Non-concave utility maximisation on the positive real axis in discrete time

0.36 MB·English

by  Laurence Carassus

Non-concave utility maximisation
on the positive real axis in discrete time
∗
Laurence Carassus† Miklós Rásonyi‡ Andrea M. Rodrigues§
5 April 23, 2015
1
0
2

r Abstract
p
We treat a discrete-time asset allocation problem in an arbitrage-free, generically
A
incompletefinancialmarket,wheretheinvestorhasapossiblynon-concaveutilityfunc-

2 tionandwealthisrestrictedtoremainnon-negative. Undereasilyverifiableconditions,
2 we establish theexistence of optimal portfolios.

] Keywords: Discrete-time models ; Dynamic programming ; Finite horizon ; Incom-
F
plete markets ; Non-concaveutility ; Optimal portfolio.
M
AMS MSC 2010: Primary 93E20, 91B70, 91B16, Secondary 91G10.
.
n
i 1 Introduction
f
-
q
[ We consider investors trading in a multi-asset and discrete-time financial market who are
aiming to maximise their expected utility from terminal wealth. If the utility function
2
u is defined on the non-negative half-line, is concave, and the problem has a finite value
v
function, then there is always such a strategy, see Rásonyi and Stettner [19]. In a general
3
2 semimartingale model one needs to assume, in addition, that the so-called “asymptotic
1 elasticity at + ”, denoted by AE (u), is less than one in order to obtain an optimal
+
3 portfolio for th∞e utility maximization problem, see Kramkov and Schachermayer [15] and
0
Remark 2.11 below. In the utility maximisation context, conditions on the asymptotic
.
1 elasticity (which were first used in Cvitanić and Karatzas [8]; Karatzas, Lehoczky, Shreve,
0
and Xu [12]; Kramkov and Schachermayer [15]) have become standard in the literature.
5
In this paper we want to remove the assumptions of concavity and smoothness that are
1
: usually made on u. Why? Several reasons can be invoked. The first one is quite clear:
v
the investor can change her perception of risk above a certain level of wealth. One can
i
X also consider the problem of optimizing performance at some level B: one penalizes loss
r under B with a loss function and one maximizes gain after B with a gain function. These
a
illustrations are typicalexamples of a piecewise concavefunction. This kind of problem has
been addressed in the complete case by Carassus and Pham [4] and in a pseudo-complete
market by Reichlin [20].
Other examples of non-concave utility functions are the so-called “S-shaped” functions.
Theseappearedincumulativeprospecttheory(orCPTforshort,seeKahnemanandTversky
∗L.CarassusthanksLPMA(UMR7599)forsupport. A.M.Rodriguesgratefullyacknowledgesthefinan-
cial support of FCT-Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Tech-
nology) through the Doctoral Grant SFRH/BD/69360/2010. Part of this research was carried out while
M.Rásonyi.andA.M.Rodrigues wereaffiliatedwiththeSchool ofMathematics, UniversityofEdinburgh,
Scotland, U.K.
†LMR (EA 4535, CNRSFR 3399ARC), Université Reims Champagne-Ardenne, Moulinde la Housse –
BP1039,51687Reimscedex2,France. E-mail:[email protected]
‡MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary. The second author is also affiliated
withPázmányPéterCatholicUniversity,Budapest,Hungary. E-mail:[email protected]
§DublinCityUniversity,Dublin,Ireland. E-mail:[email protected]
Non-concaveutility maximisation on the positive real axis in discrete time
[14]; Tversky and Kahneman [23]). This theory asserts that the problem’s mental represen-
tation is important: agents analyse their gains or losses with respect to a given stochastic
reference point B rather than with respect to zero and they take potential losses more into
account than potential gains. Note that in this paper, in contrast to cumulative prospect
theory, we do not allow investors to distort the probability measure by a transformation
function of the cumulative distributions. The case of “S-shaped” functions was studied by
Berkelaar,Kouwenberg and Post [1] in a complete market setting.
In the present paper we provide mild sufficient conditions (involving asymptotic elastic-
ity) on a possibly non-concave, non-differentiable and random utility function defined on
the non-negativehalf-linewhichguaranteethe existenceofanoptimalstrategy. Bytreating
multi-step discrete-time markets, we cover a substantial class of incomplete models which
can be fitted to arbitrary econometric data.
The caseof(non-random)utilitiesdefinedonthewholereallinewastreatedinCarassus
and Rásonyi [5], but so far there were no general results in the case of the non-negative
half-line; in the present setting we are only aware of Chapter IV of Reichlin [21], where
existence results were proved for some very specific market models.
We finish by listing some references in the case of probability distortions in the spirit of
cumulative prospect theory, because those results can be applied to our setup with weight
functionsequaltothe identity. Intheincompletediscrete-timesetting,the papersofCaras-
sus and Rásonyi [6]; Rásonyi and Rodríguez-Villarreal[17] study quite specific utility func-
tions. In continuous-time studies, all the references make the assumption that the market
is complete: see Jinand Zhou[11], CarlierandDana [7], Campi andDel Vigna [3], Rásonyi
and Rodrigues [16].
Abriefoutlineofthisarticleisasfollows. Section2isdedicatedtospecifyingthemarket
modelandtointroducingthe relevantnotations. InSection3weformulateourmainresult.
Next, in Section 4 we examine the problem in a one-step setting, whilst in Section 5 we
prove our main result, using a dynamic programming approach. For the sake of a simple
presentation, the proofs of some auxiliary results are collected in Section 6.
2 Notation and set-up
2.1 The market
In what follows, we shall consider a frictionless and totally liquid financial market model
with finite trading horizon T N, in which the current time is denoted by 0 and trading is
∈
assumed to occur only at the dates 0,1,...,T .
{ }
As usual, the uncertainty in the economy is characterised by a complete probability
space (Ω,F,P), where F is a σ-algebra on the sample space Ω, and P is the underlying
probability measure (to be interpreted as the physical probability). Moreover, all the in-
formation accruing to the agents in the economy is described by a discrete-time filtration
F= F ; t 0,1,...,T suchthatF coincideswiththefamilyofallP-nullsets. Finally,
t 0
we a{ssume a∈ls{o that F =}}F .
T
Next, we fix an integer d>0 and consider a process S = S ; t 0,1,...,T , so that
t
{ ∈{ }}
S represents the time-t prices of d traded risky assets. Denoting by Ξn the family of all
t t
F -measurable random vectors ξ : Ω Rn for each n N and each t 0,1,...,T , we
t
→ ∈ ∈ { }
assume that S Ξd for every t 0,1,...,T , i.e., S is F-adapted. We shall also assume,
t ∈ t ∈{ }
without loss of generality, that the risk-free asset in this economy has constant price equal
to one at all times. Finally, for each t 1,...,T , we define ∆S ,S S .
t t t−1
∈{ } −
We recall that a self-financing portfolio is a process φ= φ ; t 1,...,T , with φ
t t
{ ∈{ }} ∈
Ξd for all t 1,...,T , and its wealth process Πφ = Πφ; t 0,1,...,T satisfies, for
t−1 ∈{ } t ∈{ }
n o
Page 2/23
Non-concaveutility maximisation on the positive real axis in discrete time
every t 1,...,T ,
∈{ }
t
Πφ =Πφ+ φ ,∆S a.s.
t 0 h s si
s=1
X
Here , denotes scalar product in Rd and is the corresponding Euclidean norm.
h· ·i k · k
We denote by Φ the class of all self-financing portfolios. In addition, we shall impose the
followingtradingconstraint: thevalueofaportfolioshouldnotbeallowedtobecomestrictly
negative. Sowesaythataportfolioφ Φisadmissible forx 0(andwewriteφ Ψ(x ))
0 0
∈ ≥ ∈
if, for every t 1,...,T , the inequality Πφ 0 holds a.s. with Πφ = x . Because of
∈ { } t ≥ 0 0
budget constraints, such a restriction is natural and frequently imposed, see e.g. Kramkov
and Schachermayer [15]; Rásonyi and Stettner [19].
The following no arbitrage assumption stipulates that no investor should be allowed to
make a profit out of nothing and without risk, even with a budget constraint.
Assumption 2.1. The market does not admit arbitrage, i.e.,
for all x 0, if φ Ψ(x ) with Πφ x a.s., then Πφ =x a.s. (NA)
0 ≥ ∈ 0 T ≥ 0 T 0
Remark 2.2. It is proved in Proposition 1.1 of Rásonyi and Stettner [19] that (NA) is
equivalent to the classical no arbitrage condition: φ Φ, Π0,φ 0 a.s. implies that
∀ ∈ T ≥
Π0,φ = 0 a.s. where Π0,φ stands for the wealth process associated to φ when starting with
T T
a zero initial wealth i.e. Πφ =0.
0
Now fix t 1,...,T . We know that there exists a regular conditional distribution of
∆Stwithresp∈ect{toFt−1}underthephysicalmeasureP,whichweshalldenotebyP∆St|Ft−1.
By modifying on a P-null set, we may and will assume that P∆St|Ft−1(,ω) is a probability
for all ω Ω. Let Dt(ω) denote the affine hull in Rd of the support of·P∆St|Ft−1(,ω). It
∈ ·
follows from Theorem 3 in Jacod and Shiryaev [10] that, under (NA), D (ω) is actually a
t
linear space for P-almost every ω.
GivenanyF -measurablerandomvariableH 0a.s.(whichcanalsobesomeconstant
t−1
≥
x 0), we set
≥ Ξd (H), ξ Ξd : H + ξ,∆S 0 a.s. .
t−1 ∈ t−1 h ti≥
We take Ξd to be the class o(cid:8)f all random vectors ξ Ξd suc(cid:9)h that ξ(ω) D (ω) for
t−1 ∈ t−1 ∈ t
P-a.e. ω. The notation Ξ˜d (H) is self-explanatory.
t−1
e
Proposition 2.3. The following two statements are equivalent,
(i) (NA) holds true;
(ii) for every t 1,...,T , there exist F -measurable random variables β >0,κ >0
t−1 t t
∈{ }
a.s. such that, for every ξ Ξd , the inequality
∈ t−1
P(eξ,∆St βt ξ Ft−1) κt (2.1)
h i≤− k k| ≥
holds a.s.
Proof. ThisfollowsfromProposition3.3inRásonyiandStettner[18]andRemark2.2above
(see also Proposition 1.1 in Rásonyi and Stettner [19]).
Remark 2.4. We notice that the above ‘quantitative’ characterisation of (NA) holds true
onlyforF -measurable,Rd-valuedfunctionsξ whichbelongtoD a.s. Thiswillmotivate
t−1 t
the use of orthogonalprojections later on (cf. Section 4).
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Non-concaveutility maximisation on the positive real axis in discrete time
2.2 The investor
Investors’ risk preferences are described by a (possibly non-concave and non-differentiable
random) utility function.
Definition 2.5 (Non-concave random utility). A random utility (on the non-negative
half-line) is any function u: (0,+ ) Ω R verifying the following two properties,
∞ × →
(i) for every x (0,+ ), the function u(x, ): Ω R is F-measurable,
∈ ∞ · →
(ii) for a.e. ω Ω, the function u(,ω): (0,+ ) R is non-decreasing and continuous.
∈ · ∞ →
For each ω Ω for which (ii) holds, we set u(0,ω) , lim u(x,ω), and we define
x↓0
∈
u(0,ω),0 otherwise. Note that u(0,ω) may take the value .
−∞
Remark 2.6. As in this paper we restrict wealth to be non-negative, we consider utilities
which are defined only over the non-negative real line. Continuity and monotonicity are
standard assumptions. Also, as u will be used to assess the future wealth of the investor, it
maywelldependoneconomicvariablesandhenceitcanberandom,seeExample2.9below.
Lastly, unlike most studies, we do not assume concavity or smoothness of u. A possible
extension to u which is only upper semicontinuous will be subject of future research.
We proceed by noticing that, since u(,ω) is a monotone function for a.e. ω Ω, the
· ∈
limit u(+ , ) , lim u(x, ) exists a.s. (though it may not be finite), and we define
x→+∞
∞ · ·
u(+ ,ω),+ otherwise. We shall require the following.
∞ ∞
Assumption 2.7. The negative part of u at 0 has finite expectation, that is,1
E u−(0, ) <+ . (2.2)
P
· ∞
Remark 2.8. If u is deterministic, then(cid:2) the abo(cid:3)ve assumption is equivalent to u(0) > .
−∞
Thisisadmittedlyrestrictive,asitexcludesthatu(x)behaveslikeln(x)or xα(withα<0)
−
inthevicinityof0. Itstillallows,however,averylargeclassofutilities. Forthemomentwe
cannot dispense with this hypothesis as it is crucial in proving that dynamic programming
preserves the growth condition (2.6) below (see part (v’) in the proof of Theorem 3.3 in
Section 5). Note also that this assumption is the pendant of (10) in Assumption 2.9 in
Carassusand Rásonyi[5] when the domain of the utility function is equal to the whole real
line.
We continue this subsection with an important example of a randomutility function, in
the spirit of cumulative prospect theory.
Example 2.9 (Reference point). Within the CPT framework, every investor is assumed to
haveareferencepoint inwealth(alsoreferredtoasbenchmark orstatusquo intheliterature,
seee.g.BernardandGhossoub[2],HeandZhou[9],CarassusandRásonyi[6]),withrespect
to which payoffs at the terminal time T are evaluated. Therefore, investors’ decisions are
not based on the terminal level of wealth (as it is assumed in the Expected Utility Theory
of von Neumann and Morgenstern [24]), but rather on the deviation of that wealth level
fromthe referencepoint. Notethat, unlikeinCPT,oursettingdoes notinclude probability
distortions (weight functions).
Mathematically, a reference point is any fixed scalar-valued and F-measurable random
variable B 0 a.s. Thus, given a payoff X at the terminal time T and a scenario ω Ω,
≥ ∈
the investor is said to make a gain (respectively, a loss) if the deviation from the reference
level is strictly positive (respectively, strictly negative), that is, X(ω)>B(ω) (respectively,
X(ω)<B(ω)).
Note that B may be taken to be, for example, a non-negative constant (this is the case
inBerkelaar,KouwenbergandPost[1];BernardandGhossoub[2];CarassusandPham[4]).
1Here x+ , max{x,0} and x− , −min{x,0} for every x ∈ R. Furthermore, in order to make the
notationlessheavy,givenanyfunctionf : X →R,weshallwritehenceforthf±(x),[f(x)]± forallx∈X.
Page 4/23
Non-concaveutility maximisation on the positive real axis in discrete time
Thereferencepointcanalsobestochastic(forinstance,toreflectthe factthatthe investors
comparetheirperformancetothatofanotherinvestoractinginaperhapsdifferentmarket).
In this setting, the investor has a random utility defined as
u(x,ω),u(x B(ω)), x>0, ω Ω, (2.3)
− ∈
with u : ( esssupB,+ ) R a (deterministic) non-decreasing and continuous func-
− ∞ → e
tion satisfying u( esssupB)> (where we set u( esssupB),lim u(x), as
x↓−esssupB
− −∞ −
beforee). Obviously, EP[u−(0, )]<+ , so Assumption 2.7 is true for u.
· ∞
We shall maeke the following assumption on the gerowth of the function u. e
Assumption 2.10. There exist constants γ > 0 and x 0, as well as a random variable
≥
c 0 a.s. with E [c]<+ , such that for a.e. ω Ω,
P
≥ ∞ ∈
u(λx,ω) λγu(x,ω)+λγc(ω) (2.4)
≤
holds simultaneously for all λ 1 and for all x x. Furthermore, E [u+(x, )]<+ .
P
≥ ≥ · ∞
Remark 2.11. Forudeterministic,strictlyconcaveandcontinuouslydifferentiable,werecall
that
xu′(x)
AE (u),limsup
+
u(x)
x→+∞
denotestheasymptotic elasticity ofuat+ (seeKramkovandSchachermayer[15,p.943]),
∞
andwealwayshaveAE (u) 1(the readerisreferredtoKramkovandSchachermayer[15,
+
≤
Lemma 6.1]).
We know by Lemma 6.3 in Kramkov and Schachermayer [15] that AE (u) equals the
+
infimum of all realnumbers γ >0 for which there exists some x>0 such that, for allλ 1
≥
and all x x,
≥ u(λx) λγu(x) (2.5)
≤
holds.2 From the proof of Lemma 6.3 of Kramkov and Schachermayer [15], it is clear
that this characterisation of AE (u) also holds true if u is not concave (but continuously
+
differentiable). As this latter formulation (2.5) makes sense for possibly non-differentiable
and non-concave u and arbitrary γ > 0 as well, we follow Carassus and Rásonyi [5] and
define the asymptotic elasticity at + of u as
∞
AE (u),inf γ >0 : x 0 s.t. for a.e. ω,u(λx,ω) λγu(x,ω), λ 1, x x ,
+
{ ∃ ≥ ≤ ∀ ≥ ∀ ≥ }
with the usual convention that the infimum of the empty set is + .
∞
Hence, using this generalizednotion of asymptotic elasticity, we see that condition (2.4)
holds if either AE (u)< + , or u is bounded above by some integrable random constant
+
∞
C 0 a.s. and Assumption 2.7 holds. Indeed, in the first case this is trivial since (2.4) is
≥
implied by AE (u)<γ. In the second case, taking x,0, we get that for every x x and
+
≥
for every λ 1,
≥
u(λx,ω) C(ω) λγC(ω)=λγu(x,ω)+λγ[C(ω) u(x,ω)]
≤ ≤ −
λγu(x,ω)+λγ[C(ω) u(0,ω)],
≤ −
whichpermitsustodefinec(ω),[C(ω) u(0,ω)]+. Asc(ω) C(ω)+u−(0,ω),Assumption
− ≤
2.7 shows that E [c] < + . Note that in this case E [u+(x, )] = E [u+(0, )] E [C] <
P P P P
∞ · · ≤
+
∞
Wenotethat,ifuisdeterministic,concaveandboundedabove,thenAE (u) 0(again
+
≤
byKramkovandSchachermayer[15,Lemma6.1]),butthisfailsinthenon-concavecase. As
2Tobe precise, inthe cited lemmathere is strict inequalityin(2.5)and itisrequired to holdfor λ>1
only. Aseasilyseen,itworks alsoforourversion.
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Non-concaveutility maximisation on the positive real axis in discrete time
wewillsee,inExample6.1below,Assumption2.10holdstrue,buttheasymptoticelasticity
isequalto+ . Thisshowsthathavingfinite asymptoticelasticity,despite being sufficient,
∞
is not a necessary condition for a function to verify Assumption 2.10.
We immediately get that AE (u) < + (and hence Assumption 2.10 holds) provided
+
∞
that u is deterministic, continuously differentiable and there exists some p>0 such that
u′(x) u′(x)
0<liminf limsup <+ .
x→+∞ xp ≤ x→+∞ xp ∞
Indeed,ifthe aboveconditionistrueforu,thenontheonehanditispossibletofind m>0
for which there exists some x > 0 such that u′(x) > mxp for all x x. But this implies
≥
that, for all x x,
≥
x xp+1 xp+1
u(x) u(x)= u′(y) dy m − .
− ≥ p+1
Zx
On the other hand, we can find M > 0 for which there is x > 0 such that u′(x) < Mxp
M
for all x x . Defining xˇ,max x,x >0, noticing that we may assume that u(x)>0
M M
≥ { }
without loss of generality, and combining the preceding inequalities finally gives
xu′(x) Mxp+1
u(x) ≤ m (xp+1 xp+1)+u(x)
p+1 −
for all x xˇ, therefore
≥ xu′(x)
limsup <+ .
u(x) ∞
x→+∞
Inparticular,ifu′(x)isasymptoticallyequivalenttoapowerfunction(thatis,u′(x)/xp 1,
→
as x + ) then Assumption 2.10 holds. A multitude of piecewise concave or “S-shaped”
→ ∞
functions (not only piecewise power functions) can be accomodatedin this way,such as the
ones considered in Berkelaar,Kouwenberg and Post [1]; Jin and Zhou [11].
Atlast,supposethatuistheutilityofExample2.9. Iftheconditionsbelowaresatisfied:
(i) esssupB <+ ;
∞
(ii) thereexistrealnumbersγ >0,x>0andC 0suchthat,forallλ 1andallx x,
≥ ≥ ≥
u(λx) λγu(x)+λγC;
e ≤ e
(iii) the function uiscontinuouslydeifferentiableeonits domain,andtherearerealnumbers
K >0 and x>0 such that, for all x x,
≥
e
u′(x) K;
b b ≤
then u fullfills Assumption 2.10. Indeed, seteting x,max x,x +esssupB >0 yields
{ }
B(ω) B(ω)
u(λx,ω)=u λ x λγu x +eλbγC
− λ ≤ − λ
(cid:18) (cid:20) (cid:21)(cid:19) (cid:18) (cid:19)
1
e λγue(x B(ω))+λγKB(ω) 1 +λγC
≤ − − λ
(cid:18) (cid:19)
fora.e.ω,simultaneouslyforallλ 1andex x. Notethatu+(x, ) u˜+(x)andthelatter
≥ ≥ · ≤
is deterministic. Hence, choosing c , K esssupB +C (which is constant, thus trivially
integrable)gives the claimed result. We conclude by pointing out that any funcion u which
is concave for sufficiently large x satisfies the conditions (ii), (iii) above.
We may now deduce the following auxiliary result, which provides an estimateefor all
x 0, and not only for x x.
≥ ≥
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Non-concaveutility maximisation on the positive real axis in discrete time
Lemma 2.12. Under Assumption 2.10 there is a random variable C 0 a.s. such that
≥
E [C]<+ and, for a.e. ω,
P
∞
u+(λx,ω) λγu+(x,ω)+λγC(ω) (2.6)
≤
simultaneously for all λ 1 and for all x 0.
≥ ≥
Proof. See Appendix 6.
3 Main results
Theoptimalportfolioproblemconsistsinchoosingthe“best” investmentinthegivenassets:
the one which maximises the expected utility from terminal wealth.
Definition 3.1. Let Assumption 2.7 be in force. Given any x 0, the non-concave
0
≥
portfolio problem with initialwealthx onafinite horizonT is tofind φ∗ Ψ(x )suchthat
0 0
∈
v∗(x ),sup E u Πφ(), : φ Ψ(x ) =E u Πφ∗(), . (3.1)
0 P T · · ∈ 0 P T · ·
n h (cid:16) (cid:17)i o h (cid:16) (cid:17)i
We call φ∗ an optimal strategy.
Remark 3.2. (i) Note that, due to Assumption 2.7,
E u− Πφ(), E u−(0, ) <+ ,
P T · · ≤ P · ∞
h (cid:16) (cid:17)i (cid:2) (cid:3)
and the expectations in (3.1) above exist (though they may be infinite). It is also
immediatetocheckthat,thestrategyφ 0isinΨ(x )forallx 0,sothesupremum
0 0
≡ ≥
is taken over a non-empty set. In particular, v∗(x ) E [u(x , )] > , under
0 P 0
≥ · −∞
Assumption 2.7.
(ii) Onemayinquirewhytheexistenceofanoptimalφ∗ isimportantwhentheexistenceof
ε-optimalstrategiesφε (i.e., onesthat are ε-closeto the supremumoverallstrategies)
is automatic, for all ε>0.
Firstly, non-existence of an optimal strategy φ∗ usually means that an optimiser se-
quence φ1/n; n N shows a behaviour which is practically infeasible and counter-
∈
intuitive (see Example 7.3 of Rásonyi and Stettner [18]).
(cid:8) (cid:9)
Secondly, existence of φ∗ normally goes together with some compactness property
which would be needed for the constructionof eventualnumericalschemes to find the
optimiser.
Here comes the main result of the present paper. It says that the optimisation problem
(3.1) admits a solution.
Theorem 3.3. LetAssumptions2.1, 2.7 and 2.10 hold true. Assumefurtherthat, for every
x [0,+ ),
0
∈ ∞
v∗(x )<+ . (3.2)
0
∞
Then, for each x [0,+ ), there exists a strategy φ∗ Ψ(x ) satisfying
0 0
∈ ∞ ∈
E u Πφ∗(), =v∗(x ). (3.3)
P T · · 0
h (cid:16) (cid:17)i
Proof. The proof will be given in Section 5, after appropriate preparations. We give here a
brief description. A dynamic programming technique will be applied. This will allow us to
split the original problem into several sub-problems involving a random utility function U
t
at time t. At each time step, we will find a one-step optimal solution based on the natural
compactness provided by Lemma 4.7 below and on the random subsequence technique of
Page 7/23
Non-concaveutility maximisation on the positive real axis in discrete time
Kabanov and Stricker [13] (see Sections 4 and 6). Furthermore, we will prove that certain
crucial properties of U , such as continuity and the growth condition (2.6), are preserved
t
for the next iteration (i.e. for U ). These are the most involved arguments of the present
t−1
paper. Finally, we shall paste together the one-step maximisers in a natural way to get a
maximiser in (3.2).
Remark 3.4. Wewouldliketostressthat,sinceAssumption2.7isinforce,thewell-posedness
condition (3.2) is actually equivalent to the apparently stronger one
sup E u+ Πφ(), <+ . (3.4)
P T · · ∞
φ∈Ψ(x0)
h (cid:16) (cid:17)i
To see this, we recall that Πφ 0 a.s. for every φ Ψ(x ), hence
T ≥ ∈ 0
sup E u+ Πφ(), v∗(x )+ sup E u− Πφ(), v∗(x )+E u−(0, ) <+ .
P T · · ≤ 0 P T · · ≤ 0 P · ∞
φ∈Ψ(x0) φ∈Ψ(x0)
h (cid:16) (cid:17)i h (cid:16) (cid:17)i (cid:2) (cid:3)
As averysimple,yetimportantexampletowhichtheprecedingtheoremclearlyapplies,
wementionthecasewhereS satisfiesAssumption2.1andu(x,ω)isboundedabovebysome
integrable random constant C(ω), for all x, and satisfies Assumption 2.7. Another relevant
example is given by the following theorem. First, define
W , Y Ξ1: E [Y p]<+ for all p>0 . (3.5)
∈ T P | | ∞
Theorem 3.5. Let Assump(cid:8)tions 2.1, 2.7 and 2.10 hold true with(cid:9)c,u+(x, ) W. Assume
furtherthat ∆S ,1/β W foreveryt 1,...,T ,wheretheβ arether·an∈domvariables
t t t
k k ∈ ∈{ }
figuring in Proposition 2.3. Then, for every x [0,+ ), condition (3.2) is satisfied and
0
∈ ∞
there exists an optimal strategy φ∗ Ψ(x ).
0
∈
Proof. See Section 5.
4 The one-step case
Inthissection,weconsideranF-measurablefunctionY : Ω Rd,andaσ-algebraG F
containing all P-null sets of F. This setting will be applied i→n the multi-step case (see⊆the
subsequent section) with G =F and Y =∆S , for every fixed t 1,...,T .
t−1 t
∈{ }
Keeping in line with the notation of the previous section, we denote by Ξd the family of
all G-measurable functions ξ : Ω Rd.
Moreover, let PY|G : B Rd →Ω [0,1] be the unique (up to a set of measure zero)
regular conditional distribution fo×r Y→given G. By modifying it on a P-null set, we may
and will assume that PY|G((cid:0),ω(cid:1)) is a probability for each ω. Now, for each ω Ω, let
supp PY|G(,ω) represent th·e support of PY|G(,ω) (which exists and is non-emp∈ty), and
· ·
let D(ω) denote the affine hull of supp PY|G(,ω) , that is, D(ω),aff supp PY|G(,ω) .
(cid:0) (cid:1) · ·
We shall also assume the following.
(cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
Assumption 4.1. For a.e. ω Ω, D(ω) is a linear subspace of Rd.
∈
In addition, for every G-measurable random variable H : Ω R satisfying H 0 a.s.
→ ≥
(and also for any constant x 0), define the set
≥
Ξd(H), ξ Ξd: H + ξ,Y 0 a.s. .
∈ h i≥
(cid:8) (cid:9)
Finally, let Ξd denote the family of all functions ξ Ξd such that ξ(ω) D(ω) for a.e.
ω. The notation Ξ˜d(H) is self-explanatory. ∈ ∈
We shallalseoimpose the followingcondition,whichcanbe regardedas one-stepabsence
of arbitrage (cf. Proposition 2.3).
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Non-concaveutility maximisation on the positive real axis in discrete time
Assumption 4.2. There exist G-measurable random variables β,κ>0 a.s. such that
P( ξ,Y β ξ G) κ a.s. (4.1)
h i≤− k k| ≥
for all ξ Ξd. We may and will assume β 1.
∈ ≤
Assumption 4.3. Let the function V : [0,+ ) Ω R satisfy both properties below:
e ∞ × →
(i) for any fixed x [0,+ ), the function V(x, ): Ω R is measurable with respect to
F; ∈ ∞ · →
(ii) for a.e. ω Ω, the function V(,ω): [0,+ ) R is continuous and non-decreasing.
∈ · ∞ →
We shall also need the following integrability conditions.
Assumption 4.4. For every x [0,+ ),
∈ ∞
esssupE V+(x+ ξ(),Y() , ) G <+ a.s. (4.2)
P
h · · i · ∞
ξ∈Ξd(x)
(cid:2) (cid:12) (cid:3)
(cid:12)
Assumption 4.5. The conditional expectation of V−(0, ) : Ω [0,+ ) with respect to
G is finite a.s., i.e., · → ∞
E V−(0, ) G <+ a.s. (4.3)
P
· ∞
Finally, we impose the following(cid:2) growth(cid:12)con(cid:3)dition on V.
(cid:12)
Assumption 4.6. There exists a constant γ > 0 and a random variable C¯ 0 a.s. such
that E C¯ <+ and for a.e. ω, ≥
P
∞
(cid:2) (cid:3) V+(λx,ω) λγV+(x,ω)+λγC¯(ω) (4.4)
≤
simultaneously for all λ 1 and for all x 0.
≥ ≥
Next, we remark that denoting by ξˆ(ω) the orthogonal projection of ξ(ω) on D(ω) for
some ξ Ξd, we have ξˆ Ξd and ξˆ,Y = ξ,Y a.s., the readeris referredto Carassusand
∈ ∈ h i h i
Rásonyi [5, Remark 8] for further details. This means that any portfolio can be replaced
with its projectiononD without changingeither its value or its desirability to the investor.
We now recall that the set of all admissible strategies in D is bounded.
Lemma 4.7. Assume that Assumption 4.2 holds true. Given any x 0, there exists a
0
G-measurable, real-valued random variable K ,x /β x such that, f≥or every x [0,x ]
and for every ξ Ξ˜d(x), we have x0 0 ≥ 0 ∈ 0
∈
ξ K a.s. (4.5)
k k≤ x0
Proof. This is Lemma 2.1 in Rásonyi and Stettner [19].
As for the next lemma, it will allow us to apply the Fatou lemma to a sequence of
conditional expectations tending to the essential supremum in (4.7) below.
Lemma 4.8. Assumethat Assumptions4.1, 4.2, 4.3, 4.4, 4.5 and 4.6 hold true. Given any
x 0, there is a non-negative random variable L′ : Ω R such that E[L′ G] < + a.s.,
an≥d for every ξ Ξ˜d(x) the inequality → | ∞
∈
V+(x+ ξ(),Y() , ) L () (4.6)
x
h · · i · ≤ ·
holds for all x with L , xγ +1 L′, outside a fixed P-null set.
x
Proof. See Section 6. (cid:0) (cid:1)
Now a regular version of the essential supremum is shown to exist.
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Non-concaveutility maximisation on the positive real axis in discrete time
Lemma 4.9. Assume that Assumptions 4.1, 4.2, 4.3, 4.4, 4.5 and 4.6 hold true. There
exists a function G : [0,+ ) Ω R satisfying the two properties below:
∞ × →
(i) the function G(x, ) is a version of esssup E [V(x+ ξ(),Y() , ) G] for each
· ξ∈Ξd(x) P h · · i · |
x [0,+ );
∈ ∞
(ii) forP-a.e.ω Ω,thefunctionG(,ω): [0,+ ) Risnon-decreasingandcontinuous
∈ · ∞ →
on [0,+ ).
∞
Furthermore, given any G-measurable random variable H 0 a.s.,
≥
G(H(), )= esssupE [V(H()+ ξ(),Y() , ) G] a.s. (4.7)
P
· · · h · · i · |
ξ∈Ξd(H)
Proof. See Appendix 6.
Proposition 4.10. Assume that Assumptions 4.1, 4.2, 4.3, 4.4, 4.5 and 4.6 hold true. For
every G-measurable random variable H 0 a.s., there exists ξ(H)() Ξ˜d(H) with
≥ · ∈
G(H(), )=EP V H + ξ(H)(),Y() ,e G a.s. (4.8)
· · · · ·
h (cid:16) D E (cid:17)(cid:12) i
Proof. See Section 6. e (cid:12)
(cid:12)
5 The multi-step case
In this section, we shallfollow Carassusand Rásonyi[5]; RásonyiandStettner [18, 19], and
employ a dynamic programming approach to split the original optimisation problem into a
number of sub-problems at different trading dates. Our goal is to invoke the results of the
preceding section, thus allowing us to obtain an optimal solution at each stage. Combining
them in an appropriate way will yield a globally optimal investment strategy.
Proof of Theorem 3.3. We must prove that some crucial assumptions of Section 4 are pre-
served at each time step. So let us start by defining
U (x,ω),u(x,ω), x 0, ω Ω.
T
≥ ∈
We wish to apply the results of Section 4 with Y ,∆S , G ,F and V ,U .
T T−1 T
(i) Since Assumption 2.1 holds by hypothesis, Theorem 3 in Jacod and Shiryaev [10]
implies that the affine space D (ω) is a linear subspace of Rd a.s., therefore Assump-
T
tion 4.1 is verified. It follows from Proposition 2.3 that Assumption 4.2 holds as well.
(ii) WenotefurtherthatAssumption4.3isalsotruefromthedefinitionofarandomutility
function.
(iii) We now claim that Assumption 4.4 is satisfied. In order to show this, fix an arbitrary
x 0. First we check that E [U (x+ ξ(),∆S () , ) F ] is well-defined and
P T T T−1
≥ h · · i · |
finite a.s. for any given ξ Ξd (x). It is straightforward to see that the Rd-valued
∈ T−1
process defined by
ξ, if t=T,
φ ,
ξ t 0, otherwise,
(cid:26)
(cid:0) (cid:1)
is a portfolio in Ψ(x), with
E u+ Πφξ(), F =E u+(x+ ξ(),∆S () , ) F
P T · · T−1 P h · T · i · T−1
(cid:20) (cid:18) (cid:19)(cid:12) (cid:21)
(cid:12)(cid:12)(cid:12) =EP(cid:2)UT+(x+hξ(·),∆ST(·)i,·)(cid:12)(cid:12) FT−1(cid:3) a.s.
(cid:2) (cid:12) (cid:3)
(cid:12)
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