Explicit solutions for a
general class of optimal
allocation problems
Ka Chun Cheung, Jan Dhaene, Yian Rong, S.C.P. Yam
AFI_1378
Explicit solutions for a general class of optimal
allocation problems
Ka Chun Cheung∗,? , Jan Dhaene† , Yian Rong? , and S.C.P. Yam‡
? Department
of Statistics and Actuarial Science, The University of Hong Kong,
Pokfulam Road, Hong Kong.
† Actuarial Research Group, AFI Research Unit, Faculty of Business and Economics, K.U.Leuven,
Naamsestraat 69, B-3000 Leuven, Belgium.
‡ Department of Statistics, The Chinese University of Hong Kong,
Shatin, Hong Kong.
Abstract
We revisit the problem of minimizing a separable convex function with a linear
constraint and box constraints. This optimization problem arises naturally in many
applications in economics, insurance, and finance. Existing literature exclusively tackles
this problem by using the traditional Kuhn-Tucker theory, which leads to either iterative
schemes or yields explicit solutions only under some special classes of convex functions.
Instead, we present a new approach of solving this constrained minimization problem
explicitly by using the theory of comonotonicity. The key step is to apply an integral
representation result to express each convex function as the expected stop-loss of some
suitable random variable.
Keywords: Optimal allocation; Comonotonicity; Convex function; Stop-loss
1
Introduction and problem formulation
Denote by (R1 , . . . , Rn ) the portfolio of risks we are facing. A provision of d dollars is available
to be allocated among these n risks. We use the function fi (di ) to model the risk level (or
required reserve) of the risk Ri when di dollars is allocated to Ri . When more capital is
allocated to risk i, the position is considered to be safer, and hence the corresponding risk
level or required reserve is less. This means that fi should be a decreasing function. It is also
natural to assume that the decrement is diminishing per unit of growth. Accordingly, fi is
both decreasing and convex.
Alternatively, instead of treating fi as a reserve or risk function, we can think of it as a penalty
function in the sense that allocated capital is expected to be as close as possible to the loss
∗
Corresponding Author. Email: [email protected]
1
being allocated to, and that fi penalizes the deviation. Common examples include quadratic
deviations and absolute deviations. With this interpretation, it is natural to assume that fi
is convex, but not necessarily decreasing.
All the considerations above lead us to study the minimization of the total required reserve
or total penalty by determining the optimal amount d∗i needed for each risk Ri :
n
X
min
d1 +···+dn =d,0≤di ≤d
fi (di ).
i=1
To allow for more flexible constraints on the individual allocations, we consider the following
more general problem instead:
n
X
min
fi (di ),
(1)
(d1 ,...,dn )∈A(d)
i=1
where
A(d) = {(d1 , . . . , dn ) ∈ Rn | d1 + · · · + dn = d, di ∈ [li , ui ], i = 1, . . . , n}
is the set of admissible allocations, in which li , ui are given fixed constants such that li < ui
and
l1 + · · · + ln < d < u1 + · · · + un .
This condition guarantees that A(d) is non-empty nor a singleton. We require that each fi
is convex and continuous on [li , ui ], but is not necessarily decreasing. For simplicity, we also
assume that (fi )0+ (li ) and (fi )0− (ui ) are finite for all i, where f+0 and f−0 denote the right-hand
and left-hand derivative of any convex function f respectively throughout this paper. Since
A(d) is compact and the objective function is continuous, an optimal solution always exists.
The linear constraint d1 + · · · + dn = d can easily be extended to a more general linear
constraint of the form
c1 d1 + · · · + cn dn = d
(2)
where c1 , . . . , cn are some fixed constants. We may assume that all of them are non-zero, for
if ci = 0 for some i, we can minimize fi (di ) over di ∈ [li , ui ] separately as a one-dimensional
problem. With this new constraint, we have the following more general minimization problem:
n
X
min
c1 d1 +···+cn dn =d,li ≤di ≤ui
fi (di ).
(3)
i=1
Assume that the admissible set is non-empty. Let d̃i := ci di and f̃i (x) := fi (x/ci ) for all i,
then problem (3) becomes
n
X
min
f̃i (d̃i ),
d˜1 +···+d˜n =d,d˜i ∈[l˜i ,u˜i ]
i=1
where [l˜i , ũi ] = [cli , cui ] if ci > 0, and [l˜i , ũi ] = [cui , cli ] if ci < 0. With this transformation,
problem (3) can be treated as a special case of problem (1).
Capital allocation rules in the form of (1) or (3) are fairly general. It covers the various
optimization-based models proposed and studied in Dhaene et al. (2012). These problems
also arise naturally in many optimization models in economics, operation management, finance, marketing, etc. We refer to Bitran and Hax (1981), Luss and Gupta (1975), Stefanov
2
(2005) and the references therein, for various applications and discussions. Existing solution
methods include standard convex programming (Kuhn-Tucker theory) (for instance, Charnes
and Cooper (1958)), dynamic programming (for instance, Wilkinson and Gupta (1969)) and
the iterative method (e.g. Luss and Gupta (1975), Stefanov (2005)). The classical KuhnTucker theory lies at the heart of all these different approaches. Similar allocation problems
for future no-yet-realized risk or payoff, rather than for the current deterministic capital in
the current context, has also gained considerable attention in recent years, see Acciaio (2007),
Barrieu and El Karoui (2005), Filipović and Svindland (2008), Jouini et al. (2008), Kiesel
and Rüschendorf (2008, 2010), and Ludkovski and Rüschendorf (2008).
If the box constraints di ∈ [li , ui ] are removed, and each fi is defined on the whole real line,
problem (1) is just the classical “inf-convolution” of the convex functions f1 , . . . , fn in convex
analysis, which is well-studied in the literature (see, for instance, Rockafellar (1970)). The
introduction of the box constraints di ∈ [li , ui ] makes the problem more difficult. This paper
presents a completely new method to solve problem (1). Instead of using any Lagrangiantype (Kuhn-Tucker theory) technique, we first express each convex function fi as a stop-loss
transform so that the objective function becomes a sum of expected stop-losses. The box
constraint can be effectively captured by the choice of the random variables. Problem (1)
then becomes the minimization of a sum of expected stop-losses subject to a (homogeneous)
linear constraint, which can be solved effectively and explicitly by using the theory of comonotonicity.
2
Supports of comonotonic random vectors
A set A ⊂ Rn is said to be comonotonic if any two points in A can be ordered componentwise,
that is, for any (x1 , . . . , xn ) and (y1 , . . . , yn ) in A, (xi −yi )(xj −yj ) ≥ 0 for any i, j ∈ {1, . . . , n}.
A random vector (X1 , . . . , Xn ) is said to be comonotonic if there is some comonotonic set
A ⊂ Rn so that P((X1 , . . . , Xn ) ∈ A) = 1. Comonotonicity of (X1 , . . . , Xn ) is equivalent to
d
that (X1 , . . . , Xn ) = (FX−11 (U ), . . . , FX−1n (U )) for any uniform(0, 1) random variable U . In this
paper, for any given random vector X = (X1 , . . . , Xn ), we define its comonotonic modification
Xc = (X1c , . . . , Xnc ) as
Xc := (FX−11 (U ), . . . , FX−1n (U )),
where U is an arbitrary uniform(0, 1) random variable. By construction, a comonotonic
modification is always comonotonic and has the same marginal distributions as the original
random vector. For a comprehensive overview of the theory on comonotonicity, we refer to
Dhaene et al. (2002).
Throughout this paper, FX−1 denotes the left-continuous inverse of the distribution function
FX of any random variable X:
FX−1 (p) := inf{x ∈ R | FX (x) ≥ p},
0 ≤ p ≤ 1.
Similarly, the right-continuous inverse distribution function is defined as
FX−1+ (p) := sup{x ∈ R | FX (x) ≤ p},
0 ≤ p ≤ 1.
With the convention that inf ∅ = +∞ and sup ∅ = −∞, FX−1+ (0) and FX−1 (1) are the essential infimum and essential supremum of X respectively. By definition, FX−1 (0) = −∞ and
3
FX−1+ (1) = ∞ regardless of the actual distribution of X. For our later purpose, we also need
the notion of α-mixed inverse distribution function. Following Kaas et al. (2000), it is defined
as
−1(α)
FX
(p) := αFX−1 (p) + (1 − α)FX−1+ (p), 0 ≤ p ≤ 1, 0 ≤ α ≤ 1.
−1(0)
For consistency, we also adopt the convention 0 · ±∞ = 0 so that FX
−1(1)
FX (p) = FX−1 (p) for any 0 ≤ p ≤ 1.
(p) = FX−1+ (p) and
For the rest of this section, Xc = (X1c , . . . , Xnc ) denotes a fixed comonotonic random vector
with marginal distribution functions F1 , . . . , Fn , and S c is the comonotonic sum X1c +· · ·+Xnc .
It is a fundamental property about comonotonicity (see for instance Denneberg (1994) and
Dhaene et al. (2002)) that the inverse distribution function of S c can be computed explicitly
as follows:
−1(α)
FS c
−1(α)
(p) = F1
(p) + · · · + Fn−1(α) (p),
0 ≤ p ≤ 1, 0 ≤ α ≤ 1.
(4)
By definition,
msupp(Xc ) := (F1−1 (u), . . . , Fn−1 (u)) | 0 < u < 1
is a comonotonic set in Rn and is a support1 of Xc . We say that s ∈ Rn is a comonotonic
support point of Xc , if {s} ∪ msupp(Xc ) is again comonotonic. In other words, adding a
comonotonic support point to msupp(Xc ) will not destroy its comonotonicity. The collection
of all comonotonic support points of Xc will be denoted as csupp(Xc ):
csupp(Xc ) := s ∈ Rn | {s} ∪ msupp(Xc ) is comonotonic .
It is clear that msupp(Xc ) ⊂ csupp(Xc ), and that msupp(Xc ) ∪ {s} is also a comonotonic
support of Xc for every s ∈ csupp(Xc ). However, it is possible that Xc does not take on a
value at a comonotonic support point, and csupp(Xc ) may not be comonotonic.
For any d ∈ R, we denote by `(d) the hyperplane {d ∈ Rn | d1 + · · · + dn = d}. For reasons
that will become clear later, we are interested in finding the intersection `(d) ∩ csupp(Xc ),
which will be denoted as i(d, Xc ):
i(d, Xc ) := s ∈ Rn | {s} ∪ msupp(Xc ) is comonotonic and s1 + · · · + sn = d .
c
First, we remark that if d < FS−1+
(0) or d > FS−1
c
c (1), the cardinality of i(d, X ) is infinity.
−1+
−1+
−1+
For if d < FS c (0) = F1 (0) + · · · + Fn (0), it is obvious that there are infinitely many
possible ways to decompose d into a sum d1 + · · · + dn such that di ≤ Fi−1+ (0) for each i.
Given any such a decomposition, (d1 , . . . , dn ) ∈ i(d, Xc ), and hence the cardinality of i(d, Xc )
is infinity. The case for d > FS−1
c (1) is similar.
The following result characterizes the set i(d, Xc ) for the more interesting and relevant case
where FS−1+
(0) ≤ d ≤ FS−1
c
c (1).
(0) ≤ d ≤ FS−1
Proposition 2.1. Suppose that d is a real number such that FS−1+
c
c (1). Then
i(d, Xc ) = (d1 , . . . , dn ) ∈ `(d) | Fi−1 (FS c (d)) ≤ di ≤ Fi−1+ (FS c (d)) for all i .
(5)
1
By a support of a random variable or a random vector Y , we mean any Borel measurable set A such that
P(Y ∈ A) = 1.
4
P
P
Proof: We first assume that ni=1 Fi−1+ (0) < d < ni=1 Fi−1 (1). In this case, 0 < FS c (d) < 1.
Suppose d = (d1 , . . . , dn ) belongs to the set on the right hand side of (5). Then for any
i = 1, . . . , n, we have di ≥ Fi−1 (p) when p ∈ (0, FS (d)] and di ≤ Fi−1 (p) when p ∈ (FS c (d), 1),
so d ∪ msupp(Xc ) is comonotonic. Therefore, d lies in csupp(Xc ) and hence in i(d, Xc ).
Now we suppose that d = (d1 , . . . , dn ) ∈ i(d, Xc ). Notice that
n
X
−1+
Fi−1 (FS c (d)) = FS−1
(FS c (d)) =
c (FS c (d)) ≤ d ≤ FS c
i=1
n
X
Fi−1+ (FS c (d)).
(6)
i=1
Since d ∈ csupp(Xc ), either Fi−1 (FS c (d)) ≤ di for all i or Fi−1 (FS c (d)) > di for all i. The
second possibility is ruled out by the second inequality in (6) and the condition that d ∈ `(d),
unless di = Fi−1 (FS c (d)) for all i. Therefore, we must have Fi−1 (FS c (d)) ≤ di for all i. By the
same argument, we have di ≤ Fi−1+ (FS c (d)) for all i. This proves the reverse inclusion.
P
If d = FS−1+
(0) = ni=1 Fi−1+ (0) ∈ R, then it is obvious that the only way to decompose d
c
into aPsum d = d1 + · · · + dn in such a way that d ∪ msupp(Xc ) is comonotonic is given by
d = ni=1 Fi−1+ (0). In this case, i(d, Xc ) = {(F1−1+ (0), . . . , Fn−1+ (0))}. If FS c (d) = 0, the
right hand side of (5) becomes
(d1 , . . . , dn ) ∈ Rn | d1 + · · · + dn = d, −∞ ≤ di ≤ Fi−1+ (0) for all i ,
which contains (F1−1+ (0), . . . , Fn−1+ (0)) only; if FS c (d) > 0, the right hand side of (5) becomes
(d1 , . . . , dn ) ∈ Rn | d1 + · · · + dn = d, Fi−1+ (0) ≤ di ≤ Fi−1+ (FS c (d)) for all i ,
again, this set contains (F1−1+ (0), . . . , Fn−1+ (0)) only.
Pn
−1
into a sum d =
Finally, if d = FS−1
c (1) =
i=1 Fi (1) ∈ R, the only way to decompose d P
d1 +· · ·+dn in such a way that d∪msupp(Xc ) is comonotonic is given by d = ni=1 Fi−1 (1), so
that i(d, Xc ) = {(F1−1 (1), . . . , Fn−1 (1))}. Since FS c (d) = 1, the right hand side of (5) becomes
(d1 , . . . , dn ) ∈ Rn | d1 + · · · + dn = d, Fi−1 (1) ≤ di ≤ ∞ for all i ,
which contains (F1−1 (1), . . . , Fn−1 (1)) only.
To study the cardinality i(d, Xc ), we need to introduce the following subset of R:
(
)
n
n
X
X
s(Xc ) := d ∈ R d =
Fi−1 (p), p ∈ (0, 1], or d =
Fi−1+ (p), p ∈ [0, 1) .
i=1
i=1
Corollary 2.2. Suppose that d is a real number such that FS−1+
(0) ≤ d ≤ FS−1
If
c
c (1).
c
c
c
d ∈ s(X ), then card(i(d, X )) = 1; otherwise, if d 6∈ s(X ),
(a) card(i(d, Xc )) = ∞ if there are more than one of Fi−1 , i = 1, . . . , n, jump at FS c (d);
(b) card(i(d, Xc )) = 1 if exactly one of Fi−1 , i = 1, . . . , n, jumps at FS c (d).
P
Proof: For the first assertion, consider d ∈ s(Xc ), andPsuppose that d = ni=1 Fi−1 (p) for
some p ∈ (0, 1] (the argument for the case where d = ni=1 Fi−1+ (p) for some p ∈ [0, 1) is
similar). Obviously, (F1−1 (p), . . . , Fn−1 (p)) ∈ i(d, Xc ). If s = (s1 , . . . , sn ) is a different point in
5
i(d, Xc ), then by the definition of comonotonicity, either Fi−1 (p) ≤ si for all i, or Fi−1 (p) ≥ si
for all i, with the inequality
strict P
for at least one i in both possibilities. At the same
Pn being
−1
time, it is required that i=1 Fi (p) = ni=1 si = d. Clearly, such a point s does not exist,
so i(d, Xc ) is a singleton.
For the second assertion, note that d 6∈ s(Xc ) implies that FS−1+
(0) < d < FS−1
c
c (1), or equivalently 0 < FS c (d) < 1. In this case, at least one of the Fi−1 , i = 1, . . . , n, jumps at FS c (d).
For if not, Fi−1 (FS c (d)) = Fi−1+ (FS c (d)) for all i, summing over i yields FS−1
c (FS c (d)) =
−1+
−1
−1
−1
c
c
c
c
FS c (FS (d)) and hence d = FS c (FS (d)) = F1 (FS (d)) + · · · + Fn (FS (d)), which contradicts the assumption that d 6∈ s(Xc ). Now the result follows from (5).
−1(α)
∗
Corollary 2.3. Suppose that FS−1+
(0) < d < FS−1
(FS c (d)) for
c
c (1). If we define di = Fi
−1(α)
∗
∗
i = 1, . . . , n, where α ∈ [0, 1] is a solution of FS c (FS c (d)) = d, then (d1 , . . . , dn ) ∈ i(d, Xc ).
Proof: This corollary follows from (5) and the fact that Fi−1 (FS c (d)) ≤ d∗i ≤ Fi−1+ (FS c (d))
for all i.
3
A canonical optimal capital allocation problem
Before solving problem (1), we first consider a special case in which fi (di ) takes the form of
E[(Xi − di )+ ] for a given integrable random variable Xi :
min
n
X
d1 +···+dn =d
E[(Xi − di )+ ].
(7)
i=1
Notice that d1 + · · · + dn = d is the only constraint in problem (7). We do not impose any
box constraint to restrict the individual allocations.
A nice feature of problem (7) is that its optimal solutions admit an elegant closed-form
expression. Using
Dhaene et al. (2002) and Kaas et al. (2002)
Pthe notion of comonotonicity,
Pn
−1
proved that for ni=1 FX−1+
(0)
<
d
<
F
(1),
a solution to problem (7) is given by
i=1 Xi
i
−1(α)
d∗i = FXi
(FS c (d)),
i = 1, . . . , n,
(8)
−1(α)
in which α ∈ [0, 1] is a solution of the equation FS c (FS c (d)) = d, and S c := X1c + · · · + Xnc
where Xc = (X1c , . . . , Xnc ) is a comonotonic modification of (X1 , . . . , Xn ).
By adapting the geometric argument proposed in Kaas et al. (2002), we can indeed give a full
characterization of the solution set of problem (7) for any d ∈ R, which contains (d∗1 , . . . , d∗n )
in (8) as a special case.
Theorem 3.1. For any d ∈ R, the solution set of problem (7) equals i(d, Xc ).
Proof: Let U be a uniform(0, 1) random variable, Xc := (FX−11 (U ), . . . , FX−1n (U )) be a comonotonic modification of (X1 , . . . , Xn ) and define S c := FX−11 (U ) + · · · + FX−1n (U ). First, we may
replace each Xi in problem (7) by FX−1i (U ) because the actual dependence among the risks is
6
irrelevant. By the subadditivity of the positive part function, for any d1 + · · · + dn = d, we
have
" n
! #
n
n
X
X
X
E[(FX−1i (U ) − di )+ ] ≥ E
FX−1i (U ) −
di
= E[(S c − d)+ ],
i=1
i=1
i=1
+
c
so E[(S − d)+ ] is a lower bound of the objective function in (7).
Next, we show that every d = (d1 , . . . , dn ) ∈ i(d, Xc ) is a solution of problem (7). By
definition, {d} ∪ msupp(Xc ) is comonotonic, so either (FX−1i (U ) − di )+ = FX−1i (U ) − di for
all i simultaneously, or (FX−1i (U ) − di )+ = 0 for all i simultaneously. In either case, we have
P −1
(FXi (U )−di )+ = (S c −d)+ , which means that the objective function in problem (7) attains
its lower bound at d.
On the other hand, suppose that d ∈ `(d) but d 6∈ csupp(Xc ). Then there exist some
i, j ∈ {1, . . . , n} and u ∈ (0, 1) such that (FX−1i (u) − di )(FX−1j (u) − dj ) < 0. By the left
continuity P
of F −1 , this strict inequality continues to hold on [u − ε, u] for some ε > 0.
Therefore, (FX−1i (U ) − di )+ > (S c − d)+ with a strictly positive probability, and so d is not
optimal.
We remark that similar results can be found in Chen et al. (2012), in which the authors discuss
the issue of (non)-uniqueness of the optimal solution of problem (7). Without recourse to
the notion of i(d, Xc ), they show directly that the set of all optimal solutions is given by the
set on the right-hand side of (5).
The following result is a direct consequence of Theorem 3.1, Corollary 2.2 and Corollary 2.3.
Corollary 3.2. If d ∈ s(Xc ), then problem (7) admits a unique solution, and if FS−1+
(0) <
c
−1+
−1
∗
∗
d < FS c (1), then any optimal solution (d1 , . . . , dn ) of problem (7) satisfies FXi (0) ≤ d∗i ≤
FX−1i (1).
4
Representing convex functions as stop-loss transforms
In this section, we show that problem (1) is indeed equivalent to problem (7) for some suitable
random variables X1 , . . . , Xn , and hence the solution set of problem (1) equals i(d, Xc ) by
Theorem 3.1.
Recall that each fi in problem (1) is assumed to be convex and continuous on [li , ui ], with
finite right-hand derivative at li and finite left-hand derivative at ui .
Proposition 4.1. Let g : [l, u] → R be a continuous and convex function with
0
0
(l) ≤ g−
(u) ≤ 1.
−1 ≤ g+
(9)
Then there exists a random variable X with P(l ≤ X ≤ u) = 1 such that
g(x) = g(l) + x + l + 2E[(X − x)+ ] − 2E(X) for any x ∈ [l, u].
Moreover, the distribution function of X is given by


if
0
0
FX (x) = (1 + g+ (x))/2 if


1
if
7
x < l,
l ≤ x < u,
x ≥ u.
(10)
0
(x) is right-continuous and increasing on [l, u). The condition
Proof: By convexity, g+
0
0
g+ (l) ≥ −1 and g− (u) ≤ 1 ensures that the expression in (10) is a genuine distribution
function. We denote by X an arbitrary random variable with such distribution function. It
is clear that P(l ≤ X ≤ u) = 1, and there is a possible jump at l and at u. For any x ∈ [l, u],
we have
x + 2E[(X − x)+ ] − 2E(X)
= E |X − x| − E(X)
Z u−l
Z
Z x−l
FX−l (t) dt +
(1 − FX−l (t)) dt −
=
0
Z
x−l
u−l
(1 − FX−l (t)) dt − l
0
x−l
(2FX−l (t) − 1) dt − l
=
0
Z
=
x−l
0
g+
(t + l) dt − l
0
= g(x) − g(l) − l.
Rearranging this equation yields the desired result.
By the same argument, we have the following variant of Proposition 4.1 in which the upper
end point u of the domain of g is infinity:
Proposition 4.2. Let g : [l, ∞) → R be a decreasing convex function with
0
0
−1 ≤ g+
(l) ≤ lim g+
(x) = 0.
x→∞
Let L := limx→∞ g(x) ∈ R. Then there exists a random variable with P(x ≥ l) = 1 such that
g(x) = L + E[(X − x)+ ] for any x ≥ l.
Moreover, the distribution function of X is given by
(
0
if
FX (x) =
0
1 + g+
(x) if
x < l,
x ≥ l.
A common requirement in these two propositions is that the convex function concerned has
bounded derivative on the relevant domain. Such requirement enables us to rescale linearly
the right-hand derivative into a distribution function of a random variable. Our methodology
indeed remains valid even if the derivative is not bounded. In that case, instead of rescaling
the right-hand derivative into a distribution function of a random variable, we can directly
treat the right-hand derivative as the distribution function of a Radon measure on R, see
page 16 of Föllmer and Schied (2004) and page 545 of Revuz and Yor (1999). Comonotonicity
of measurable maps on a general measurable space can be defined in exactly the same way
as it is defined for random variables on a probability space, Property (4) on the additivity
of the inverse distribution functions of comonotonic sums is also valid. We choose not to
pursue such generality in order to put the focus on the ideas and techniques rather than on
technicalities. Interested readers can easily work out the details for the general case.
8
5
Solution set of minimization problem (1)
Now we return to our optimal capital allocation problem (1):
min
d∈A(d)
n
X
fi (di ),
i=1
where the functions fi are convex and continuous on [li , ui ] with −∞ < (fi )0+ (li ) ≤ (fi )0− (ui ) <
∞, and
A(d) = {(d1 , . . . , dn ) ∈ Rn | d1 + · · · + dn = d, di ∈ [li , ui ], i = 1, . . . , n}
is the set of admissible allocations with l1 + · · · + ln < d < u1 + · · · + un . Since condition (9)
of Proposition 4.1 may not be satisfied by fi , a simple rescaling is needed. To this end, take
ν to be any number that is strictly larger than ν ∗ , which is define by
ν ∗ := max |(fi )0+ (li )| ∨ |(fi )0− (ui )| ∈ R,
(11)
1≤i≤n
and define functions
f̃i (x) := fi (x)/ν
for x ∈ [li , ui ] and i = 1, . . . , n.
(12)
The functions f̃i constructed in this way satisfy all conditions of Proposition 4.1, and hence
there exist random variables X1 , . . . , Xn such that for i = 1, . . . , n,
f̃i (x) = f̃i (li ) + x + li + 2E[(Xi − x)+ ] − 2E(Xi ),
x ∈ [li , ui ].
From (12) and Proposition 4.1, the distribution function of Xi is given by


if x < li ,
0
0
FXi (x) = (1 + (f̃i )+ (x)/ν)/2 if li ≤ x < ui ,


1
if x ≥ ui .
(13)
Moreover, as ν is chosen to be larger than ν ∗ , each Xi has a point mass at both of its essential
infimum li and essential supremum ui . With the above transformation, we have
n
X
fi (di ) = C + 2ν
i=1
n
X
E[(Xi − di )+ ]
i=1
for any (d1 , . . . , dn ) ∈ A(d), where C is some constant which is independent of (d1 , . . . , dn ).
Therefore, problem (1) is equivalent to the following problem:
min
d∈A(d)
n
X
E[(Xi − di )+ ],
(14)
i=1
in the sense that the two problems have the same optimal solution sets.
One can immediately notice the similarity between problem (7) and problem (14). The only
difference between them is that problem (14) requires that di ∈ [li , ui ] for all i while problem
(7) does not. However, from Corollary 3.2 and (13) , it is known that any optimal solution
(d∗1 , . . . , d∗n ) of problem (7) satisfies d∗i ∈ [FX−1+
(0), FX−1i (1)] = [li , ui ], so the additional box
i
constraint on the individual allocations is automatically fulfilled. Combining with Theorem
3.1, we have the following result:
9
Theorem 5.1. The solution set of problem (1) equals i(d, Xc ), where the marginal distribution of Xi is given by (13). Moreover, if d ∈ s(Xc ), the problem (1) admits a unique
solution.
In the remainder of this section, S c denotes the comonotonic sum X1c + · · · + Xnc , where
(X1c , . . . , Xnc ) is a comonotonic modificaton of (X1 , . . . , Xn ) with marginal distributions given
by (13).
Corollary 5.2. The solution set to problem (1) is given by
c (d)) for all i .
(d1 , . . . , dn ) | d1 + · · · + dn = d, FX−1i (FS c (d)) ≤ di ≤ FX−1+
(F
S
i
Moreover, 0 < FS c (d) < 1.
Proof: Since ν is chosen to be strictly larger ν ∗ defined in (12), FX−1+
(0) = li and FX−1i (1) =
i
P
P
−1
ui for each i and so FS−1+
(0) =
c
i li < d <
i ui = FS c (1) = nd. This implies that
0 < FS c (d) < 1. Now the result follows from Theorem 5.1 and Proposition 2.1.
Corollary 5.3. Problem (1) has a unique solution given by (FX−11 (FS c (d)), . . . , FX−1n (FS c (d)))
if each fi is strictly convex
Although this corollary is a standard result in the theory of convex minimization, here we
will give a new and simple proof by using the theory of comonotonicity.
Proof: When each fi is strictly convex, each FXi defined in (13) is strictly increasing on
[li , ui ], and hence FX−1i does not contain any discontinuity. In particular, this implies that
FX−1i (FS c (d)) = FX−1+
(FS c (d)) and so by Corollary 5.2, problem (1) has a unique solution
i
−1
given by (FX1 (FS (d)), . . . , FX−1n (FS (d))).
The next result can be found in Bitran and Hax (1981). Instead of proving it using KuhnTucker theory, we demonstrate that it is a direct consequence of Corollary 5.2.
Corollary 5.4. Suppose that fi is strictly increasing on [li , ui ] for i ∈ JI ⊂ {1, . . . , n} and is
strictly decreasing on [li , ui ] for i ∈ JD ⊂ {1, . . . , n}. If (d∗1 , . . . , d∗n ) is an optimal solution of
problem (1), then either d∗i = li for all i ∈ JI , or d∗i = ui for all i ∈ JD , or both.
Proof: If fi is strictly increasing on [li , ui ], the corresponding FXi in (13) jumps at li from 0
(p) = li on an interval containing KI ⊃ (0, 1/2].
to FXi (li ) > 1/2, and hence FX−1i (p) = FX−1+
i
Similarly, if fi is strictly decreasing on [li , ui ], the corresponding FXi in (13) jumps at ui
from FXi (ui −) < 1/2 to 1, and hence FX−1i (p) = FX−1+
(p) = ui on an interval containing
i
KD ⊃ [1/2, 1). From Corollary 5.2, if FS c (d) ∈ KI , then d∗i = li for all i ∈ JI ; if FS c (d) ∈ KD ,
then d∗i = ui for all i ∈ JD . As KI ∪ KD = (0, 1), the result follows.
6
Connection with infimum-convolution
If the box constraints di ∈ [li , ui ] for all i in problem (1) are removed (but d1 + · · · + dn = d
is kept), and the domain of each real-valued convex function fi is R instead of [li , ui ], then
(∧ni=1 fi )(d) :=
min
d1 +···+dn =d
10
n
X
i=1
fi (di )
(15)
is called the inf-convolution of f1 , . . . , fn in convex analysis (cf. Rockafellar (1970)). Notice
that the minimization in (15) may not have a solution in general. The next well-known result
gives necessary and sufficient conditions for the optimality of d = (d1 , . . . , dn ).
Proposition 6.1. Consider some (d∗1 , . . . , d∗n ) with d∗1 + · · · + d∗n = d. Then
(∧ni=1 fi )(d) =
n
X
fi (d∗i )
i=1
if and only if
∂f1 (d∗1 ) ∩ · · · ∩ ∂fn (d∗n ) 6= ∅.
(16)
Here, ∂fi (di ) := [(fi )0− (di ), (fi )0+ (di )] is the subdifferential of fi at di . The objective here is
to prove this result using the perspective of comonotonicity and the theory we developed in
previous sections.
Proof: We first prove the “if” part. Fix some (d∗1 , . . . , d∗n ) with d∗1 + · · · + d∗n = d and some
u∗ such that
u∗ ∈ ∂f1 (d∗1 ) ∩ · · · ∩ ∂fn (d∗n ).
Choose any u1 , . . . , un , l1 , . . . , ln such that li < d∗i < ui for all i, and let Xc = (X1c , . . . , Xnc )
be a comonotonic random vector with marginal distributions given by (13) for some large
enough ν. By (13), (fi )0− (d∗i ) = ν(2FXi (d∗i −) − 1) and (fi )0+ (d∗i ) = ν(2FXi (d∗i ) − 1), and so
FXi (d∗i −) ≤ u∗∗ ≤ FXi (d∗i ) for all i,
where u∗∗ := (u∗ /ν + 1)/2. This can be rewritten as
FX−1i (u∗∗ ) ≤ di ≤ FX−1+
(u∗∗ ) for all i.
i
In particular, this implies that (d∗1 , . . . , d∗n ) ∈ i(d, Xc ). By Theorem 5.1, (d∗1 , . . . , d∗n ) solves
the problem
n
X
min
fi (di ).
d1 +···+dn =d,li ≤di ≤ui
i=1
However, as the li ’s and the ui ’s can be chosen arbitrarily small and large respectively, we
conclude that
n
n
X
X
min
fi (di ) =
fi (d∗i ).
d1 +···+dn =d
i=1
i=1
This proves the“if” part.
P
For the “only if” part, suppose that d∗1 + · · · + d∗n = d and (∧ni=1 fi )(d) = ni=1 fi (d∗i ). Choose
any u1 , . . . , un , l1 , . . . , ln such that li < d∗i < ui for all i. Then (d∗1 , . . . , d∗n ) solves the problem
n
X
min
d1 +···+dn =d,li ≤di ≤ui
fi (di ).
i=1
It then follows from Theorem 5.1 and Proposition 2.1 that FX−1i (FS c (d)) ≤ d∗i ≤ FX−1+
(FS c (d))
i
c
c
for all i, where (X1 , . . . , Xn ) is a comonotonic random vector with marginal distributions given
by (13) for some large enough ν, and S c := X1c + · · · + Xnc . From the proof of the “if” part,
we have
ν(2FS c (d) − 1) ∈ ∂f1 (d∗1 ) ∩ · · · ∩ ∂fn (d∗n ).
This shows that the right hand intersection is non-empty.
11
7
Two examples
In this section, we provide two carefully worked-out examples to illustrate how the theory
developed thus far can be used to solve concrete problems.
Example 1 Consider the following optimal capital allocation problem:
min
n
X
d1 +···+dn =d,di ≥li
− si [1 − exp(−mi di )] ,
(17)
i=1
where si , mi are some strictly positive constants. It is assumed that
d > L := l1 + · · · + ln
(18)
in order to avoid that the problem is trivial or ill-posed. To simplify the notation, we define
θi :=
ln mi si
,
mi
Θi :=
i
X
aj ,
Ai := 1 − mi si exp(−mi li ),
j=1
Mi :=
i
X
1
,
m
j
j=1
Li :=
n
X
lj ,
j=i+1
for i = 1, . . . , n. Note that Ln := 0 by convention. Without loss of generality, we assume
that A1 ≤ · · · ≤ An .
Proposition 7.1. For any given d > L, define
(
)
i
X
i∗ := inf i ∈ {1, . . . , n − 1} | d ≤ L +
Mj ln(1 − Aj ) − ln(1 − Aj+1 ) ∧ n,
(19)
j=1
with the convention that inf ∅ = ∞. Then the optimal solution to problem (17) is given by
(
li ,
i = i∗ + 1, . . . , n
d∗i =
i∗ −d
, i = 1, . . . , i∗ .
θi − Θi∗m+L
i Mi∗
Proof:
We first notice that every fi (di ) := −si [1 − exp(−mi di )] is strictly decreasing
and strictly convex. Without loss of generality, we may assume that s1 , . . . , sn have been
rescaled properly such that −1 ≤ (fi )0+ (li ) for all i. Since limx→∞ fi (x) exists in R and
limx→∞ (fi )0+ (x) = 0 for all i, it follows from Proposition 4.2 and Corollary 5.3 that problem
(17) has a unique solution (d∗1 , . . . , d∗n ) given by d∗i = FX−1i (FS c (d)) where S c := FX−11 (U ) +
· · · + FX−1n (U ) for any uniform(0, 1) random variable U and
(
0,
x < li ,
FXi (x) =
1 − mi si exp(−mi x), li ≤ x.
It remains to compute FS c (d) and FX−1i .
The inverse of this distribution function is given by
(
li ,
0 < p ≤ Ai
1
mi si
−1
=
ln
∨ li .
FXi (p) = 1
mi si
m
1
−
p
,
A
≤
p
<
1,
ln
i
i
mi
1−p
12
Also, for any p ∈ (0, 1), we find that
FS−1
= FX−11 (p) + · · · + FX−1n (p)
c (p)


l1 + · · · + ln ,




m1 s1
1


 m1 ln 1−p + l2 + · · · + ln ,
1
1 s1
2 s2
ln m1−p
+ m12 ln m1−p
+ l3 + · · · + ln ,
=
m1


.

..




 1 ln m1 s1 + · · · + 1 ln mn sn ,
m1
1−p
mn
1−p
0 < p ≤ A1
A1 ≤ p ≤ A2
A2 ≤ p ≤ A3
(20)
An ≤ p < An+1 := 1.
Notice that FS−1
c is continuous and is piecewise linear on (0, A1 ], [A1 , A2 ], . . . , [An , An+1 ). Simple algebraic manipulation shows that
FS−1
c (Ai+1 )
=L+
i
X
Mj ln(1 − Aj ) − ln(1 − Aj+1 ),
i = 1, . . . , n − 1,
j=1
which is the expression in (19). The definition of i∗ in (19) enables us to locate the exact
“layer” that FS c (d) belongs to, so that Ai∗ < FS c (d) ≤ Ai∗ +1 . By solving the equation
FS−1
c (FS c (d)) = d for FS c (d) using this particular layer in (20), we obtain
Θi∗ + Li∗ − d
, d > L.
(21)
FS c (d) = 1 − exp
Mi∗
Therefore, the optimal solution to problem (17) is given by
1
mi si
−1
∗
di = FXi (FS c (d)) =
ln
∨ li ,
mi 1 − FS c (d)
i = 1, . . . , n.
Simplifying this expression yields the desired result.
For instance, if
L<d≤
1
m 1 s1
ln
+ l2 + · · · + ln ,
m1 1 − A2
then i∗ = 1 and A1 < FS c (d) ≤ A2 . Therefore, the optimal solution is given by
(
1 −d
= d − (l2 + · · · + ln ), i = 1,
θ1 − Θ1m+L
1 M1
d∗i =
li ,
i = 2, . . . , n.
As another illustration, suppose that d is sufficiently large such that
d>
n−1
X
1
m1 s1
1
m n sn
ln
+ ··· +
ln
=L+
Mj ln(1 − Aj ) − ln(1 − Aj+1 ) ,
m1 1 − An
mn 1 − An
j=1
then i∗ = n and An < FS c (d) < 1. Applying Proposition 7.1 yields that
d∗i = θi −
Θn − d
> li ,
mi Mn
13
i = 1, . . . , n.
Example 2 In Dhaene et al. (2012), the following optimal capital allocation was considered:
n
X
ζi (Yi − di )2
min
E
,
d1 +···+dn =d
ν
i
i=1
where ζ1 , . . . , ζn are positive random variables with mean 1, ν1 , . . . , νn are given strictly
positive numbers summing to 1, and Y1 , . . . , Yn are some square integrable random variables.
We refer to Dhaene et al. (2012) for a detailed interpretation of this model. In that paper,
it is shown that the optimal allocations are given by
!
n
X
∗
di = E[ζi Yi ] + νi d −
E[ζj Yj ] , i = 1, . . . , n.
j=1
Here, we want to add the box constraints di ∈ [0, d] for all i to the minimization problem
above. More precisely, we would like to apply the theory developed in the previous sections
to solve the following problem:
n
X
ζi (Yi − di )2
.
(22)
min
E
d1 +···+dn =d,0≤di ≤d
νi
i=1
To simplify our notation, we define ci := E[ζi Yi ] for i = 1, . . . , n and assume without loss of
generality that
c1
cn
≥ ··· ≥ .
(23)
ν1
νn
Proposition 7.2. For any given d > 0, define
(
)
i
X
ci+1
cj
∗
−
i := inf i ∈ {1, . . . , n − 1} d ≤
νj
∧ n,
νj
νi+1
j=1
(24)
with the convention that inf ∅ = ∞. Then the optimal solution to problem (22) is given by
(
Pi∗
ν
P i∗ i
d
−
c
i = 1, . . . , i∗ ,
j=1 j + ci ,
∗
ν
j
i=1
di =
0,
i = i∗ + 1, . . . , n.
Proof: We first let
ζi (Yi − x)2
fi (x) := E
,
νi
x ∈ R, i = 1, . . . , n.
Notice that the box constraints di ∈ [0, d] for all i can be replaced by di ∈ [0, ui ] for all i as
long as each ui is larger than d. In particular, we choose u1 , . . . , un greater than d such that
(f1 )0+ (u1 ) = · · · = (fn )0+ (un ) ≥ max (fi )0+ (0),
1≤i≤n
which is equivalent to
−2E[ζi Yi ] 2(u1 − E[ζ1 Y1 ])
2(un − E[ζn Yn ])
.
= ··· =
≥ max 1≤i≤n
ν1
νn
νi
14
Moreover, we take ν to be the common value on the left-hand side of the inequality above.
By Theorem 4.1, problem (22) is equivalent to
n
X
min
E[(Xi − di )+ ],
d1 +···+dn =d,0≤di ≤ui
i=1
where the distribution function of Xi is given by


0
FXi (x) = (1 + f+0 (x)/ν)/2


1
if
if
if
x < 0,
0 ≤ x < ui ,
x ≥ ui .
Our choice of u1 , . . . , un guarantees that none of the distribution function FXi has a point
mass at ui . Inverting the distribution function above yields that
(
0,
0 < p ≤ Ai
−1
FXi (p) =
= (ννi (p − 1/2) + ci )+
ννi (p − 1/2) + ci , Ai ≤ p < 1,
ci
for i = 1, . . . , n. From assumption (23), A1 ≤ · · · ≤ An . Let S c be
where Ai := 1/2 − νν
i
the comonotonic sum FX−11 (U ) + · · · + FX−1n (U ), where U is any uniform(0, 1) random variable.
Then for p ∈ (0, 1),
FS−1
= FX−11 (p) + · · · + FX−1n (p)
c (p)

0,






νν1 (p − 1/2) + c1 ,
νν1 (p − 1/2) + c1 + νν2 (p − 1/2) + c2 ,
=

.


..



νν1 (p − 1/2) + c1 + · · · + ννn (p − 1/2) + cn ,
0 < p ≤ A1
A1 ≤ p ≤ A2
A2 ≤ p ≤ A3
An ≤ p < 1.
By a similar argument as in the proof of Proposition 7.1, we obtain
Pi∗
1 d − j=1 cj
.
FS c (d) = +
P∗
2
ν ii=1 νj
(25)
From Corollary 5.3, the optimal solution to problem (22) is given by
d∗i = FX−1i (FS c (d)) = (ννi (FS c (d) − 1/2) + ci )+ ,
i = 1, . . . , n.
Putting the expression of FS c (d) in (25) in this formula yields the desired result.
As an illustration, consider the case where
c1
c2
0 < d ≤ ν1
−
.
ν1 ν2
Then i∗ = 1 and d∗1 = d and d∗2 = · · · = d∗n = 0. As another illustration, if
n−1
X
cn
cj
−
< d,
νj
νj
νn
j=1
then i∗ = n and
d∗i = νi
d−
n
X
!
cj
+ ci ,
i = 1, . . . , n,
j=1
which is the optimal solution obtained in Dhaene et al. (2012) without any box constraints.
15
8
8.1
Some variants
Minimization of weighted sum of stop-loss premiums
Consider the following variant of minimization problem (7):
min
d1 +···+dn =d,li ≤di ≤ui
n
X
νi E[(Yi − di )+ ],
(26)
i=1
where ν1 , . . . , νn are some strictly positive constants that are not all equal, and Y1 , . . . , Yn are
some integrable random variables with possibly unbounded support.
We first remark that the support of each Yi in problem (26) can be assumed to be contained
in [li , ui ] without loss of generality. For if di ∈ [li , ui ],
E[(Yi − di )+ ] = E[(Yi ∨ li − di )+ ] = E[((Yi ∨ li ) ∧ ui − di )+ ] + E[(Yi − ui )+ ],
and thus Yi can be replaced by (Yi ∨ li ) ∧ ui in problem (26) without changing the solution
set. For the remainder of this section, we assume that the support of each Yi is contained in
[li , ui ].
To solve problem (26), one may simply treat it as a special case of problem (1) by writing
fi (di ) := νi E[(Yi − di )+ ],
di ∈ [li , ui ],
and proceed as in Section 4 to express fi (di ) as an affine function of E[(Xi − di )+ ] for some
suitable Xi so that the leading coefficients are equalized.
In what follows, we present a simple trick to accomplish this transformation by using suitable
Bernoulli variables to “absorb” the coefficients νi . To explain this approach, we first assume,
without loss
P of generality, that each νi is strictly less than 1. If not, we may simply replace
νi by νi / νi . Let Z1 , . . . , Zn be Bernoulli variables which are independent of Y1 , . . . , Yn so
that
(
1 with probability νi
Zi =
0 with probability 1 − νi ,
and Xi := Zi (Yi − li ) + li . Then for any li ≤ di ≤ ui ,
νi E[(Yi − di )+ ] = P(Zi = 1)E[((Yi − li ) − (di − li ))+ ]
= E[(Zi (Yi − li ) − (di − li ))+ ]
= E[(Xi − di )+ ]
by the assumed independence between Zi and Xi .
8.2
Nonlinear constraints
In problem (1), the linear constraint d1 + · · · + dn = d can be replaced by a non-linear
constraint of the form
h1 (d1 ) + · · · + hn (dn ) = d,
16
where each hi is a 1−1 function so that the inverse h−1
i is well-defined (on a suitable domain).
−1
Let d̃i := hi (di ) and f̃i (x) = fi (hi (x)) for all i. If each h−1
is convex and fi is increasing
i
−1
and convex, or hi is concave and fi is decreasing and convex, then f̃i is convex as well, and
hence the corresponding minimization problem is reduced to the form of problem (1) again.
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18
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