆ supp(P ) ∪ supp(Q).
(b) For any P, Q ∈ Ma (R) we have
supp(P ∨cx Q) ⊆ conv(supp(P ) ∪ supp(Q)),
supp(P ∧cx Q) ⊆ supp(P ) ∪ supp(Q).
STOCHASTIC ORDER RELATIONS AND LATTICES
9
(c) For any P, Q ∈ M∗ (R+ ) we have
supp(P ∨mrl Q) ⊆ supp(P ) ∪ supp(Q),
supp(P ∧mrl Q) ⊆ supp(P ) ∪ supp(Q).
Proof. We show the case ∧cx . The other cases are similar. Let K = FP ∧cx Q
(2)
and fix x 6∈ supp(P ) ∪ supp(Q). Then there is a neighborhood Ux of x where F̄P
(2)
(2)
(2)
(2)
and F̄Q are affine. Hence K = vex(min(F P , F Q )) is also affine on Ux and thus
x 6∈ supp(P ∧cx Q).
As a consequence of Theorem 3.7 we get the following result. The proof is omitted.
Theorem 3.8.
(a) For any measurable subset S ⊂ R the partially ordered set (M(S), ≤st ) is a sublattice of (M(R), ≤st ).
(b) For any convex subset S ⊂ R and any a ∈ S the partially ordered set (Ma (S), ≤cx )
is a sublattice of (Ma (R), ≤cx ).
(c) For any measurable subset S ⊂ R+ the partially ordered set (M∗ (S), ≤mrl ) is a
sublattice of (M∗ (R+ ), ≤mrl ).
Remark 6. It is well known that on M(R) we have
≤lr ⊂ ≤hr ⊂ ≤st ⊂ ≤icx ,
and on Ma (R) we have
≤disp ⊂ ≤cx .
For some of the orders examined in this section we have a stronger comparability
result than the one stated in Remark 1, that is the infima and suprema are comparable
with respect to the stronger order, as the following proposition shows.
Proposition 3.9.
(a) P ∨icx Q ≤st P ∨st Q,
(b) P ∧st Q ≤st P ∧icx Q,
(c) P ∨st Q ≤hr P ∨hr Q,
(d) P ∧hr Q ≤hr P ∧st Q.
Proof.
(2)
(2)
(2)
(a) Define H = FP ∨icx Q . Then H (x) = max(FP (x), FQ (x)). As H(x) =
(2)
− d+ H (x)/ dx, we have that H(x) equals either FP (x) or FQ (x), and
therefore
H(x) ≤ max(FP (x), FQ (x)) = F P ∨st Q (x),
which implies the desired result.
(2)
(2)
(2)
(b) Define K = FP ∧icx Q . Then K (x) = vex(min(FP , FQ ))(x). For fixed
x we have that K(x) either equals F P (x) or F Q (x), or there exists a largest
interval [a, b) containing x on which K
(2)
(2)
is affine, hence K is constant. Since
(2)
(2)
in the latter case K
equals either F P or F Q in the point a and it is
smaller than both these functions between a and b, we then have
K(x) = K(a) ≥ min(F P (a), F Q (a)) ≥ min{F P (x), F Q (x)} = F P ∧st Q (x),
which implies the desired result.
10
A. MÜLLER AND M.SCARSINI
(c) Define R = P ∨st Q. Then FR (x) = max(FP (x), FQ (x)). Let µ be a dominating measure of P and Q and let rP and rQ be the corresponding hazard
rate functions. Then rR (x) equals either rP (x) or rQ (x) µ-a.s.. Therefore
rR (x) ≤ max{rP (x), rQ (x)} = rP ∨hr Q (x) µ − a.s.
and therefore R ≤hr P ∨hr Q. The proof of (d) is similar.
Theorem 3.10. The following sets of probability measures on R are lattices, if
they are endowed with the order ≤lr :
(i) the set of all probability measures with a common finite support,
(ii) the set of all probability measures on a bounded interval (a, b) having a strictly
positive Lebesgue density f such that log f is of locally bounded variation.
Proof.
(i) In general P ≤lr Q holds if they both have densities fP , fQ with respect
to some dominating measure such that log fP ≤↑ log fQ . Therefore, if both
probability measures have a common finite support {x1 , . . . , xn } with x1 <
. . . < xn then we get, by (5.2) and (5.3),
½
¾
(fP ∨lr Q )(xi+1 )
fQ (xi+1 ) fP (xi+1 )
= max
,
(3.1)
(fP ∨lr Q )(xi )
fQ (xi )
fP (xi )
and
(fP ∧lr Q )(xi+1 )
= min
(fP ∧lr Q )(xi )
½
fQ (xi+1 ) fP (xi+1 )
,
fQ (xi )
fP (xi )
¾
(3.2)
(ii) We have
log(fP ∨lr Q ) = c(log fP ∨↑ log fQ )
where c is such that
Rb
a
(fP ∨lr Q )(x) dx = 1. Similarly,
log(fP ∧lr Q ) = c0 (log fP ∧↑ log fQ )
where c0 is such that
Rb
a
(fP ∧lr Q )(x) dx = 1.
Remark 7. For a probability measure P with values in {0, 1, ..., N } the so called
equilibrium rate function eP is defined as
eP (n) :=
P ({n − 1})
,
P ({n})
n = 1, ..., N.
It is well known that eP uniquely determines P and that P ≤lr Q holds if and only if
eP (n) ≥ eQ (n) for all n = 1, ..., N , see e.g. [34], p. 435ff. Thus the proof of part (i)
of Theorem 3.10 is not surprising. It just states that
eP ∨lr Q (n) = min{eP (n), eQ (n)}
and
eP ∧lr Q (n) = max{eP (n), eQ (n)}.
Example 2. (a) Let S = {1, 2, 3} and let
fP (1) = fP (3) =
1
,
4
fP (2) =
1
,
2
STOCHASTIC ORDER RELATIONS AND LATTICES
11
and let
fQ (1) = fQ (2) = fQ (3) =
1
.
3
It follows from (3.1) that
fP ∨lr Q (3) = fP ∨lr Q (2) = 2fP ∨lr Q (1).
Normalization yields
fP ∨lr Q (3) = fP ∨lr Q (2) =
2
5
and
fP ∨lr Q (1) =
1
.
5
2
5
and
fP ∧lr Q (3) =
1
.
5
Similarly, one obtains
fP ∧lr Q (1) = fP ∧lr Q (2) =
(b) Let S = [0, 1] and consider the following example for Lebesgue-densities:
(
4x,
x ≤ 1/2,
and fQ (x) = 1, 0 ≤ x ≤ 1.
fP (x) =
4 − 4x, x > 1/2
From (5.1) it follows that
d
log fP ∨lr Q (x) =
dx
(
1/x, 0 < x ≤ 1/2,
0,
x > 1/2.
Normalization yields
(8
fP ∨lr Q (x) =
3 x,
4
3,
x ≤ 1/2,
x > 1/2.
Similarly, one obtains
(4
fP ∧lr Q (x) =
3,
8
3 (1
x ≤ 1/2,
− x), x > 1/2.
(c) The set of all probability measures with support N0 , endowed with the order ≤lr ,
is not a lattice. To see this, let
fP (n) = (1/2)k+1 , n = 2k, 2k + 1, k = 1, 2, ...
and
fQ (n) = (1/2)k+1 , n = 2k − 1, 2k, k = 1, 2, ...
A density h with h/fP increasing and h/fQ increasing would have to be increasing
on the whole of N0 which is impossible. Therefore the set {P, Q} has no upper bound
with respect to ≤lr . A very similar argument can be used to show that the set of all
probability measures on R having Lebesgue densities, endowed with the order ≤lr , is
not a lattice. Let
fP (x) = (1/2)k+3 , 2k ≤ |x| < 2k + 2, k = 0, 1, 2...
12
A. MÜLLER AND M.SCARSINI
and
(
1/4,
fQ (x) =
(1/2)k+3 ,
|x| < 1,
2k − 1 ≤ |x| < 2k + 1, k = 1, 2, ...
Then a density h with h/fP increasing and h/fQ increasing would have to be increasing on the whole of R which is impossible.
4. Lattices of measures on Rd . In this section we will study the lattice structure of the orders defined in Subsection 2.4.
Theorem 4.1. The following ordered sets of probability measures are lattices:
(a) for any copula C, the set (M(C) (Rd ), ≤st ),
(b) for any conditionally increasing copula C, and for all a ∈ Rd , the set
(C)
(Ma (Rd ), ≤dcx ),
(c) for any conditionally increasing copula C, and for all a ∈ Rd , the set
(C)
(Ma (Rd ), ≤plcx ).
The following lemmas will be needed.
Lemma 4.2 ([29, 30]). Let P, Q ∈ M(C) (Rd ). Then P ≤st Q iff, for i ∈
{1, . . . , d}, Pi ≤st Qi .
Lemma 4.3 ([24]). Let C be a conditionally increasing copula, and let P, Q ∈
M(C) (Rd ). Then P ≤dcx Q iff, for i ∈ {1, . . . , d}, Pi ≤cx Qi .
Proof. [Proof of Theorem 4.1]
(a) By Lemma 4.2 and Theorem 3.1, we obtain that the ordered set (M(C) (Rd ), ≤st )
is a product of lattices, hence a lattice.
(b) By Lemma 4.3 and by Theorem 3.3, we obtain that, if C is CI, then the set
(C)
(Ma (Rd ), ≤dcx ) is a product of lattices, hence a lattice,
(c) As P ≤dcx Q implies P ≤plcx Q which in turn implies Pi ≤cx Qi for all marginals,
it follows from Lemma 4.3 that the orderings ≤dcx and ≤plcx are equivalent on
the set of all probability measures with a fixed conditionally increasing copula C.
Thus the result follows immediately from part (b).
Example 3. Theorem 4.1 can be helpful for solving some multivariate versions
of optimization problems with stochastic ordering constraints of the type described
in (1.1). Let X = (X1 , . . . , Xd ) describe a portfolio of d risks, corresponding e.g. to
different lines of business. Consider a portfolio optimization problem of the following
type:
max f (X)
subject to X ≤plcx Y(i) ,
i = 1, ..., n,
(4.1)
+
CX = C .
Notice that X ≤plcx Y is equivalent to
d
X
j=1
wj Xj ≤cx
d
X
wj Yj
for all w1 , . . . , wd ≥ 0,
j=1
which can be interpreted as follows: a portfolio consisting of the risks (X1 , . . . , Xd )
is less risky than a portfolio consisting of the benchmark risks (Y1 , . . . , Yd ), for all
possible portfolio weights w1 , . . . , wd . The assumption that the copula of X is given
STOCHASTIC ORDER RELATIONS AND LATTICES
13
by the upper Fréchet bound C + (or in other words that the risks are comonotonic)
is a quite common assumption in the calculation of risk measures for portfolios, see
e.g. [6]. The two main reasons for this are first that comonotonicity typically yields
a worst case bound, and second that many risk measures are easy to evaluate in the
case of comonotonicity.
It follows from Theorem 4.1 that the problem (4.1) is equivalent to
max f (X)
subject to X ≤plcx Y(1) ∧plcx . . . ∧plcx Y(n) ,
(4.2)
+
CX = C ,
which in turn is equivalent to
max f (X)
(1)
subject to Xj ≤cx Yj
(n)
∧cx . . . ∧cx Yj
,
j = 1, ..., d,
(4.3)
+
CX = C .
[17] were the first to recognize that, for d > 1, the stochastic order on Rd does
not generate a lattice.
Proposition 4.4 ([17]). The ordered set (M(Rd ), ≤st ) is not a lattice.
The counterexample that they use in their proof is the following. Consider for
d = 2 the probability measures
P =
1
1
δ(0,0) + δ(1,1) ,
2
2
Q=
1
1
δ(0,1) + δ(1,0) .
2
2
Given the upper sets
A = {(x1 , x2 ) : x1 ≥ 1, x2 ≥ 1},
B = {(x1 , x2 ) : x1 ≥ 0, x2 ≥ 0, max{x1 , x2 } ≥ 1},
any upper bound (with respect to ≤st ) R for P and Q has to satisfy
R(A) ≥
1
,
2
R(B) = 1.
The measures
R1 =
1
1
δ(0,1) + δ(1,1)
2
2
and R2 =
1
1
δ(1,0) + δ(1,1)
2
2
are upper bounds for {P, Q} with respect to ≤st . But R̃ ≤st R1 , R̃(A) ≥ 1/2, and
R̃(B) = 1 imply R̃ = R1 , therefore R1 is a smallest upper bound. By symmetry it
can be shown that R2 is a smallest upper bound with respect to ≤st . Therefore no
supremum exists for {P, Q}.
Remark 8. Even if (a) of Theorem 4.1 holds, in general two distributions in the
set of all distributions on Rd do not have a supremum with respect to ≤st , even if
they have a common copula. Consider for instance N (0, 1) × δ0 and δ0 × N (0, 1),
where N (0, 1) is the standard normal distribution. Notice that any distribution with
marginals 1/2·(N + (0, 1)+δ0 ) is an upper bound with respect to ≤st , where we denote
by N + (0, 1) a standard normal distribution conditioned to be positive.
Proposition 4.5. For any a ∈ Rd the ordered set (Ma (Rd ), ≤cx ) is not a
lattice.
14
A. MÜLLER AND M.SCARSINI
Proof. Without loss of generality we will consider the case a = 0. Any other case
can be obtained by translation. Let
1
1
δ(−1,−1) + δ(1,1) ,
2
2
1
1
Q = δ(−1,1) + δ(1,−1) .
2
2
P =
The measure
R=
1
1
1
1
δ(−2,0) + δ(0,−2) + δ(2,0) + δ(0,2)
4
4
4
4
(4.4)
dominates both P and Q in (M0 (Rd ), ≤cx ). This can be seen by using the idea of
fusion studied by [9, 10].
In fact P can be obtained from R by fusing 14 δ(−2,0) + 41 δ(0,−2) into 12 δ(−1,−1) , and
1
1
1
4 δ(2,0) + 4 δ(0,2) into 2 δ(1,1) .
Similarly Q can be obtained from R by fusing 14 δ(−2,0) + 14 δ(0,2) into 12 δ(−1,1) , and
1
1
1
4 δ(2,0) + 4 δ(0,−2) into 2 δ(1,−1) .
Consider now a measure
S=
5
3
3
3 .
δ 3 3 + δ 3
+ δ
11 ( 2 , 2 ) 11 ( 2 ,−4) 11 (−4, 2 )
Since any measure with support in the convex hull of
µ
¶ µ
¶ µ
¶
3 3
3
3
,
,
, −4 , −4,
2 2
2
2
and expectation (0, 0) is convexely dominated by S (see [10]), we have that S is an
upper bound for {P, Q}.
On the other hand R and S are not comparable on (M0 (Rd ), ≤cx ), since the
convex hulls of their supports are not ordered by inclusion (again see [10]).
If P ∨cx Q existed, then it would have to be dominated by both R and S, hence
its support would have to be contained in the intersection of the convex hulls of the
supports of R and S (indicated in grey in figure 1). Assume that this is possible.
Then, in order to dominate P the measure P ∨cx Q would have to deposit mass
1/2 on the segment B = ( 12 , 23 ), ( 32 , 12 ). In order to dominate Q it would have to
deposit mass 1/2 on the segment A = (−2, 0), (− 12 , 32 ) and mass 1/2 on the segment
C = (0, −2), ( 32 , − 12 ), see figure 1. Since the three segments are disjoint, this leads to
a contradiction.
Hence {P, Q} have no supremum in (M0 (Rd ), ≤cx ).
Proposition 4.6. The ordered set (M∗ (Rd ), ≤lcx ) is not a lattice.
In order to prove the above proposition we need the following definition and result,
for which the reader is referred to [23].
Definition 4.7. Given a probability measure P ∈ M∗ (Rd ), we define `(P ) its
lift-zonoid
½µ
¶
¾
Z
`(P ) = conv
P (B),
x P (dx) : B ∈ B(Rd ) .
B
Lemma 4.8 ([22]). For a ∈ Rd , and for P, Q ∈ Ma (Rd ), the following two
conditions are equivalent:
15
STOCHASTIC ORDER RELATIONS AND LATTICES
3
2
B
1
A
0
-4
-3
-2
-1
0
1
2
-1
C
-2
-3
-4
-5
Fig. 4.1. Graphical illustration to the Proof of Proposition 4.5.
(a) P ≤lcx Q,
(b) `(P ) ⊆ `(Q).
Lemma 4.9. The class of zonoids in Rd+1
having one common vertex in 0 and
+
another one in (1, µ), ordered by inclusion, is not a lattice.
Proof. Given two sets A, B ∈ Rd+1 , let A ⊕ B = {s + t : s ∈ A, t ∈ B} be their
Minkowski sum. Consider the following zonotopes in R4 :
Z1 = 0, a1 ⊕ 0, a2 ⊕ 0, a3 ,
Z2 = 0, b1 ⊕ 0, b2 ⊕ 0, b3 ,
Z3 = 0, c1 ⊕ 0, c2 ⊕ 0, c3 ,
Z4 = 0, d1 ⊕ 0, d2 ⊕ 0, d3 ,
where
¶
1 2 2 5
a1 =
, , ,
,
3 9 9 9
¶
µ
1 4 4 1
, , ,
,
b1 =
3 9 9 9
µ
¶
1 3 4 2
c1 =
, , ,
,
3 9 9 9
µ
¶
1 3 2 4
d1 =
, , ,
,
3 9 9 9
µ
µ
¶
1 2 5 2
a2 =
, , ,
,
3 9 9 9
µ
¶
1 4 1 4
b2 =
, , ,
,
3 9 9 9
µ
¶
1 2 3 4
c2 =
, , ,
,
3 9 9 9
µ
¶
1 4 3 2
d2 =
, , ,
,
3 9 9 9
µ
¶
1 5 2 2
a3 =
, , ,
,
3 9 9 9
µ
¶
1 1 4 4
b3 =
, , ,
,
3 9 9 9
µ
¶
1 4 2 3
c3 =
, , ,
,
3 9 9 9
µ
¶
1 2 4 3
d3 =
, , ,
.
3 9 9 9
16
A. MÜLLER AND M.SCARSINI
Fig. 4.2. Graphical illustration to the Proof of Lemma 4.9.
It is not difficult to verify that the zonotopes Z1 , Z2 , Z3 , Z4 have one vertex in 0
and the other in 1 := (1, 1, 1, 1). Let S2 be the simplex
S2 = {(x1 , x2 , x3 ) : x1 , x2 , x3 ≥ 0,
3
X
xj = 1}.
j=1
¡
¢
Then each of the above zonoids is generated by segments of the type 0, 13 , x1 , x2 , x3 ,
with (x1 , x2 , x3 ) ∈ S2 .
It is enough to look at the simplex S2 to notice that both Z3 and Z4 are included
in Z1 ∩ Z2 , so they are lower bounds for {Z1 , Z2 }. Therefore an infimum of {Z1 , Z2 }
would have to contain Z3 and Z4 . We observe that each of the six segments that
generate Z3 and Z4 is extreme for the convex hull of Z3 and Z4 , which coincides with
the intersection of Z1 and Z2 , see figure 2. Therefore any zonoid that includes both
Z3 and Z4 would have to contain all six generating segments among its generators,
but then it would not have a vertex in 1. This proves that the set {Z1 , Z2 } has no
infimum.
Proof. [Proof of Proposition 4.6] It is enough to combine Lemma 4.8 and Lemma 4.9.
Proposition 4.10. The ordered set (M(Rd ), ≤sm ) is not a lattice.
Proof. Let
1
(δ(2,1) + δ(1,3) + δ(3,2) ),
3
1
Q = (δ(1,2) + δ(3,1) + δ(2,3) ).
3
P =
17
STOCHASTIC ORDER RELATIONS AND LATTICES
The measures
1
(δ(1,1) + δ(3,2) + δ(2,3) ),
3
1
S = (δ(1,2) + δ(2,1) + δ(3,3) ).
3
R=
are upper bounds for {P, Q}. To prove this notice that for instance
Z
Z
1
f dR − f dP = (f (x ∨ y) + f (x ∧ y) − f (x) − f (y)) ,
3
for x = (2, 1), y = (1, 3). Similar results hold for the other cases. The definition of
supermodularity implies that R, S are upper bounds for {P, Q}.
The distribution function FP ∨sm Q would have to satisfy
FP , FQ ≤ FP ∨sm Q ≤ FR , FS ,
which implies
FP ∨sm Q (1, 1) = 0,
FP ∨sm Q (1, 2) =
1
,
3
FP ∨sm Q (2, 1) =
1
,
3
FP ∨sm Q (2, 2) =
1
.
3
This is not possible.
The argument in the above proof can be used to prove also the following proposition.
Proposition 4.11. The ordered sets (M(Rd ), ≤dcx ), (M(Rd ), ≤lo ) and
(M(Rd ), ≤uo ) are not lattices.
A different argument showing that (M(Rd ), ≤lo ) is not a lattice can be derived
by Example 2.1 in [27].
5. Properties of the order ≤↑ . The relation ≤↑ on BV(S) induces a partial
order ≤↑ on BV/∼ (S), the set of all equivalence classes F/∼ defined by the equivalence
relation
F ∼ G if G − F
is constant.
Then (BV/∼ (S), ≤↑ ) is a lattice (see e.g. section 8.6 in [1]). It is easy to see that this
can be extended to arbitrary measurable subsets S ⊂ R, denoting by BVloc (S) the
set of functions F : S → R that are of bounded variation on closed bounded subsets.
The subset BV+
loc/∼ (S), ≤↑ ) of all right-continuous functions of local bounded
variation forms a sublattice, which is strongly related with the set of all signed measures of local bounded variation endowed with the natural partial order. Indeed, let
S(S) be the set of all signed measures on (S, B(S)) of local bounded variation. Define
on it the order relation ¹ as follows:
µ¹ν
if µ(B) ≤ ν(B)
for all B ∈ B(S).
Then (S(S), ¹) is a lattice, where µ ∨ ν and µ ∧ ν are given as follows. Let ρ be a
dominating measure of µ and ν (e.g. take ρ = |µ| + |ν|), and denote by
fµ =
dµ
dρ
and
fν =
dν
dρ
18
A. MÜLLER AND M.SCARSINI
the corresponding Radon-Nikodym derivatives. Then µ ∨ ν and µ ∧ ν are the signed
measures with the Radon-Nikodym derivatives
d(µ ∨ ν)
= max{fµ , fν }
dρ
and
d(µ ∧ ν)
= min{fµ , fν }.
dρ
The mapping from (BV+
loc/∼ (S), ≤↑ ) to (S(S), ¹), assigning to (the equivalence class
of) a distribution function F the corresponding signed measure µF with
µF ((a, b]) = F (b) − F (a) for all a, b ∈ S, a < b,
is a lattice isomorphism, see [1], Theorem 9.61. There are two important special cases,
where this lattice isomorphism can be used to derive explicit formulas for F ∨↑ G and
F ∧↑ G. If S is an open set, and F and G are differentiable, then
(F ∨↑ G)0 (s) = max{F 0 (s), G0 (s)}
and
(F ∧↑ G)0 (s) = min{F 0 (s), G0 (s)},
s ∈ S.
(5.1)
If S = N0 , then
(F ∨↑ G)(s+1)−(F ∨↑ G)(s) = max{F (s+1)−F (s), G(s+1)−G(s)},
s ∈ N0 , (5.2)
and
(F ∧↑ G)(s+1)−(F ∧↑ G)(s) = min{F (s+1)−F (s), G(s+1)−G(s)},
s ∈ N0 . (5.3)
The following special cases of spaces of functions endowed with the order ≤↑ are
needed in Section 3. Let F log (R) be the set of all functions f : R → R ∪ {−∞}, which
are decreasing and right-continuous with limx→−∞ f (x) = 0, in other words F log (R)
is the set of logarithms of survival function of distributions with support R ∪ {+∞},
where we define log(0) = −∞, see Remark 5.
Lemma 5.1. The partially ordered set (F log (R), ≤↑ ) is a lattice.
Proof. For f, g ∈ F log (R) define
Sf,g = {x ∈ R : f (x) > −∞, g(x) > −∞}
=: (−∞, αf,g ).
Then (BV+
loc/∼ (Sf,g ), ≤↑ ) is a lattice as described above, and therefore f ∧↑ g(x) and
f ∨↑ g(x) are well-defined for x ∈ Sf,g . This can be extended to the whole of R by
defining
f ∧↑ g(x) := −∞,
for x ≥ αf,g
and


limt↑αf,g f ∨↑ g(t) + g(x) − g(αf,g ),
f ∨↑ g(x) := limt↑αf,g f ∨↑ g(t) + f (x) − f (αf,g ),


−∞,
for x ≥ αf,g , g(x) > −∞,
for x ≥ αf,g , f (x) > −∞,
for f (x) = g(x) = −∞.
As monotonicity and right-continuity are preserved under the lattice operations, it is
straightforward to see that (F log (R), ≤↑ ) becomes a lattice under these operations.
Now let Q be the set of all quantile functions, i.e. the set of all right-continuous
increasing functions f : (0, 1) → R. The proof of the following result is immediate.
Lemma 5.2. The partially ordered set (Q, ≤↑ ) is a lattice.
STOCHASTIC ORDER RELATIONS AND LATTICES
19
Acknowledgments. The idea of the paper originates from a question posed by
Nicolas Vieille during a talk of one of the authors. The authors thank Dirk Potthast
for drawing the figures, Wolfgang Weil and Markus Kiderlen for insightful discussions
on zonoids, and Gerd Herzog and Moshe Shaked for some useful references. Helpful
comments of two referees are gratefully acknowledged.
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