MATHEMATICS OF OPERATIONS RESEARCH
Vol. 26, No. 4, November 2001, pp. 723–740
Printed in U.S.A.
STOCHASTIC COMPARISON OF
RANDOM VECTORS WITH A COMMON COPULA
ALFRED MÜLLER and MARCO SCARSINI
We consider two random vectors X and Y, such that the components of X are dominated in the
convex order by the corresponding components of Y. We want to ﬁnd conditions under which this
implies that any positive linear combination of the components of X is dominated in the convex order
by the same positive linear combination of the components of Y. This problem has a motivation in
the comparison of portfolios in terms of risk. The conditions for the above dominance will concern
the dependence structure of the two random vectors X and Y, namely, the two random vectors will
have a common copula and will be conditionally increasing. This new concept of dependence is
strictly related to the idea of conditionally increasing in sequence, but, in addition, it is invariant
under permutation. We will actually prove that, under the above conditions, X will be dominated
by Y in the directionally convex order, which yields as a corollary the dominance for positive linear
combinations. This result will be applied to a portfolio optimization problem.
1. Introduction. Given two random vectors X and Y of, say, the same dimension d,it
is often useful to compare a positive linear combination of their components with respect
to some ordering. This is, for instance, the case when each vector represents a portfolio of
returns on d investments, and the weights represent the relative prices of the securities in
the portfolio. The study of orderings for linear combinations of random vectors appears for
instance in Muliere and Scarsini (1989) and Scarsini and Shaked (1990) with reference to
the usual stochastic ordering, and in Arnold (1987) and Koshevoy and Mosler (1996, 1998)
with reference to the Lorenz ordering.
The following problem is of interest: Under which conditions does the existence of some
marginal orderings among the components of the vectors imply an ordering among a positive
linear combination of the components? This means that only the marginals shall be ordered,
whereas the dependence structure (i.e., the copula) shall be ﬁxed.
If the order that we consider is the usual stochastic order, the solution to the problem
has been given by Scarsini (1988), who proved the following result: If X and Y have
a common copula, then the stochastic order among the marginals implies the stochastic
order among the vectors (which in turn obviously implies the stochastic order among the
positive combinations). Scarsini (1998) proved that the result is false if the stochastic order is
replaced by the convex order. Therefore, to solve our problem for the convex order, we need
to consider some variation on the idea of convexity. Several extensions of convexity (and
its corresponding orders) from the univariate to the multivariate case have been proposed.
In particular, the so-called directional convexity has proved useful in several applications in
applied probability, see, e.g., Shaked and Shanthikumar (1990), Meester and Shanthikumar
(1993, 1999), and Bäuerle and Rolski (1998).
The usual deﬁnition of convexity does not take into account a possible order structure
on the space, whereas directional convexity does. Therefore this property seems to be
Received April 14, 1999; revised March 8, 2000, and March 15, 2001.
MSC 2000 subject classiﬁcation. Primary: 60E15; secondary: 60E05, 62H20.
OR/MS subject classiﬁcation. Primary: Probability.
Key words. Directionally convex order, local mean preserving spread, copula, conditionally increasing random
vectors, convex ordering, portfolio optimization.
1526-5471 electronic ISSN, © 2001, INFORMS