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Mathematical Finance, Vol. 18, No. 1 (January 2008), 171–183
A CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING
FROM PORTFOLIO SELECTION
HANQING JIN,ZUO QUAN XU,AND XUN YUZHOU
The University of Oxford
A continuous-time financial portfolio selection model with expected utility maxi-
mization typically boils down to solving a (static) convex stochastic optimization prob-
lem in terms of the terminal wealth, with a budget constraint. In literature the latter is
solved by assuming a priori that the problem is well-posed (i.e., the supremum value
is finite) and a Lagrange multiplier exists (and as a consequence the optimal solution
is attainable). In this paper it is first shown that, via various counter-examples, neither
of these two assumptions needs to hold, and an optimal solution does not necessarily
exist. These anomalies in turn have important interpretations in and impacts on the
portfolio selection modeling and solutions. Relations among the non-existence of the
Lagrange multiplier, the ill-posedness of the problem, and the non-attainability of an
optimal solution are then investigated. Finally, explicit and easily verifiable conditions
are derived which lead to finding the unique optimal solution.
KEY WORDS: portfolio selection, convex stochastic optimization, Lagrange multiplier, well-
posedness, attainability.
1. INTRODUCTION
Given a probability space (, F,P), consider the following constrained stochastic opti-
mization problem:
Maximize Eu(X)
subject to E[Xξ]=a,X≥0isarandom variable,
(1.1)
where a>0isaparameter, ξ>0agiven scalar-valued random variable, u(·):IR
+→ IR +
a twice differentiable, strictly increasing, strictly concave function with u(0) =0,u(0+)=
+∞,u(+∞)=0. Define V(a)=supE[Xξ]=a,X≥0is a r.v. Eu(X).
It is well known that many continuous-time financial portfolio selection problems
with expected utility maximization boil down to solving problem (1.1). In the context
of a portfolio model, u(·)isthe utility function (all the assumed properties on u(·)have
economic interpretations), ξis the so-called pricing kernel or state price density, ais the
initial wealth (hence the first constraint is the budget constraint), and Xis the terminal
wealth to be determined. Once an optimal X∗to (1.1) is found, the portfolio replicating
X∗is the optimal portfolio for the original dynamic portfolio choice problem, if the
The study is supported by the RGC Earmarked Grants CUHK 4175/03E, CUHK418605, and Croucher
Senior Research Fellowship.
Manuscript received February 2006; final revision received July 2006.
Address correspondence to Xun Yu Zhou, Mathematical Institute and Nomura Centre for Mathematical
Finance, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK; e-mail: zhouxy@maths.ox.ac.uk.
C
2008 The Authors. Journal compilation C
2008 Blackwell Publishing Inc., 350 Main St., Malden, MA 02148,
USA, and 9600 Garsington Road, Oxford OX4 2DQ, UK.
171
172 H. JIN, Z. Q. XU, AND X. Y. ZHOU
market is complete. For details see, e.g., Cvitanic and Karatzas (1992), Karatzas (1997),
Karatzas and Shreve (1998), and Korn (1997).
In literature (1.1) is usually solved by the Lagrange method, which is summarized in
the following theorem.
THEOREM 1.1. If (1.1) admits an optimal solution X∗whose objective value is finite,
then there exists λ>0such that X∗=(u)−1(λξ ).Conversely, if E[(u)−1(λξ )ξ]=a<+∞
and E[u((u)−1(λξ ))] <+∞,then X ∗=(u)−1(λξ)is optimal for (1.1) with parameter a.
This theorem provides an efficient scheme to find the optimal solution for Problem (1.1):
For any a>0, solve the Lagrange equation E[(u)−1(λξ)ξ]=a—if one could —to deter-
mine a Lagrange multiplier λ, and then X∗=(u)−1(λξ)isthe optimal (automatically
unique as the utility function is strictly concave) solution for (1.1), if Eu(X∗)isfinite.
However, there are many issues about Problem (1.1) that are left untouched by the
preceding theorem/scheme. To elaborate, in general there are the following progressive
issues related to an optimization problem such as (1.1):
(i) Feasibility:whether there is at least one solution satisfying all the constraints
involved. For (1.1), since X=a/ξ is a feasible solution, the feasibility is not an
issue.1
(ii) Well-posedness:whether the supremum value of the problem with a non-empty
feasible set is finite (in which case the problem is called well-posed) or +∞
(ill-posed). An ill-posed problem is a mis-formulated one: the trade-off is not
set right so one could always push the objective value to be arbitrarily high.2
(iii) Attainability:whether a well-posed problem admits an optimal solution. It may
or may not.
(iv) Uniqueness:whether an attainable problem has a unique optimal solution. It
is not an issue for (1.1), since uniqueness holds automatically due to the strict
concavity of the utility function.
Clearly, Theorem 1.1 covers only the case when the problem is well-posed and the
attainability holds, by assuming a priori that a Lagrange multiplier exists [indeed, in the
context of portfolio selection the existing work always assumes that the Lagrange multi-
plier exists; see theorem 2.2.2 on page 7 of Karatzas (1997) and page 65 of Korn (1997)].3
Moreover, in theorem 2.2.2 on page 7 of Karatzas (1997) and assumption 6.2 on page 773
of Cvitanic and Karatzas (1992), it is assumed up front that the underlying problem is
well-posed.4In this paper, we will first show, through various counter-examples, that none
of the aforementioned assumptions that have all along been taken for granted needs to
hold true. Then, we will address the following questions: When does the Lagrange mul-
tiplier exist? What if it does not? What does it have to do with the well-posedness and
1Feasibility could be by itself an interesting problem if more complex constraints are involved. See
Section 3 of Bielecki et al. (2005) for an example.
2Again, well-posedness is an important, sometimes very difficult, problem in its own right; see
Jin and Zhou (2008) for a behavioral portfolioselection model where the well-posedness becomes an eminent
issue. Also see Korn and Kraft (2004) for more ill-posed examples.
3In these references it is assumed that f(λ)=E[(u)−1(λξ)ξ]<+∞ for any λ>0, which is equivalent to
the existence of the Lagrange multiplier for any a>0; see Section 2 for details.
4Some of the references cited here deal with models with consumptions; yet the essence of the Lagrange
method remains the same.
CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION 173
attainability? What are the conditions ensuring the existence of a unique optimal solution
for (1.1) for a given a>0orforany a>0?
The aim of this paper is to give a thorough treatment of (1.1), including answers to
the above questions. In particular, Section 2 reveals the possibility of non-existence of
the Lagrange multiplier. Section 3 studies the implications of the non-existence of the
Lagrange multiplier, and Section 4 shows the possibility of ill-posedness even with the
existence of the Lagrange multiplier. Finally, Section 5 presents easily verifiable conditions
for uniquely solving (1.1).
2. NON-EXISTENCE OF LAGRANGE MULTIPLIER
It is possible that the Lagrange multiplier simply does not exist, which will be demon-
strated in this section via several examples.
First off, define
f(λ)=E[(u)−1(λξ)ξ],λ>0.(2.1)
Then f(·)isnon-increasing (notice that f(·)may take value +∞). The following lemma
is evident given the monotonicity of (u)−1(·) and the monotone convergence theorem.
LEMMA 2.1. If f (λ0)<+∞ for some λ0>0, then f (·)is continuous on (λ0,+∞)and
right continuous at λ0,with f (+∞)=0.
It follows from Lemma 2.1 that if f(λ0)<+∞ for some λ0>0, then the Lagrange
multiplier exists for any 0 <aa0:=E[(u)−1(λ0ξ)ξ]. In particular, if
f(λ)<+∞ ∀λ>0,(2.2)
then the Lagrange multiplier exists for any a>0. This is why in existing literature (2.2)
is usually assumed up front [see, e.g. (2.2.11) on page 37 of Karatzas (1997) and (24) on
page 65 of Korn (1997)]. Now, we are to show that this assumption may not hold even
for simple cases.
EXAMPLE 2.1. Take u(x)=√x,x≥0,P(ξ≤t)=1−e−t,t≥0. In this
example, u(x)=1
2√x,(u)−1(y)=(2y)−2, and f(λ)=E[(u)−1(λξ)ξ]=1
4λ2Eξ−1=+∞
for any λ>0. Therefore E[(u)−1(λξ)ξ]=aadmits no solution for any a>0.
In the above example the Lagrange multiplier does not exist for any a >0. In the
following examples, Lagrange multipliers exist for some a >0, and do not for other
a>0.
EXAMPLE 2.2. Define p(x)=ex−1−x−x2/2!−x3/3!
x2=+∞
n=2
xn
(n+2)! ,g(x)=p(1
x),h(x)=
g−1(x),x>0. Take
u(x)=
xh(x)+1/h(x)
0
p(y)
y2dy,x>0,
0,x=0,
and P(ξ≥t)=1−e−1/t,t>0; or 1/ξ follows the exponential distribution with
parameter 1.
In this example, p(·)isstrictly increasing with p(0+)=0,p(+∞)=+∞; hence g(·)
is strictly decreasing with g(0+)=+∞,g(+∞)=0, and h(·)iswell-defined and strictly
decreasing with h(0+)=+∞,h(+∞)=0. All these functions are smooth.
174 H. JIN, Z. Q. XU, AND X. Y. ZHOU
For the utility function u(·), notice that x
0
p(y)
y2dy =x
0∞
n=0
yn
(n+4)! dy =+∞
n=0
xn+1
(n+4)!(n+1)
is well-defined for any x>0, and
lim
x→0+xh(x)=lim
y→+∞ g(y)y=lim
y→+∞ p1
yy=0,
which means that u(·)isright-continuous at 0. Furthermore, for any x>0
u(x)=h(x)+xh(x)−p(1/h(x))
1/h(x)2
h(x)
h(x)2
=h(x)+xh(x)−p(1/h(x)) h(x)
=h(x)+xh(x)−g(h(x)) h(x)
=h(x).
Therefore u(·)isconcave and u(0+)=h(0+)=+∞,u(+∞)=h(+∞)=0. Moreover,
u(x)=h(x), and (u)−1(y)=g(y)=+∞
n=2
1
(n+2)!yn.Onthe other hand, from the distribu-
tion of ξit follows easily that Eξ−n=n!forany n∈IN.
Now let us calculate f(λ)=E[(u)−1(λξ)ξ]forany λ>0:
f(λ)=E[g(λξ)ξ]
=E+∞
n=2
1
(n+2)!λnξ−(n−1)
=+∞
n=2
(n−1)!
(n+2)!λn
=+∞
n=2
1
(n+2)(n+1)n1
λn
.
By the convergence of series, we know that f(λ)<+∞ if and only if λ≥1.
Define a1=f(1) =E[(u)−1(ξ)ξ]=+∞
n=2
1
(n+2)(n+1)n=1
12 . Then for any 0 <a≤a1,
we can find a Lagrange multiplier λ≥1 such that E[(u)−1(λξ )ξ]=a.Onthe other hand,
the Lagrange multiplier is non-existent when a>a1.
In the preceding examples ξis related to the exponential distribution, whereas in
applying to portfolio selection ξis typically lognormal. The next example shows such a
case.
EXAMPLE 2.3. Take a positive random variable ξsatisfying 0 <E[ξ−(n−1)]<+∞∀n≥
1 and limn→+∞ E[ξ−(n−1)]
E[ξ−n]=0(e.g., when ξis lognormal). Define an=1
n2E[ξ−(n−1)],n≥2,
and p(x)=+∞
n=2anxn,g(x)=p(1
x),h(x)=g−1(x),x>0. Take
u(x)=
xh(x)+1/h(x)
0
p(y)
y2dy,x>0,
0,x=0.
Exactly the same analysis as in Example 2.2 yields that u(·)isautility function satisfying
all the required conditions, with u(x)=h(x) and (u)−1(x)=g(x)=+∞
n=2anx−n.
CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION 175
Now, for any λ>0,
f(λ)=E[g(λξ)ξ]=E+∞
n=2
anλ−nξ−(n−1)=+∞
n=2
1
n2λn.
Hence f(λ)<+∞ if and only if λ≥1. As a result, the Lagrange multiplier exists if and
only if 0 <a≤a1,where a1=f(1) =E[(u)−1(ξ)ξ]=+∞
n=2
1
n2=π2−6
6.
3. IMPLICATIONS OF NON-EXISTENCE OF LAGRANGE MULTIPLIER
So, if the Lagrange multiplier does not exist, what can we say about the underlying
optimization problem (1.1)? Theorem 1.1 implies that the non-existence of the Lagrange
multiplier is an indication of either the ill-posedness or the non-attainability of (1.1). In
this section we elaborate on this.
THEOREM 3.1. If E[(u)−1(λξ )ξ]=+∞for any λ>0, then V(a)=+∞for any a >0.
Proof . Fix λ0>0 and a>0. Since E[(u)−1(λ0ξ)ξ]=+∞, one can find a set A∈F
such that E[(u)−1(λ0ξ)ξ1A]∈(a,+∞). Define h(λ)=E[(u)−1(λξ)ξ1A], λ∈[λ0,+∞).
Then h(·)isnon-increasing and continuous on [λ0,+∞) with h(+∞)=0; hence there
exists λ1>λ
0such that h(λ1)=a.
Denote X1=(u)−1(λ1ξ)1A,which is a feasible solution for Problem (1.1) with pa-
rameter a, and V(a)≥E[u(X1)1A]≥E[X1u(X1)1A]=E[(u)−1(λ1ξ)λ1ξ1A]=λ1a>
λ0a. (Here we have used the fact that u(x)≥xu(x)∀x>0owing to the concavity of
u(·) and that u(0) =0.) Since λ0>0isarbitrary, we arrive at V(a)≥limλ0→+∞ λ0a=
+∞.
This theorem indicates that if the Lagrange multiplier does not exist for all a >0,
then (1.1) is ill-posed for all a >0. Example 2.1 exemplifies such a case. Now, if the
Lagrange multiplier does not exist for only some a (such as in Examples 2.2 and 2.3), is it
still possible that (1.1) is well-posed for the same a?Tostudy this, we need the following
lemma.
LEMMA 3.1. V(a)<+∞,∀a>0if and only if ∃a>0such that V(a)<+∞.
Proof .Itsuffices to prove that if V(a)<+∞ for some a>0 then V(b)<+∞ for
any b>0.
For b≥a,wehave
V(b)=sup
E[Xξ]=b,X≥0
Eu(X)=sup
E[Xξ]=a,X≥0
Eu b
aX
≤sup
E[Xξ]=a,X≥0
b
aEu(X)=b
aV(a)<+∞,
where the first inequality is due to the concavity of u(·) and u(0) =0.
For any 0 <b<a,
V(b)=sup
E[Xξ]=b,X≥0
Eu(X)=sup
E[Xξ]=a,X≥0
Eu b
aX
≤sup
E[Xξ]=a,X≥0
Eu(X)=V(a)<+∞,
where the first inequality is due to u(·) being increasing. The proof is complete.
176 H. JIN, Z. Q. XU, AND X. Y. ZHOU
COROLLARY 3.1. If V(a)<+∞ for some a >0, then there exists a0>0such that
Problem (1.1) admits a unique optimal solution for all 0<a≤a0.
Proof .Itfollows from Theorem 3.1 that there exists λ0with E[(u)−1(λ0ξ)ξ]<+∞;
consequently the Lagrange multiplier exists for any 0 <a≤a0:=E[(u)−1(λ0ξ)ξ]by
Lemma 2.1. On the other hand, Lemma 3.1 yields that V(a)<+∞ for all a; hence
the desired result follows by virtue of Theorem 1.1.
Now let us continue with Example 2.3.
EXAMPLE 3.1. In Example 2.3, take λ=2. We have proved that a2:=E[(u)−1(2ξ)ξ]<
+∞. Denote X∗=(u)−1(2ξ). Then
Eu(X∗)=Eu(g(2ξ))
=E2ξg(2ξ)+1/(2ξ)
0
p(y)
y2dy
=2a2++∞
n=2
an
n−1E[(2ξ)−(n−1)]
=2a2++∞
n=2
2−(n−1)
n2(n−1)
<+∞.
Theorem 1.1 suggests that X∗is the unique optimal solution for (1.1) with parameter a2
and, in particular, V(a2)=Eu(X∗)<+∞.ByLemma 3.1, we know V(a)<+∞ for any
a>0, i.e., (1.1) is well-posed for any a>0.
However, we have proved in Example 2.3 that E[(u)−1(λξ)ξ]=aadmits no solution
for any a>a1. Therefore Problem (1.1) with parameter a>a1is well-posed; yet it admits
no optimal solution (i.e., the problem is not attainable).
4. ILL-POSEDNESS WHEN LAGRANGE MULTIPLIER EXISTS
The last section demonstrated that one of the possible consequences of the non-existence
of a Lagrange multiplier is the ill-posedness of the underlying optimization problem.
This section aims to show via an example that Problem (1.1) may be ill-posed even if the
Lagrange multiplier does exist for any a >0.
EXAMPLE 4.1. Let
u(x)=√x,0≤x≤1,
1−ln 2 +ln(1 +x),x>1,
and ξbe a positive random variable such that E[ln 1
ξ]=+∞.Itiseasy to check that u(·)
has all the required properties, and
(u)−1(x)=
1
x−1,0<x≤0.5,
1
4x2,x>0.5.
CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION 177
Hence
f(λ)=E[(u)−1(λξ)ξ]=1
λE[(1 −λξ)1λξ ≤0.5]+E1
4λ2ξ
1λξ>0.5≤3
2λ<+∞ ∀λ>0.
As a result, the Lagrange multiplier exists for any a>0. However, for any λ>0,
Eu(u)−1(λξ)=E[(1 −ln 2 −ln(λξ ))1λξ ≤0.5]
+E1
2λξ
1λξ>0.5≥Eln 1
ξ
1λξ≤0.5−ln(λ)=+∞.
REMARK 4.1. In existing literature it is usually assumed, either explicitly [see, e.g.,
(2.2.13) on page 37 of Karatzas (1997)] or implicitly, that the problem is well-posed for
all a. The preceding example proves that the well-posedness is not guaranteed even when
the Lagrange multiplier exists.
5. OPTIMAL SOLUTION
Having discussed on the ill-posedness and non-attainability, we are now in a position to
study the optimal solution of (1.1). The problems with Theorem 1.1 are two-fold. On
one hand, the required conditions that the Lagrange equation E[(u)−1(λξ)ξ]=aadmits
a positive solution and that E[u((u)−1(λξ ))] <+∞ do not necessarily hold (as already
demonstrated), and on the other hand even if the conditions do hold, they are implicit
and/or hard to verify. In this section, we will present conditions that are explicit and easy
to use.
Recall that f(λ)=E[(u)−1(λξ )ξ],λ>0. If f(λ)=+∞for any λ>0, then it follows
from Theorem 3.1 that V(a)=+∞for any a>0, which is a pathological case. Hence we
assume that there exists a λ>0 such that f(λ)<+∞. Denote λ0=inf{λ>0: f(λ)<
+∞} <+∞ and a0=f(λ0+) (notice that a0=+∞is possible, and a0=f(λ0)when
λ0>0).
PROPOSITION 5.1. Suppose λ0<+∞.We have the following conclusions.
(i) If a0<+∞, then Problem (1.1) with parameter a>0 admits a unique optimal
solution if and only if E[u((u)−1(λ0ξ))] <+∞ and a≤a0.
(ii) If a0=+∞, then Problem (1.1) admits a unique optimal solution for any a>
0ifandonly if E[u((u)−1(ξ))] <+∞.
Proof .Inview of Theorem 1.1 and Lemma 3.1 (i) is clear. To prove (ii), if a0=+∞,
by Lemma 2.1, f(·)iscontinuous on (λ0,+∞) with f(λ0+)=+∞and f(+∞)=0; hence
the Lagrange multiplier exists for any a>0. Now, if E[u((u)−1(ξ))] <+∞, using u(x)≥
xu(x) with x=(u)−1(ξ), we have
+∞ >E[u((u)−1(ξ))] ≥E[(u)−1(ξ)ξ]=:a1.
It follows from Theorem 1.1 that V(a1)=E[u((u)−1(ξ))] <+∞. Lemma 3.1 further
yields V(a)<+∞,∀a>0. The desired result is now a consequence of Theorem 1.1.
Now we derive some sufficient conditions, explicit in terms of u(·)orξ,for the existence
of a unique optimal solution to (1.1). First we have the following simple case.
178 H. JIN, Z. Q. XU, AND X. Y. ZHOU
THEOREM 5.1. If ε=essinf ξ>0, then Problem (1.1) admits a unique optimal solution
for any a >0.
Proof .Givena>0. For any feasible solution Xof Problem (1.1),
Eu(X)≤u(EX)≤uE[Xξ]
ε=ua
ε.
Therefore V(a)<+∞.
Meanwhile, for any λ>0,
f(λ)=E[(u)−1(λξ)ξ]≤1
λE[u((u)−1(λξ))] ≤1
λu((u)−1(λε)) <+∞.
This proves the existence of the Lagrange multiplier λ>0forany a>0. By Theorem 1.1,
Xλ=(u)−1(λξ)isthe unique optimal solution for (1.1).
Let us make some preparations for our main result.
Define R(x)=−xu (x)
u(x)≥0asthe Arrow–Pratt index of risk aversion of the utility func-
tion u(·).
LEMMA 5.1. If lim infx→+∞R(x)>0, then lim supx→+∞
u(kx)
u(x)<1forany k>1.
Proof . Because lim infx→+∞R(x)>0, there exist M>0,K>0, such that R(x)≥K
for any x≥M.Forany x≥M,k>1,
u(kx)
u(x)−1=u(kx)−u(x)
u(x)
=kx
x
u(y)dy
u(x)
=−kx
x
R(y)u(y)/ydy
u(x)
≤−kx
x
R(y)u(kx)/ydy
u(x)
=−u(kx)
u(x)kx
x
R(y)/ydy
≤−
u(kx)
u(x)Kkx
x
1/ydy
=−u(kx)
u(x)Kln k.
Therefore u(kx)
u(x)≤1
1+Kln kwhich implies lim supx→+∞
u(kx)
u(x)≤1
1+Kln k<1.
LEMMA 5.2. limsupx→0+
(u)−1(λx)
(u)−1(x)<+∞ for any 0<λ<1if and only if
lim supx→+∞
u(kx)
u(x)<1for any k >1.
CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION 179
Proof .Wefirst claim that limsupx→0+
(u)−1(λx)
(u)−1(x)<+∞ for any 0 <λ<1ifandonly
if ∃0<¯
λ<1 such that lim supx→0+
(u)−1(¯
λx)
(u)−1(x)<+∞.
To prove this claim, suppose lim supx→0+
(u)−1(¯
λx)
(u)−1(x)<+∞ for some 0 <¯
λ<1. Then
lim sup
x→0+
(u)−1(¯
λ2x)
(u)−1(x)
=lim sup
x→0+
(u)−1(¯
λ2x)
(u)−1(¯
λx)
(u)−1(¯
λx)
(u)−1(x)
≤lim sup
x→0+
(u)−1(¯
λ2x)
(u)−1(¯
λx)lim sup
x→0+
(u)−1(¯
λx)
(u)−1(x)
<+∞.
From induction it follows limsupx→0+
(u)−1(¯
λnx)
(u)−1(x)<+∞ for any n∈IN . Since
lim supx→0+
(u)−1(λx)
(u)−1(x)is non-increasing in λ, lim supx→0
(u)−1(λx)
(u)−1(x)<+∞ for any 0 <λ<1.
Similarly, one can prove that lim supx→+∞
u(kx)
u(x)<1forany k>1ifandonly if ∃¯
k>1
such that lim supx→+∞
u(¯
kx)
u(x)<1.
Now, suppose L=lim supx→0+
(u)−1(1
2x)
(u)−1(x)<+∞ (notice that L≥1). Then there exists
δ>0 such that for any x∈(0,δ],
(u)−1(1
2x)
(u)−1(x)≤2L
⇒1
2x≥u(2L(u)−1(x))
⇒1
2≥u(2L(u)−1(x))
u((u)−1(x))
⇒u(2Ly)
u(y)≤1
2,∀y≥(u)−1(δ)
⇒lim sup
x→+∞
u(2Lx)
u(x)≤1
2.
Therefore lim supx→+∞
u(kx)
u(x)<1forany k>1.
The proof for the other direction is similar.
Recall that we have defined f(λ)=E[(u)−1(λξ )ξ] and λ0=inf{λ>0: f(λ)<+∞}.
PROPOSITION 5.2. Suppose one of the following conditions is satisfied:
(i) lim infx→+∞R(x)>0.
(ii) lim supx→+∞
u(kx)
u(x)<1for some k >1.
(iii) lim supx→0
(u)−1(λx)
(u)−1(x)<+∞ for some λ∈(0, 1).
Then the Lagrange multiplier exists for any a >0if and only if λ0<+∞.
Proof . The necessity is obvious. To prove the sufficiency, note that if λ0<+∞, then
there exists λ1>0 such that f(λ1)<+∞,which by the monotonicity of f(·) further
implies that f(λ)<+∞∀λ>λ
1.Forany λ∈(0,λ
1], denote k=λ/λ1∈(0,1].
180 H. JIN, Z. Q. XU, AND X. Y. ZHOU
Since one of the three given conditions is satisfied, by Lemmas 5.1 and 5.2 it must
have 1 ≤L=lim supx→0
(u)−1(kx)
(u)−1(x)<+∞. Hence there exists δ>0 such that (u)−1(kx)
(u)−1(x)<2L
for any x∈(0,λ
1δ]. Now, for any λ>0,
E(u)−1(λξ)ξ1ξ≤δ=E(u)−1(λξ )
(u)−1(λ1ξ)(u)−1(λ1ξ)ξ1ξ≤δ
≤2LE[(u)−1(λ1ξ)ξ1ξ≤δ]
≤2Lf (λ1),
E(u)−1(λξ)ξ1ξ>δ=1
λE(u)−1(λξ)(λξ )1ξ>δ
≤1
λEu((u)−1(λξ))1ξ>δ
≤1
λu((u)−1(λδ)).
Hence,
f(λ)=E[(u)−1(λξ)ξ]
=E(u)−1(λξ)ξ1ξ≤δ+E(u)−1(λξ )ξ1ξ>δ
≤2Lf (λ1)+1
λu((u)−1(λδ))
<+∞.
This shows that in fact λ0=0, and hence the equation f(λ)=aadmits a positive solution
λ(a)forany a>0.
REMARK 5.1. The preceding proof also shows that under the condition of Proposition
5.2, the following claims are equivalent:
(i) The Lagrange multiplier exists for any a>0.
(ii) λ0<+∞.
(iii) λ0=0.
(iv) f(1) <+∞ .
(v) f(λ)<+∞∀λ>0.
THEOREM 5.2. Under the condition of Proposition 5.2, Problem (1.1) admits a unique
optimal solution for any a >0if and only if E[u((u)−1(ξ))] <+∞.
Proof .Itsuffices to prove the sufficiency. If E[u((u)−1(ξ))] <+∞, then f(1) =
E[(u)−1(ξ)ξ]≤E[u((u)−1(ξ))] <+∞.Thus λ0=0 and a0=f(λ0+)=+∞.Itfollows
from Proposition 5.1 then that Problem (1.1) admits a unique optimal solution.
The conditions in the preceding theorem, lim infx→+∞ −xu (x)
u(x)>0 and E[u((u)−1×
(ξ))] <+∞,are very easy to verify. For example, a commonly used utility function is
u(x)=xα,0<α<1. The two conditions are satisfied when ξis lognormal.
REMARK 5.2. Example 3.1 shows that the conclusion of Theorem 5.2 can be false in
the absence of its condition.
CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION 181
COROLLARY 5.1. If E[ξ−α]<+∞∀α≥1,then, under the condition of Proposition 5.2,
Problem (1.1) admits a unique optimal solution for any a >0.
Proof .Itsuffices to prove that E[u((u)−1(ξ))] <+∞ holds automatically. Un-
der the condition of Proposition 5.2, there is L≥2 such that (u)−1(x)<
L(u)−1(2x)∀x∈(0,1). Denote L0=supx∈[1
2,1](u)−1(x)<+∞.Forany x∈(0, 1),
find n∈IN s o that 1
2≤2nx<1. Then (u)−1(x)<L(u)−1(2x)<L2(u)−1(22x)<
···<Ln(u)−1(2nx)≤LnL0≤L−log2xL0=x−log2LL0.Byvirtue of the fact that
u(+∞)=0, we may assume that u(x)≤L1x∀x≥(u)−1(1). Therefore for any x∈
(0, 1), we have u((u)−1(x)) ≤L1(u)−1(x)<L0L1x−log2L. Finally, E[u((u)−1(ξ))] ≤
E[u((u)−1(ξ))1ξ<1]+u((u)−1(1)) ≤L0L1E[ξ−log2L]+u((u)−1(1)) <+∞.
REMARK 5.3. If ξis lognormal, then the assumption that E[ξ−α]<+∞∀α≥1 holds
automatically. [In the contextof portfolio selection with the prices of the underlying stocks
followinggeometric Brownian motion, ξis typically a lognormal random variable—under
certain conditions of course; for details see Remark 3.1 in Bielecki et al. (2005)]. On the
other hand, this assumption could be weakened to that E[ξ−α0]<+∞ for certain α0(the
value of which could be precisely given). We leave the details to the interested readers.
Recall that in Section 4 we presented an example where Problem (1.1) is ill-posed even
though the Lagrange multiplier exists for any a>0. The following result shows that this
will not occur for certain ξ.
Let F(·)bethe probability distribution function of ξ.Inview of Theorem 5.1, we
assume essinf ξ=0, which in turn ensures F(x)>0∀x>0.
THEOREM 5.3. If lim infx→0+xF(x)
F(x)>0,and E[(u)−1(λξ )ξ]=a>0for some λ>0,
then Problem (1.1) with parameter a is well-posed and admits a unique optimal solution.
Proof . Since lim infx→0+xF(x)
F(x)>0, there exist M>0 and K>0 such that xF (x)
F(x)≥1
K
for any 0 <x≤M. Then
E[u((u)−1(λξ))1ξ<M]
=M
0
u((u)−1(λx)) dF(x)
=M
0x
M
du((u)−1(λy)) dF(x)+M
0
u((u)−1(λM)) dF(x)
=λM
0x
M
yd[(u)−1(λy)] dF(x)+u((u)−1(λM))F(M)
=λM
0x(u)−1(λx)−M(u)−1(λM)+M
x
(u)−1(λy)dydF(x)
+u((u)−1(λM))F(M)
=λM
0
x(u)−1(λx)dF(x)+λM
0M
x
(u)−1(λy)dy dF (x)
+[u((u)−1(λM)) −λM(u)−1(λM)]F(M)
=λM
0
x(u)−1(λx)dF(x)+λM
0y
0
dF(x)(u)−1(λy)dy
+[u((u)−1(λM)) −λM(u)−1(λM)]F(M)
182 H. JIN, Z. Q. XU, AND X. Y. ZHOU
=λM
0
x(u)−1(λx)dF(x)+λM
0
F(y)(u)−1(λy)dy
+[u((u)−1(λM)) −λM(u)−1(λM)]F(M)
≤λM
0
x(u)−1(λx)dF(x)+KλM
0
yF(y)(u)−1(λy)dy
+[u((u)−1(λM)) −λM(u)−1(λM)]F(M)
≤λ(1 +K)a+[u((u)−1(λM)) −λM(u)−1(λM)]F(M)
<+∞.
Consequently,
E[u((u)−1(λξ))]
=Eu((u)−1(λξ))1ξ<M+Eu((u)−1(λξ))1ξ≥M
≤Eu((u)−1(λξ))1ξ<M+u((u)−1(λM))
<+∞.
The desired result follows then from Theorem 1.1.
REMARK 5.4.. The condition lim infx→0+xF(x)
F(x)>0 implicitly requires that F(·)be
differentiable in the neighborhood of 0. Notice that this requirement is purely technical
so as to make the result neater. Once could replace the condition lim infx→0+xF(x)
F(x)>0
by aweaker one without having to assume the differentiability of F(·) (as hinted by the
preceding proof—the details are left to the interested reader). On the other hand, the
condition is satisfied if ξis lognormal.
Combining Theorems 1.1 and 5.3, we have immediately:
COROLLARY 5.2. Suppose lim infx→0+xF(x)
F(x)>0.Then Problem (1.1) with parameter
a>0admits an optimal solution if and only if the Lagrange multiplier λexists corresponding
to a, in which case the unique optimal solution is X∗=(u)−1(λξ ).
The following synthesized result gives easily verifiable conditions under which Prob-
lem (1.1) is completely solved.
THEOREM 5.4. We have the following conclusions.
(i) If lim infx→+∞ −xu (x)
u(x)>0,then the following statements are equivalent:
(ia) Problem (1.1) is well-posed for any a >0.
(ib) Problem (1.1) admits a unique optimal solution.
(ic) E[u((u)−1(ξ))] <+∞.
(id) ∃λ>0 such that E[u((u)−1(λξ))] <+∞.
Moreover, when one of (ia)–(id) holds the optimal solution to (1.1) with
parameter a >0is X∗=(u)−1(λ(a)ξ),where λ(a)is the Lagrange multiplier
corresponding to a.
(ii) If lim supx→0+(−xF(x)
F(x))<0,then Problem (1.1) is well-posed for any a >0
if and only if E[(u)−1(λξ)ξ]<+∞ for some λ>0, in which case there exists
0<a0≤+∞so that (1.1) admits a unique optimal solution X∗=(u)−1(λ(a)ξ)
for any a >0(if a0=+∞)orforany 0<a≤a0(if a0<+∞).
Proof .
(i) If (1.1) is well-posed for any a>0, then Theorem 3.1 yields that f(λ0)<+∞
for some λ0>0. It follows from Proposition 5.2 and Theorem 1.1 that (1.1)
CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION 183
admits a unique optimal solution for any a>0. The desired equivalence is then
a consequence of Theorem 5.2 and Theorem 1.1.
(ii) The first conclusion (“if and only if”) follows from Theorems 3.1 and 5.3. For the
second conclusion, let λ0=inf{λ>0: f(λ)<+∞} <+∞ and a0=f(λ0+).
Then the Lagrange multiplier exists for any a>0 (if a0=+∞)orforany
0<a≤a0(if a0<+∞), and Corollary 5.2 completes the proof.
REMARK 5.5. Portfolio selection is essentially an endeavor that an investor, given a
market (represented by ξor its distribution function F(·)), tries to make the best out of
his initial wealth (namely a) taking advantage of the availability of the market, where the
“best” is measured by her preference (i.e., the utility function u(·)). We have shown that
these entities, namely F(·), a, and u(·), must coordinate well, otherwise one may end up
with a wrong model. The assumptions stipulated in Theorem 5.4 tell precisely how this
well-coordination can be translated into mathematical conditions.
6. CONCLUDING REMARKS
The stochastic optimization problem studied in this paper, though interesting in its
own right, has profound applications in financial asset allocation among others. It is
demonstrated that many assumptions that have been taken for granted, such as the well-
posedness of the problem, existence of the Lagrange multiplier, and existence of an op-
timal solution, may be invalid in the first place. In particular, the issue of well-posedness
is equally important, if not more important, than that of finding an optimal solution
from a modeling point of view. Attainability of optimal solutions is another important
matter: if an optimal solution is not attainable, as is the case with Example 3.1, then
one has to resort to finding an asymptotically optimal solution. Mathematically, both
the ill-posedness and the non-attainability are symptomized by the non-existence of the
Lagrange multiplier, as analyzed in details in this paper.
It is worth noting that the results of this paper have been utilized in solving a sub
problem of the continuous-time behavioral portfolio selection model Jin and Zhou (2008),
where the ill-posedness is more a rule than an exception.
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