Affiliation:
1. Department of Mathematics, Carnegie Mellon University, Wean Hall, 5000 Forbes Ave , Pittsburgh , PA 15213 , USA
2. Department of Statistics, Columbia University, 1255 Amsterdam Avenue , New York , NY 10027 , USA
Abstract
Abstract
For probability measures
μ
,
ν
\mu ,\nu
, and
ρ
\rho
, define the cost functionals
C
(
μ
,
ρ
)
≔
sup
π
∈
Π
(
μ
,
ρ
)
∫
⟨
x
,
y
⟩
π
(
d
x
,
d
y
)
and
C
(
ν
,
ρ
)
≔
sup
π
∈
Π
(
ν
,
ρ
)
∫
⟨
x
,
y
⟩
π
(
d
x
,
d
y
)
,
C\left(\mu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\mu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y)\hspace{1.0em}{\rm{and}}\hspace{1em}C\left(\nu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\nu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y),
where
⟨
⋅
,
⋅
⟩
\langle \cdot ,\cdot \rangle
denotes the scalar product and
Π
(
⋅
,
⋅
)
\Pi \left(\cdot ,\cdot )
is the set of couplings. We show that two probability measures
μ
\mu
and
ν
\nu
on
R
d
{{\mathbb{R}}}^{d}
with finite first moments are in convex order (i.e.,
μ
≼
c
ν
\mu {\preccurlyeq }_{c}\nu
) iff
C
(
μ
,
ρ
)
≤
C
(
ν
,
ρ
)
C\left(\mu ,\rho )\le C\left(\nu ,\rho )
holds for all probability measures
ρ
\rho
on
R
d
{{\mathbb{R}}}^{d}
with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of
∫
f
d
ν
−
∫
f
d
μ
\int f{\rm{d}}\nu -\int f{\rm{d}}\mu
over all 1-Lipschitz functions
f
f
, which is obtained through optimal transport (OT) duality and the characterization result of OT (couplings) by Rüschendorf, by Rachev, and by Brenier. Building on this result, we derive new proofs of well known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.
Subject
Applied Mathematics,Modeling and Simulation,Statistics and Probability