Optimal investment with possibly non-concave utilities and no-arbitrage: a measure theoretical approach Miklós R ´ asonyi

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Optimal investment with possibly non-concave utilities and no-arbitrage: a measure theoretical approach Miklós

R ´ asonyi

Romain Blanchard, Laurence Carassus, Miklos Rasonyi

To cite this version:

Romain Blanchard, Laurence Carassus, Miklos Rasonyi. Optimal investment with possibly non- concave utilities and no-arbitrage: a measure theoretical approach Miklós R ´ asonyi. Mathematical Methods of Operations Research, Springer Verlag, 2018. �hal-01883419�

Optimal investment with possibly non-concave utilities and no-arbitrage: a measure theoretical approach

Romain Blanchard

LMR, Universit´e Reims Champagne-Ardenne (URCA), France romain.blanchard@etudiant.univ-reims.fr

Laurence Carassus

Research Center, Pˆole Universitaire de Vinci and LMR, URCA, France laurence.carassus@univ-reims.fr

Mikl´os R ´asonyi

MTA Alfr´ed R´enyi Institute of Mathematics, Hungary rasonyi@renyi.mta.hu

January 20, 2018

Abstract

We consider a discrete-time financial market model with finite time horizon and investors with utility functions d efined on the non-negative half-line. We allow these functions to be random, non- concave and non-smooth. We use a dynamic programming framework together with measurable selection arguments to establish both the characterization of the no-arbitrage property for such markets and the existence of an optimal portfolio strategy for such investors.

Key words: no-arbitrage condition ; non-concave utility functions; optimal investment AMS 2000 subject classification: Primary 93E20, 91B70, 91B16 ; secondary 91G10, 28B20

1 Introduction

We consider investors trading in a multi-asset and discrete-time financial market. We revisit two classical problems: the characterization of no arbitrage and the maximisation of expected utility from the terminal wealth of an investor. There are strong connections between them: on one hand, no- arbitrage is a necessary condition for the existence of optimal strategies, see e.g. Pliska [1997], on the other hand, utility maximization is a means to construct dual objects (e.g. equivalent martingale measures), see Rogers [1994] in the frictionless case and Sass and Smaga [2014] for the case under transaction costs.

We consider a possibly non-concave and non-smooth random utility function U, defined on the non-negative half-line. For instance,U may be “S-shaped” (i.e. convex up to a certain level of wealth and concave beyond that) but our results apply to a broader class of utility functions, e.g. to piecewise concave ones. We provide sufficient conditions which guarantee the existence of an optimal strategy.

Such problems lie outside mainstream optimization theory and constitute an area of intensive study in recent years, see e.g. Bensoussan et al. [2015], He and Zhou [2011], Jin and Zhou [2008], Carlier and Dana [2011].

We are working in the setting of Carassus et al. [2015] and extend its results in several direc- tions. First, we remove a restrictive integrability hypothesis [Carassus et al., 2015, Assumption 2.7]:

EU⁻(·,0) < ∞. Recall that the non-random version of this assumption: U(0) < −∞ does not hold

true for the most widely used (concave) utility functions (logarithmic,U(x) =−x^α forα <0). Hence Carassus et al. [2015] excluded some of the most important cases. Note also thatU(0) =−∞has the rather natural interpretation of an investor being infinitely averse of bankruptcy. We propose here an alternative, weaker integrability condition.

Second, we introduce a weaker version of the asymptotic elasticity assumption (see Assumption 4.10 below). In particular, Assumption 4.10 holds true for concave functions (see Remark 4.15) and therefore our result extends the one obtained in R ´asonyi and Stettner [2006] to random utility func- tions.

Next, instead of using some Carath´eodory utility function U as in Carassus et al. [2015] (i.e U measurable inω and continuous inx), we considerU which is measurable inω and only upper semi- continuous (usc in the rest of the paper) inx. Allowing non-continuousU is unusual in the financial mathematics literature (see, however, Sass [2005]) but it is common in optimization theory. We high- light that this generalisation has a potential to model the investor’s behaviour which can change suddenly after reaching a desired wealth level. Such a change can be expressed by a jump ofU.

Next, we do not require that the value function is finite for all initial wealth as it is usually postulated in the maximisation of expected utility theory; instead we only assume the less restrictive and more tractable Assumption 4.7.

Finally, in contrast to the preceding literature, we do not need complete sigma-algebras for our arguments to work. This is an interesting technical aspect that can be useful in robust utility maxi- mization where multiple probabilities are considered which lead to varying families of null sets.

We use methods that are different from the ones in R ´asonyi and Stettner [2005], R ´asonyi and Stettner [2006], Carassus and R ´asonyi [2016] and Carassus et al. [2015], where similar multistep problems were treated. To solve our optimisation problem, we use dynamic programming as in those papers but here we propose a different approach which provides simpler proofs. As in Nutz [2016], we consider first a one period case where strategies are (deterministic) elements ofR^d. Then we use dynamic programming and measurable selection arguments, namely the Aumann Theorem (see, for example, Corollary 1 in Sainte-Beuve [1974]) to solve the multi-period problem. We will also often rely on the Projection Theorem (see for example Theorem 3.23 in Castaing and Valadier [1977]). Our modelisation of(Ω,F,F, P)is more general than in Nutz [2016] since, as opposed to Nutz [2016], we consider only one probability measure so we don’t have to postulate Borel spaces or analytic sets.

The same methodology is used to reprove classical results on the characterization of no-arbitrage (see R ´asonyi and Stettner [2005] and Jacod and Shiryaev [1998]) in our context of possibly incomplete sigma-algebras, we present this in Section 3 below.

Our paper can also be regarded as a review of problems in non-concave optimisation which arise in finance so detailed proofs are given. The paper is organized as follows: in section 2 we introduce our setup; section 3 contains the results on no-arbitrage; section 4 presents the main theorem on expected utility maximisation from terminal wealth; section 5 establishes the existence of an optimal strategy for the one period case; we prove our main theorem on utility maximisation in section 6. Finally, section 7 collects some technical results and proofs as well as elements about the measurability of random sets.

2 Set-up

Fix a time horizon T ∈ Nand let (Ωt)1≤t≤T be a sequence of non-empty sets equipped with the re- spective sigma-algebras (G_t)1≤t≤T. For t = 1, . . . , T, we denote by Ω^t the t-fold Cartesian product Ω^t= Ω₁×. . .×Ω_t.An element ofΩ^twill be denoted byω^t= (ω₁, . . . , ω_t)for(ω₁, . . . , ω_t)∈Ω₁×. . .×Ω_t. We also denote byF_tthe product sigma-algebra on Ω^t,F_t =G₁⊗. . .⊗ G_t. For the sake of simplicity, we setΩ⁰ := {ω₀}and F₀ =G₀ = {∅,Ω⁰}. We will omit the dependency inω0 in the rest of the paper.

We denote byFthe filtration(F_t)0≤t≤T.

LetP1be a probability measure onF₁andqt+1be a stochastic kernel onG_t+1×Ω^tfort= 1, . . . , T−1.

Namely we assume that for allω^t ∈ Ω^t, B ∈ G_t+1 → qt+1(B|ω^t) is a probability on G_t+1 and for all B ∈ G_t+1,ω^t ∈Ω^t →q_t+1(B|ω^t)isF_t-measurable. Here we DO NOT assume thatG₁ contains the null sets ofP₁ or thatG_t+1contains the null sets of q_t+1(.|ω^t)for allω^t∈Ω^t. Then we define forA∈ F_tthe probabilityPtby Fubini’s Theorem for stochastic kernels (see Lemma 7.1) via

P_t(A) = Z

Ω1

Z

Ω2

· · · Z

Ωt

1_A(ω₁, . . . , ω_t)q_t(dω_t|ω^t−1)· · ·q₂(dω₂|ω¹)P₁(dω₁). (1) Finally, (Ω,F,F, P) := (Ω^T,F_T,F, PT) will be our basic filtered probability space. The expectation underP_twill be denoted byE_Pt ; whent=T, we simply writeE.

We introduceF_t^T, the filtration onΩ^T associated toF_t, defined by F_t^T =G₁⊗. . .⊗ G_t⊗ {∅,Ω_t+1}. . .⊗ {∅,Ω_T}.

LetΞ^T_t be the set ofF_t^T-measurable random variables fromΩ^T toR^d. LetXt: Ω^T →Ωt,Xt(ω1, . . . , ω_T) = ω_t be the coordinate mapping corresponding tot. ThenF_t^T =σ(X₁, . . . , X_t). Soh ∈Ξ^T_t if and only if there exists some g ∈ Ξt such that h = g(X1, . . . , Xt). This implies that h(ω^T) = g(ω^t). For ease of notation we will identifyhandgand alsoF_t,F_t^T,ΞtandΞ^T_t.

Remark 2.1 If we choose Polish spaces forΩt,t = 1, . . . , T then any probability measure P on Ωcan be decomposed in the form of (1) (see Dellacherie and Meyer [1979] III.70-7).

The positive (resp. negative) part of a number or a random variable X is denoted by X⁺ (resp.

X⁻). We will also writef^±(X)for(f(X))^±for any random variableXand (possibly random) function f.

In the rest of the paper we will use generalised integral: for somef_t: Ω^t→R∪ {±∞},F_t-measurable, such thatR

Ω^tf_t⁺(ω^t)Pt(dω^t)<∞orR

Ω^tf_t⁻(ω^t)Pt(dω^t)<∞, we define Z

Ω^t

f_t(ω^t)P_t(dω^t) :=

Z

Ω^t

f_t⁺(ω^t)P_t(dω^t)− Z

Ω^t

f_t⁻(ω^t)P_t(dω^t)∈R∪ {−∞,∞}.

We refer to Lemma 7.1, Definition 7.2 and Proposition 7.3 of the Appendix for more details and prop- erties. In particular, ifft is non-negative or ifft is such thatR

Ω^tf_t⁺(ω^t)Pt(dω^t) < ∞ (this will be the two cases of interest in the paper) we can apply Fubini’s Theorem¹to get

Z

Ω^t

ft(ω^t)Pt(dω^t) = Z

Ω1

Z

Ω2

· · · Z

Ωt

ft(ω1, . . . , ωt)qt(dωt|ω^t−1)· · ·q2(dω2|ω¹)P1(dω1).

We will denote by N_Pt the set of P_t negligible sets of Ω^t i.e N_Pt = {N ⊂ Ω^t, ∃M ∈ F_t, N ⊂ M andPt(M) = 0}. LetF_t ={A∪N, A∈ F_t, N ∈ N_Pt}and Pt(A∪N) = Pt(A) forA∪N ∈ F_t. Then it is well known thatPt is a measure onF_twhich coincides with Pton F_t,(Ω^t,F_t, Pt) is a complete probability space andP_trestricted toN_Ptis zero. Fort= 0, . . . , T−1, letΞ_tbe the set ofF_t-measurable random variables mappingΩ^ttoR^d.

The following lemma makes the link between conditional expectation and kernel.

Lemma 2.2 Let0≤s≤t≤T. Leth∈Ξtsuch thatR

Ω^th⁺dPt<∞then E(h|F_s) = ϕ(X₁, . . . , X_s)P_s–a.s., where

ϕ(ω1, . . . , ωs) = Z

Ωs+1×...×Ωt

h(ω1, . . . , ωs, ωs+1, . . . ωt)qt(ωt|ω^t−1). . . qs+1(ωs+1|ω^s).

1From now, we call Fubini’s theorem the Fubini theorem for stochastic kernel (see eg Lemma 7.1, Proposition 7.3).

Proof. For the sake of completeness, the proof is reported in Section 7.3 of the Appendix. 2 Let {S_t, 0 ≤ t ≤ T} be a d-dimensional F_t-adapted process representing the price of d risky securities in the financial market in consideration. There exists also a riskless asset for which we assume a constant price equal to1. The notation∆St:=St−St−1 will often be used. Ifx, y∈R^dthen the concatenationxy stands for their scalar product. The symbol| · |denotes the Euclidean norm on R^d(or onR).

Trading strategies are represented by d-dimensional predictable processes (φt)1≤t≤T, where φⁱ_t denotes the investor’s holdings in assetiat timet; predictability means thatφt∈Ξt−1. The family of all predictable trading strategies is denoted byΦ.

We assume that trading is self-financing. As the riskless asset’s price is constant 1, the value at timetof a portfolioφstarting from initial capitalx∈Ris given by

V_t^x,φ =x+

t

X

i=1

φi∆Si.

Example 2.3 For a better understanding of the present, abstract setting one may consider a simple trinomial tree model where Ω_t = {−1,0,1}, G_t := 2^Ωt for each t = 1, . . . , T. Let S_t := Pt

j=1X_j, t= 1, . . . , T where theX_j are the respective coordinate mappings. The kernelsq_t+1(·|ω^t)then assign probabilities to the 3 possible outcomes{−1,0,1} conditionally to the past pathω^t = (ω1, . . . , ωt). P1

is just a probability on{−1,0,1}. Strategies can be identified with functionsφt: Ω^t−1 →Rexpressing the holding in the risky asset in the interval(t−1, t].

3 No-arbitrage condition

The following absence of arbitrage condition or NA condition is standard, it is equivalent to the ex- istence of a risk-neutral measure in discrete-time markets with finite horizon, see e.g. Dalang et al.

[1990].

(NA)IfV_T^0,φ≥0P-a.s. for someφ∈ΦthenV_T^0,φ= 0P-a.s.

Definition 3.1 Let(Ω,F)be a measurable space and(T,T) a topological space. A random setR is a set valued function that assigns to eachω ∈ Ωa subsetR(ω) ofT. We writeR : Ω T. We say that R is measurable if for any open setO ∈T {ω ∈ Ω, R(ω)∩O 6=∅} ∈ F. The graph ofR is defined as Gr(R) :={(ω, t)∈Ω×T : t∈R(ω)}.

We now provide classical tools and results about the (NA) condition and its “concrete” local charac- terization, see Proposition 3.6, that we will use in the rest of the paper. We start with the setD^t+1(see Definition 3.1) whereD^t+1(ω^t)is the smallest affine subspace ofR^dcontaining the support of the dis- tribution of∆S_t+1(ω^t, .)underq_t+1(.|ω^t). IfD^t+1(ω^t) =R^dthen, intuitively, there are no redundant as- sets. Otherwise, forφt+1 ∈Ξt, one may always replaceφt+1(ω^t,·)by its orthogonal projectionφ^⊥_t+1(ω^t,·) on D^t+1(ω^t) without changing the portfolio value since φ_t+1(ω^t)∆S_t+1(ω^t,·) = φ^⊥_t+1(ω^t)∆S_t+1(ω^t,·), q_t+1(·|ω^t) a.s., see Remark 5.2 and Lemma 7.16 below as well as Remark 9.1 of F¨ollmer and Schied [2002].

Definition 3.2 Let0≤t≤T be fixed. We define the random setDe^t+1: Ω^tR^dby De^t+1(ω^t) :=\ n

A⊂R^d, closed, q_t+1 ∆St+1(ω^t, .)∈A|ω^t) = 1o

. (2)

For ω^t ∈ Ω^t,De^t+1(ω^t) ⊂ R^dis the support of the distribution of∆S_t+1(ω^t,·)underq_t+1(·|ω^t). We also define the random setD^t+1 : Ω^tR^dbyD^t+1(ω^t) :=Aff

De^t+1(ω^t)

, where Aff denotes the affine hull of a set.

Lemma 3.3 Let 0 ≤ t ≤ T be fixed. Then De^t+1 and D^t+1 are both non-empty, closed-valued and F_t-measurable random sets (see Definition 3.1). In particular,Graph(D^t+1)∈ F_t⊗ B(R^d).

Proof. The proof is reported in Section 7.3 of the Appendix. 2

In Lemma 3.4, which is used in the proof of Lemma 3.5 for projection purposes, we obtain a well- known result : for ω^t ∈ Ω^t fixed and under a local version of (NA), D^t+1(ω^t) is a vector subspace of R^d(see for instance Theorem 1.48 of F¨ollmer and Schied [2002]). Then in Lemma 3.5 we prove that under the (NA) assumption, forPt almost allω^t,D^t+1(ω^t) is a vector subspace ofR^d. We also provide a local version of the (NA) condition (see (4)).

Lemma 3.4 Let0≤t≤T andω^t∈Ω^tbe fixed. Assume that for allh∈D^t+1(ω^t)\{0}

qt+1(h∆St+1(ω^t,·)≥0|ω^t)<1.

Then0∈D^t+1(ω^t)and the setD^t+1(ω^t)is actually a vector subspace ofR^d.

Proof. The proof is reported in Section 7.3 of the Appendix. 2

Lemma 3.5 Assume that the (NA) condition holds true. Then for all 0 ≤ t ≤ T −1, there exists a full measure set Ω^t_{N A1} such that for allω^t ∈ Ω^t_{N A1}, 0 ∈D^t+1(ω^t),i.e D^t+1(ω^t) is a vector space of R^d. Moreover, for allω^t∈Ω^t_{N A1} and allh∈R^dwe get that

q_t+1(h∆S_t+1(ω^t,·)≥0|ω^t) = 1⇒q_t+1(h∆S_t+1(ω^t,·) = 0|ω^t) = 1. (3) In particular, ifω^t∈Ω^t_{N A1}andh∈D^t+1(ω^t)we obtain that

qt+1(h∆St+1(ω^t,·)≥0|ω^t) = 1⇒h= 0. (4) Proof. Let0≤t≤T be fixed. We introduce the following random set:

Π^t:=

ω^t∈Ω^t, ∃h∈D^t+1(ω^t), h6= 0, q_t+1(h∆St+1(ω^t,·)≥0|ω^t) = 1 .

Assume for a moment thatΠ^t∈ F_tand that Pt(Π^t) = 0. Letω^t ∈Ω^t\Π^t. The fact that0 ∈D^t+1(ω^t) directly follows from the definition ofΠ^t and from Lemma 3.4. We now prove (3). Leth ∈R^dbe fixed such that q_t+1(h∆S_t+1(ω^t,·) ≥ 0|ω^t) = 1. If h = 0 there is nothing to prove. If h ∈ D^t+1(ω^t)\ {0}, ω^t ∈ Π^t which is impossible. Now if h /∈ D^t+1(ω^t) and h 6= 0, let h⁰ be the orthogonal projection of h onD^t+1(ω^t). We first show thatqt+1(h⁰∆St+1(ω^t,·) ≥ 0|ω^t) = 1. Indeed, if it were not the case the setB :={h⁰∆St+1(ω^t,·)<0}would verifyqt+1(B|ω^t)>0. SetL^t+1(ω^t) := D^t+1(ω^t)⊥

.Using Lemma 7.16, since(h−h⁰)∈L^t+1(ω^t),q_t+1(A|ω^t) = 1whereA:={(h−h⁰)∆S_t+1(ω^t,·) = 0}. We would therefore obtain thatqt+1(A∩B|ω^t)>0which implies thatqt+1(h∆St+1(ω^t, .)≥0|ω^t)<1, a contradiction. Thus qt+1(h⁰∆St+1(ω^t,·)≥0|ω^t) = 1. Ifh⁰6= 0, ash⁰ ∈D^t+1(ω^t),ω^t∈Π^twhich is again a contradiction. Thus h⁰ = 0and asA∩ {h⁰∆S_t+1(ω^t,·) = 0} ⊂ {h∆S_t+1(ω^t,·) = 0},q_t+1(h∆S_t+1(ω^t,·) = 0|ω^t) = 1.

AsΩ^t\Π^t ∈ F_t there existsΩ^t_{N A1} ∈ F_t andN^t ∈ N_Pt such thatΩ^t\Π^t= Ω^t_{N A1} ∪N^t and Pt(Ω^t_{N A1}) = Pt(Ω^t\Π^t) = 1. SinceΩ^t_{N A1} ⊂Ω^t\Π^t, it follows that for allω^t∈Ω^t_{N A1},0∈D^t+1(ω^t)and for allh∈R^d, (3) holds true.

We prove (4). Assume now thatω^t∈Ω^t_{N A1} andh∈D^t+1(ω^t)are such thatqt+1(h∆St+1(ω^t,·)≥0|ω^t) = 1. Using (3) and Lemma 7.16 we get thath∈L^t+1(ω^t). Soh∈D^t+1(ω^t)∩L^t+1(ω^t) ={0}and (4) holds true.

It remains to prove thatΠ^t∈ F_tandPt(Π^t) = 0. We introduce the random set H^t(ω^t) :=

h∈D^t+1(ω^t), h6= 0, q_t+1(h∆St+1(ω^t,·)≥0|ω^t) = 1 .

ThenΠ^t=

ω^t∈Ω^t, H^t(ω^t)6=∅ =proj_|ΩtGraph(H^t).

We prove now thatGraph(H^t)∈ F_t⊗ B(R^d). Indeed, we can rewrite that Graph(H^t) =Graph(D^t+1)\ n

(ω^t, h)∈Ω^t×R^d, q_t+1(h∆S_t+1(ω^t,·)≥0|ω^t) = 1o \

Ω^t×R^d\{0}

. As from Lemma 7.7,

(ω^t, h)∈Ω^t×R^d, qt+1(h∆St+1(ω^t,·)≥0|ω^t) = 1 ∈ F_t ⊗ B(R^d) and from Lemma 3.3, Graph(D^t+1) ∈ F_t⊗ B(R^d), we obtain that Graph(H^t) ∈ F_t ⊗ B(R^d). The Projection Theorem applies and Π^t = {H^t 6= ∅} = proj_|ΩtGraph(H^t) ∈ F_t. From the Aumann Theorem there exists aF_t-measurable selectorht+1 : Π^t→ R^dsuch that ht+1(ω^t)∈H^t(ω^t)for everyω^t∈Π^t. We now extendh_t+1onΩ^tby setting h_t+1(ω^t) = 0forω^t∈Ω^t\Π^t. It is clear thath_t+1 remainsF_t-measurable.

Applying Lemma 7.8, there exists h_t+1 : Ω^t → R^d which isF_t-measurable and satisfies h_t+1 = h_t+1 Pt-almost surely. Then if we set

ϕ(ω^t) =q_t+1(h_t+1(ω^t)∆S_t+1(ω^t, .)≥0|ω^t), ϕ(ω^t) =q_t+1(h_t+1(ω^t)∆S_t+1(ω^t, .)≥0|ω^t),

we get from Proposition 7.7 that ϕ is F_t-measurable and from Proposition 7.4 iii) that ϕ is F_t- measurable. Furthermore as{ω^t∈Ω^t, ϕ(ω^t)6=ϕ(ω^t)} ⊂ {ω^t∈Ω^t, ht(ω^t)6=ht+1(ω^t)},ϕ=ϕ Pt-almost surely. This implies that R

Ω^tϕdP_t = R

Ω^tϕdP_t. Now we define the predictable process (φ_t)1≤t≤T by φ_t+1 =h_t+1 andφ_i = 0fori6=t+ 1. Then

P(V_T^0,φ≥0) = P(h_t+1∆S_t+1 ≥0) =P_t+1(h_t+1∆S_t+1≥0)

= Z

Ω^t

ϕ(ω^t)P_t(dω^t) = Z

Ω^t

ϕ(ω^t)P_t(dω^t)

= Z

Π^t

qt+1 ht(ω^t)∆St+1(ω^t,·)≥0|ω^t

Pt(dω^t) + Z

Ω^t\Π^t

qt+1 0×∆St+1(ω^t,·)≥0|ω^t

Pt(dω^t)

= Pt(Π^t) +Pt(Ω^t\Π^t) = 1,

where we have used that if ω^t ∈ Π^t, ht+1(ω^t) ∈ H^t(ω^t) and otherwise ht+1(ω^t) = 0. With the same arguments we obtain that

P(V_T^0,φ>0) =Pt(ht+1∆St+1 >0)

= Z

Π^t

qt+1 ht+1(ω^t)∆St+1(ω^t,·)>0|ω^t

Pt(dω^t) + Z

Ω^t\Π^t

qt+1 0>0|ω^t

Pt(dω^t)

= Z

Π^t

qt+1 ht+1(ω^t)∆St+1(ω^t,·)>0|ω^t

Pt(dω^t).

Letω^t∈Π^tthenq_t+1 h_t+1(ω^t)∆S_t+1(ω^t,·)>0|ω^t

>0. Indeed, if it is not the case then q_t+1 h_t+1(ω^t)∆S_t+1(ω^t,·)≤0|ω^t

= 1. Asω^t∈Π^t,h_t+1(ω^t)∈D^t+1(ω^t)andq_t+1 h_t+1(ω^t)∆S_t+1(ω^t,·)≥0|ω^t

= 1, Lemma 7.16 applies andht+1(ω^t)∈L^t+1(ω^t). Thus we get thatht+1(ω^t)∈L^t+1(ω^t)∩D^t+1(ω^t) ={0}, a contradiction. So ifPt(Π^t)>0we obtain that P(V_T^0,φ >0)>0. This contradicts the (NA) condition

and we obtainP_t(Π^t) = 0, the required result. 2

Similarly as in R ´asonyi and Stettner [2005] and Jacod and Shiryaev [1998], we prove a “quantitative”

characterization of (NA).

Proposition 3.6 Assume that the (N A) condition holds true and let 0 ≤ t ≤ T. Then there exists Ω^t_{N A}∈ F_twithP_t(Ω^t_{N A}) = 1andΩ^t_{N A}⊂Ω^t_{N A1} (see Lemma 3.5 for the definition ofΩ^t_{N A1}) such that for allω^t∈Ω^t_{N A}, there exists aF_t-measurableω^t→αt(ω^t)∈(0,1]such that for allh∈D^t+1(ω^t)

q_t+1 h∆S_t+1(ω^t,·)≤ −α_t(ω^t)|h||ω^t

≥α_t(ω^t). (5)

Proof. Letω^t∈Ω^t_{N A1}be fixed (Ω^t_{N A1} is defined in Lemma 3.5).

Step 1 : Proof of (5).Introduce the following set forn≥1 A_n(ω^t) :=

h∈D^t+1(ω^t), |h|= 1, q_t+1

h∆S_t+1(ω^t,·)≤ −1 n|ω^t

< 1 n

. (6)

Let n0(ω^t) := inf{n≥ 1, An(ω^t) = ∅}. Note that ifD^t+1(ω^t) = {0}, thenn0(ω^t) = 1 < ∞. We assume now that D^t+1(ω^t) 6= {0} and we prove by contradiction that n₀(ω^t) < ∞. Assume that n₀(ω^t) =

∞ i.e for all n ≥ 1, An(ω^t) 6= ∅. We thus get hn(ω^t) ∈ D^t+1(ω^t) with |h_n(ω^t)| = 1 and such that qt+1 hn(ω^t)∆St+1(ω^t,·)≤ −_n¹|ω^t

< ¹_n.By passing to a sub-sequence we can assume thathn(ω^t)tends to someh^∗(ω^t)∈D^t+1(ω^t)(recall that the setD^t+1(ω^t)is closed by definition) with|h^∗(ω^t)|= 1. Then {h^∗(ω^t)∆S_t+1(ω^t,·<0} ⊂lim inf_nB_n(ω^t), whereB_n(ω^t) :={h_n(ω^t)∆S_t+1(ω^t,·)≤ −1/n}. Furthermore as1_{lim infnBn(ω}^t₎= lim infn1_Bn(ω^t₎, Fatou’s Lemma implies that

qt+1 h^∗(ω^t)∆St+1(ω^t,·)<0|ω^t

≤ Z

Ωt+1

1_{lim infnBn(ω}^t₎(ωt+1)qt+1(ωt+1|ω^t)

≤lim inf

n

Z

Ωt+1

1_Bn(ω^t₎(ωt+1)qt+1(ωt+1|ω^t) = 0.

This implies that qt+1 h^∗(ω^t)∆St+1(ω^t,·)≥0|ω^t

= 1, and thus from (4) in Lemma 3.5 we get that h^∗(ω^t) = 0which contradicts|h^∗(ω^t)| = 1. Thusn₀(ω^t) < ∞ and we can set for ω^t ∈ Ω^t_{N A1},α_t(ω^t) =

1

n0(ω^t).It is clear thatα_t∈(0,1]. Then for allω^t∈Ω^t_{N A1}, for allh∈D^t+1(ω^t)with|h|= 1, by definition ofA_n0(ω^t₎(ω^t)we obtain

qt+1 h∆St+1(ω^t,·)≤ −α_t(ω^t)|ω^t

≥αt(ω^t). (7)

Step 2 : Measurability.

We now construct a functionα_t which is F_t-measurable and satisfies (5) as well. To do that we use the Aumann Theorem again as in the proof of Lemma 3.5 but this time applied to the random set An: Ω^tR^dwhereAn(ω^t)is defined in (6) ifω^t∈Ω^t_{N A1}andAn(ω^t) =∅otherwise.

We prove thatgraph(A_n)∈ F_t⊗B(R^d). From Lemma 7.7, the function(ω^t, h)→q_t+1 h∆S_t+1(ω^t,·)≤ −_n¹|ω^t isF_t⊗ B(R^d)-measurable. From Lemma 3.3,Graph(D^t+1)∈ F_t⊗ B(R^d)and the result follows from

Graph(An) =Graph(D^t+1)\

Ω^t_{N A1}× {h∈R^d, |h|= 1}

\

(ω^t, h)∈Ω^t×R^d, qt+1

h∆St+1(ω^t,·)≤ −1 n|ω^t

< 1 n

.

Using the Projection Theorem, we get that {ω^t ∈ Ω^t, A_n(ω^t) 6= ∅} ∈ F_t. We now extend n₀ to Ω^t by settingn0(ω^t) = 1ifω^t∈/Ω^t_{N A1}. Then{n₀ ≥1}= Ω^t∈ F_t⊂ F_tand fork >1

{n₀ ≥k}= Ω^t_{N A1}∩ \

1≤n≤k−1

{A_n6=∅} ∈ F_t,

this implies that n₀ and thus α_t is F_t-measurable. Using Lemma 7.8, we get someF_t-measurable function αt such that αt = αt Pt almost surely, i.e there exists M^t ∈ F_t such that Pt(M^t) = 0 and {α_t 6= α_t} ⊂ M^t. We setΩ^t_{N A} := Ω^t_{N A1}T

Ω^t\M_t

. ThenP_t(Ω^t_{N A}) = 1 and asα_t isF_t-measurable it remains to check that (5) holds true.

For ω^t ∈ Ω^t_{N A}, αt(ω^t) = αt(ω^t) (recall that ω^t ∈ Ω^t\Mt) and since ω^t ∈ Ω^t_{N A1}, (7) holds true and consequently (5) as well. It is also clear thatα_t(ω^t)∈(0,1]and the proof is completed. 2 Remark 3.7 The characterization of (NA) given by (5) works only forh∈D^t+1(ω^t). This is the reason why we will have to project the strategyφt+1 ∈ΞtontoD^t+1(ω^t)in our proofs.

Example 3.8 Getting back to Example 2.3, (NA) holds iff, for allt= 1, . . . , T −1andω^t ∈Ω^t, either bothqt+1({1}|ω^t) >0and qt+1({−1}|ω^t) >0or qt+1({0}|ω^t) = 1 hold, and either bothP1({1}) >0 and P1({−1})>0orP1({0}) = 1hold.

4 Utility problem and main result

We now describe the investor’s risk preferences by a possibly non-concave, random utility function.

Definition 4.1 A random utility is any function U : Ω ×R → R∪ {±∞} satisfying the following conditions

• for everyx∈R, the functionU(·, x) : Ω→R∪ {±∞}isF-measurable,

• for allω∈Ω, the functionU(ω,·) : R→R∪ {±∞}is non-decreasing and usc onR,

• U(·, x) =−∞, for allx <0.

We introduce the following notations.

Definition 4.2 For allx≥0, we denote byΦ(x)the set of all strategiesφ∈Φsuch thatP_T(V_T^x,φ(·)≥ 0) = 1 and byΦ(U, x) the set of all strategiesφ ∈ Φ(x) such that EU(·, V_T^x,φ) exists in a generalised sense,i.e.eitherEU⁺(·, V_T^x,φ(·))<∞orEU⁻(·, V_T^x,φ(·))<∞.

Remark 4.3 Under (NA), ifφ∈Φ(x)then we have thatPt(V_t^x,φ(·)≥0) = 1for all1≤t≤T see Lemma 7.17.

We now formulate the problem which is our main concern in the sequel.

Definition 4.4 Letx≥0. Theportfolio optimization problemon a finite horizonT with initial wealth xis

u(x) := sup

φ∈Φ(U,x)

EU(·, V_T^x,φ(·)). (8)

Remark 4.5 Assume that there exists some P-full measure set Ωe ∈ F such that for all ω ∈ Ω,e x → U(ω, x)is non-decreasing and usc on[0,+∞)We setU : Ω×R→R∪ {±∞}

U(ω, x) :=U(ω, x)1

Ω×[0,+∞)e (ω, x) + (−∞)1_{Ω×(−∞,0)}(ω, x).

Then U satisfies Definition 4.1, see Lemma 7.9 for the second item. Moreover, the value function does not change u(x) = sup_φ∈Φ(U,x)EU(·, V_T^x,φ(·)), and if there exists some φ^∗ ∈ Φ(U, x) such that u(x) =EU(·, V_T^x,φ∗(·)), thenφ^∗ is an optimal solution for (8).

Remark 4.6 Let U be a utility function defined only on (0,∞) and verifying for every x ∈ (0,∞), U(·, x) : Ω → R∪ {±∞} is F-measurable and for all ω ∈ Ω, U(ω,·) : (0,∞) → R∪ {±∞} is non- decreasing and usc on(0,∞). We may extendU onRby setting, for allω ∈Ω,U(ω,0) = limx→0U(ω, x) and forx <0,U(ω, x) =−∞. Then, as before,U verifies Definition 4.1 and the value function has not changed.

We now present conditions onU which allow to assert that ifφ∈Φ(x)thenEU(·, V_T^x,φ(·))is well- defined and that there exists some optimal solution for (8).

Assumption 4.7 For allφ∈Φ(U,1),EU⁺

·, V_T^1,φ(·)

<∞.

Assumption 4.8 Φ(U,1) = Φ(1).

Remark 4.9 Assumptions 4.7 and 4.8 are connected but play a different role. Assumption 4.8 guar- antees thatEU

·, V_T^1,φ(·)

is well-defined for allΦ ∈ Φ(1)and allows us to relax Assumption 2.7 of Carassus et al. [2015] on the behavior ofU around0, namely thatEU⁻(·,0)<∞. Then Assumption 4.7 (together with Assumption 4.10) is used to show thatu(x)<∞for allx >0.

In Proposition 6.1, we will show that under Assumptions 4.7, 4.8 and 4.10, EU⁺

·, V_T^x,φ(·)

< ∞ for allx ≥0andφ∈Φ(x). ThusΦ(U, x) = Φ(x). Note that if there exists someΦ∈Φ(U, x) such that EU⁺

·, V_T^x,φ(·)

=∞andEU⁻

·, V_T^x,φ(·)

<∞thenu(x) =∞and the problem is ill-posed.

We propose some examples where Assumptions 4.7 or 4.8 hold true. Example ii) illustrates the distinction between Assumptions 4.7 and 4.8 and justifies why we do not merge both assumptions and postulate thatEU⁺

·, V_T^1,φ(·)

<∞, for allφ∈Φ(1).

i) IfU is bounded above then both Assumptions are trivially true. We get directly thatΦ(U, x) = Φ(x)for allx≥0.

ii) Assume that EU⁻(·,0) < ∞ holds true. Let x ≥ 0 and φ ∈ Φ(x) be fixed. Using that U⁻ is non-increasing for allω∈Ωwe get that

EU⁻(·, V_T^x,φ(·))≤EU⁻(·,0)<+∞,

ThusEU(·, V_T^x,φ(·))is well-defined,Φ(U, x) = Φ(x)and Assumption 4.8 holds true.

iii) Assume that there exists somexˆ≥1such thatU(·,xˆ−1)≥0P-almost surely and bu(ˆx) := sup

φ∈Φ(ˆx)

EU(·, V_T^x,φˆ (·))<∞,

where we set forφ∈Φ(ˆx)\Φ(U,x),ˆ EU(·, V_T^ˆx,φ(·)) =−∞. Letφ∈Φ(1)be fixed. Then using that U is non-decreasing for allω∈Ω, we have thatP-almost surely

U(·, V_T^1,φ(·) + ˆx−1)≥U(·,xˆ−1)≥0.

ThereforeU(·, V_T^1,φ(·) + ˆx−1) = U⁺(·, V_T^1,φ(·) + ˆx−1)P-almost surely. Now using that U⁺ is non-decreasing for allω ∈Ωwe get that for allφ∈Φ(1)

EU⁺(·, V_T^1,φ(·))≤EU⁺(·, V_T^1,φ(·) + ˆx−1) =EU(·, V_T^1,φ(·) + ˆx−1)≤bu(ˆx)<+∞

and Assumptions 4.7 and 4.8 are satisfied. Instead of stipulating thatbu(ˆx) <∞it is enough to assume thatEU(·, V_T^x,φˆ (·))<∞for allφ∈Φ(ˆx).

iv) We will prove in Theorem 4.17 that under the (NA) condition and Assumption 4.10, Assumptions 4.7 and 4.8 hold true ifEU⁺(·,1)< +∞ and if for all0 ≤ t≤ T |∆S_t|, _α¹

t ∈ W_t (see (14) for the definition ofW_t).

Assumption 4.10 We assume that there exist some constants γ ≥ 0, K > 0, as well as a random variableC satisfyingC(ω)≥ 0for allω ∈ ΩandE(C) <∞ such that for allω ∈Ω,λ≥1and x∈ R, we have

U(ω, λx) ≤ Kλ^γ

U

ω, x+1 2

+C(ω)

. (9)

Remark 4.11 First note that the constant ¹₂ in (9) has been chosen arbitrarily. Indeed, assume there exists some constantx≥0such that for allω ∈Ω,λ≥1andx∈R

U(ω, λx) ≤ Kλ^γ(U(ω, x+x) +C(ω)). (10)

Using the monotonicity ofU, we can always assume x > 0. Set for allω ∈ Ω and x ∈ R, U(ω, x) = U(ω,2xx). Then for allω∈Ω,λ≥1and x∈R, we have that

U(ω, λx) =U(ω,2λxx)≤Kλ^γ(U(ω,2xx+x) +C(ω)) =Kλ^γ

U

ω, x+ 1 2

+C(ω)

, andU satisfies (9). It is clear that ifφ^∗ is an optimal solution for the problem

u(x) := sup_{φ∈Φ(U ,}^x

2x)EU(·, V

x 2x,φ

T (·))then2xφ^∗ is an optimal solution for (8). Note as well that, since K >0and C≥0, it is immediate to see that for allω ∈Ω,λ≥1andx∈R

U⁺(ω, λx) ≤ Kλ^γ

U⁺

ω, x+ 1 2

+C(ω)

. (11)

Remark 4.12 We now provide some insight on Assumption 4.10. As the inequality (9) is used to control the behaviour ofU⁺(·, x) for large values ofx, the usual assumption in the non-concave case (see Assumption 2.10 in Carassus et al. [2015]) is that there exists somexˆ≥0such thatEU⁺(·,x)ˆ <∞ as well as a random variableC1 satisfyingE(C1)<∞andC1(ω)≥0for allω²such that for allx≥x,ˆ λ≥1andω∈Ω

U(ω, λx)≤λ^γ(U(ω, x) +C1(ω)). (12) We prove now that if (12) holds true then (10) is verified withx = ˆx,K = 1 andC =C1. Indeed, for x≥0, using the monotonicity ofU, we have for allω∈Ωandλ≥1that

U(ω, λx)≤U(ω, λ(x+ ˆx))≤λ^γ(U(ω, x+ ˆx) +C1(ω)).

Therefore (10) is a weaker assumption than (12). Note as well that if we assume that (12) holds true for allx >0, then if0< x <1andω∈Ωwe have

U(ω,1)≤ 1

x γ

(U(ω, x) +C1(ω)),

and U(ω,0) := limx→0, x>0U(ω, x) ≥ −C₁(ω). This excludes for instance the case where U is the logarithm. Furthermore, this also implies thatEU⁻(·,0)≤EC1 <∞and we are back to Assumption 2.7 of Carassus et al. [2015]

Alternatively, recalling the way the concave case is handled (see Lemma 2 in R ´asonyi and Stettner [2005]), we could have introduced that there exists a random variableC2 satisfying E(C2) < ∞ and C₂≥0such that for allx∈R,ω∈Ω

U⁺(ω, λx)≤λ^γ U⁺(ω, x) +C2(ω)

. (13)

We have not done so as it is difficult to prove that this inequality is preserved through the dynamic programming procedure when considering non-concave functions unless we assume thatEU⁻(·,0)<

∞as in Carassus et al. [2015].

Remark 4.13 If there exists some set ΩAE ∈ F with P(ΩAE) = 1 such that (9) holds true only for ω ∈ΩAE, then setting as in Remark 4.5, U(ω, x) := U(ω, x)1Ω_AE×R(ω, x), U satisfies (9) and the value function in (8) does not change. We also assume without loss of generality thatC(ω) ≥0for all ω in (9). Indeed, ifC ≥0P-a.s, we could considerCe:=CI_C≥0. Then Assumption 4.10 would hold true with Ceinstead ofC.

2In the cited paperC1≥0a.s but this is not an issue, see Remark 4.13 below

Remark 4.14 In the case where (12) holds true, we refer to remark 2.5 of Carassus and R ´asonyi [2016]

and remark 2.10 of Carassus et al. [2015] for the interpretation ofγ : for C₁ = 0, it can be seen as a generalization of the “asymptotic elasticity” ofU at+∞(see Kramkov and Schachermayer [1999]). So (12) requires that the (generalized) asymptotic elasticity at+∞is finite. In this case and ifU is differ- entiable there is a nice economic interpretation of the “asymptotic elasticity” as the ratio of “marginal utility”: U⁰(x) and the “average utility”: ^U(x)_x , see again Section 6 of Kramkov and Schachermayer [1999] for further discussions. The caseC₁>0allows bounded utilities. In Carassus et al. [2015] it is proved that unlike in the concave case, the fact thatU is bounded from above (and therefore satisfies (10)) does not imply that the asymptotic elasticity is bounded.

We propose now an example of an unbounded utility function satisfying (10) and such that

lim sup_x→∞ ^xU_U(x)^0(x) = +∞. This shows (as the counterexample of Carassus et al. [2015]), that Assump- tion 4.10 is less strong that the usual “asymptotic elasticity”. LetU :R→Rbe defined by

U(x) =−∞1_(−∞,0)(x) +X

p≥0

p1_[p,p+1− 1

2p+1)(x) +f_p(x)1_[p+1− 1

2p+1,p+1)(x) wherefp(x) = 2^p+1x+ (p+ 1) 1−2^p+1

forp∈N. ThenU satisfies Definition 4.1 and we have U⁰(x) =X

p≥0

2^p+11_[p+1− ¹

2p+1,p+1)(x).

We prove that (10) holds true. Note that for all x ≥0 we have x−1 ≤ U(x) ≤x+ 1. Letx ≥0 and λ≥1be fixed. Then we get that

U(λx)≤λx+ 1≤λ(U(x+ 1) + 1) + 1≤λ(U(x+ 1) + 2),

and (10) is true with K = x = 1 and C = 2. Now for k ≥ 0, let xk = k+ 1 − _2k+2¹ . We have U(x_k) =f_k(x_k) =k+¹₂ and

x_kU⁰(x_k)

U(xk) = 2^k+1 k+ 1−_2k+2¹

k+¹₂ →_k→∞+∞.

Remark 4.15 We propose further examples where Assumption 4.10 holds true.

i) Assume that U is bounded from above by some integrable random constant C1 ≥ 0 and that EU⁻(·,¹₂)<∞. Then for allx≥0,λ≥1,ω ∈Ωwe have

U(ω, λx)≤C₁(ω)≤λU

ω, x+1 2

+λ

C₁(ω)−U

ω, x+1 2

≤λU

ω, x+1 2

+λ

C₁(ω) +U⁻

ω,1 2

,

and (9) holds true forx ≥0with K = 1,γ = 1and C(·) =C₁(·) +U⁻(·,¹₂). AsU(·, x) =−∞for x <0, (9) is true for allx∈R.

ii) Assume thatU satisfies Definition 4.1 and that the restriction ofU to[0,∞)is concave and non- decreasing and thatEU⁻(·,1)< ∞. We use similar arguments as in Lemma 2 in R ´asonyi and Stettner [2006]. Indeed, letx≥2,λ≥1be fixed we have

U(ω, λx)≤U(ω, x) +U⁰(ω, x)(λx−x)≤U(ω, x) + U(ω, x)−U(ω,1)

x−1 (λ−1)x

≤U(ω, x) + 2(λ−1) (U(ω, x)−U(ω,1))

≤U(ω, x) + 3(λ− 1

3) (U(ω, x)−U(ω,1))

≤3λ U(ω, x) +U⁻(ω,1) ,

where we have used the concavity ofU for the first two inequalities and the fact thatx≥2and U is non-decreasing for the other ones. Thus from the proof that (12) implies (10), we obtain that (10) holds true withK = 3,γ= 1,x= 2andC(·) =U⁻(·,1).

We can now state our main result.

Theorem 4.16 Assume the (NA) condition and that Assumptions 4.7, 4.8 and 4.10 hold true. Let x≥0. Then,u(x)<∞and there exists some optimal strategyφ^∗∈Φ(U, x)such that

u(x) =EU(·, V_T^x,φ∗(·)).

Moreoverφ^∗_t(·)∈D^t(·)a.s. for all0≤t≤T.

We will use dynamic programming in order to prove our main result. We will combine the approach of R ´asonyi and Stettner [2005], R ´asonyi and Stettner [2006], Carassus and R ´asonyi [2016], Carassus et al. [2015] and Nutz [2016]. As in Nutz [2016], we will consider a one period case where the initial filtration is trivial (so that strategies are inR^d) and thus the proofs are much simpler than in the other cited papers. The price to pay is that in the multi-period case where we use intensively measurable selection arguments (as in Nutz [2016]) in order to obtain Theorem 4.16. In our model, there is only one probability measure, so we don’t have to introduce Borel spaces and analytic sets. Thus our modelisation of(Ω,F,F, P) is more general than the one of Nutz [2016] restricted to one probability measure. As we are in a non concave setting we use similar ideas to those of Carassus and R ´asonyi [2016] and Carassus et al. [2015].

Finally, as in R ´asonyi and Stettner [2005], R ´asonyi and Stettner [2006], Carassus and R ´asonyi [2016] and Carassus et al. [2015], we propose the following result as a simpler but still general setting where Theorem 4.16 applies. We introduce for all0≤t≤T

W_t:=

X: Ω^t→R∪ {±∞}, F_t-measurable, E|X|^p<∞ for allp >0 (14) Theorem 4.17 Assume the (NA) condition and that Assumption 4.10 hold true. Assume furthermore thatEU⁺(·,1)<+∞ and that for all0 ≤t≤T |∆S_t|, _α¹

t ∈ W_t. Letx ≥0. Then, for allφ ∈Φ(x) and all0≤t≤T,V_t^x,φ∈ W_t. Moreover, there exists some optimal strategyφ^∗ ∈Φ(U, x)such that

u(x) =EU(·, V_T^x,φ∗(·))<∞ Proof of the main theorems will appear in Section 6.

Remark 4.18 The assumptions of Theorem 4.17 clearly hold for the model of Example 2.3 above when U is deterministic satisfying Assumption 4.10. However, they also hold for many models with an infiniteΩ.

5 One period case

Let(Ω,H, Q)be a probability space (we denote byEthe expectation underQ) andY(·)anH-measurable R^d-valued random variable. LetD⊂R^dbe the smallest affine subspace ofR^dcontaining the support of the distribution ofY(·). We assume thatDcontains 0. The condition corresponding to (NA) in the present setting is

Assumption 5.1 There exists some constant0< α≤1such that for allh∈D

Q(hY(·)≤ −α|h|)≥α. (15)

Remark 5.2 below is exactly Remark 8 of Carassus and R ´asonyi [2016] (see also Lemma 2.6 of Nutz [2016]).

Remark 5.2 Leth∈R^dand leth⁰ ∈R^dbe the orthogonal projection ofhonD. Thenh−h⁰ ⊥Dhence {Y(·)∈D} ⊂ {(h−h⁰)Y(·) = 0}. It follows that

Q(hY(·) =h⁰Y(·)) =Q((h−h⁰)Y(·) = 0)≥Q(Y(·)∈D) = 1 by the definition ofD. HenceQ(hY(·) =h⁰Y(·)) = 1.

Assumption 5.3 We consider arandom utilityV : Ω×R→Rsatisfying the following conditions

• for everyx∈R, the functionV(·, x) : Ω→RisH-measurable,

• for everyω∈Ω, the functionV(ω,·) :R→Ris non-decreasing and usc onR,

• V(·, x) =−∞, for allx <0.

Letx≥0be fixed. We define

H_x:=n

h∈R^d, Q(x+hY(·)≥0) = 1o

, (16)

Dx:=H_x∩D. (17)

It is clear thatH_x andDx are closed subsets ofR^d. We define v(x) = (−∞)1_(−∞,0)(x) + 1[0,+∞)(x) sup

h∈Hx

EV (·, x+hY(·)). (18) From Remark 5.2,

v(x) = (−∞)1_(−∞,0)(x) + 1_[0,+∞)(x) sup

h∈Dx

EV(·, x+hY(·)). (19)

Remark 5.4 It will be shown in Lemma 5.9 that under Assumptions 5.1, 5.3, 5.5 and 5.7, for allh∈ H_x, E(V(·, x+hY(·))is well-defined and more precisely thatEV⁺(·, x+hY(·))<+∞. So, under this set of assumptions,Φ(V, x), the set ofh∈ H_x such thatEV(·, x+hY(·))is well-defined, equalsH_x.

We present now the assumptions which allow to assert that there exists some optimal solution for (18). First we introduce an “asymptotic elasticity” assumption.

Assumption 5.5 There exist some constantsγ ≥ 0, K > 0, as well as some H-measurable C with C(ω)≥0for allω∈ΩandE(C)<∞, such that for allω ∈Ω, for allλ≥1,x∈Rwe have

V(ω, λx)≤Kλ^γ

V

ω, x+1 2

+C(ω)

. (20)

Remark 5.6 The same comments as in Remark 4.13 apply. Furthermore, note that sinceK >0 and C≥0we also have that for allω∈Ω, allλ≥1andx∈R

V⁺(ω, λx)≤Kλ^γ

V⁺

ω, x+1 2

+C(ω)

. (21)

We introduce now some integrability assumption onV⁺. Assumption 5.7 For everyh∈ H₁,

EV⁺(·,1 +hY(·))<∞. (22)

The following lemma corresponds to Lemma 2.1 of R ´asonyi and Stettner [2006] in the deterministic case.