Laurence Carassus

Miklós Rásonyi

On optimal investment for a behavioural investor in multiperiod incomplete market models

Keywords: Optimisation, existence and well-posedness in behavioral finance, "S-shaped" utility function, distorsion, Choquet integral

We provide easily verifiable conditions for the well-posedness of the optimal investment problem for a behavioral investor in an incomplete discrete-time multiperiod financial market model, for the first time in the literature. Under suitable assumptions we also establish the existence of optimal strategies.

Introduction

A classical optimization problem of mathematical finance is to find the investment strategy that maximizes the expected von Neumann-Morgenstern utility of the portfolio value of some economic agent, see e.g. Chapter 2 of [START_REF] Föllmer | Stochastic Finance: An Introduction in Discrete Time[END_REF]. In mathematical terms, Eu(X) needs to be maximized in X where u is a concave increasing function and X runs over values of admissible portfolios.

In the paper [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF], based on experimentation, the authors contest the expected utility paradigm and propose the cumulative prospect theory. This theory asserts that: first, agents behave differently on gains and on losses. Second, agents overweight small probabilities and underweight large probabilities. Third, agents evaluate assets in comparison with some benchmark rather than based on final wealth positions. This can be translated into mathematics by the following assumptions: investors use an "S-shaped" utility function u (i.e. u(x) = u + (x), x ≥ 0; u(x) = -u -(-x), x < 0 where u + , u -: R + → R are concave and increasing) and they also distort the probability measure: instead of expectations, Choquet integrals appear. Furthermore, maximization of their objective function takes place over the random variables X -B where B is a fixed (stochastic) reference point and X, again, runs over the values of admissible portfolios.

That paper triggered an avalanche of subsequent investigations, especially in the economics literature. See the references of [START_REF] Jin | Behavioural portfolio selection in continuous time[END_REF] for a sample relevant to our present discussions. Mathematical treatments rarely went beyond simple, one-step models (we only know of [START_REF] Berkelaar | Optimal portfolio choice under loss aversion[END_REF], [START_REF] Jin | Behavioural portfolio selection in continuous time[END_REF] and [START_REF] Carlier | Optimal demand for contingent claims when agents have law invariant utilities[END_REF] where multiperiod models are studied; in [START_REF] Berkelaar | Optimal portfolio choice under loss aversion[END_REF], however, no probability distortion is considered). The reason for this, as pointed out in [START_REF] Jin | Behavioural portfolio selection in continuous time[END_REF], is the presence of massively difficult obstacles: the objective function is non-concave and the probability distortions make it impossible to use dynamic programming and the related machinery based on the Bellmann equation.

In [START_REF] Jin | Behavioural portfolio selection in continuous time[END_REF] a rather specific continuous-time market model (driven by Brownian motion) is treated. As this model is complete (all "reasonable" random variables can be realized by continuous trading), the behavioral investment problem can be reduced to a (still very difficult) static optimization over a set of random variables.

The arguments of both [START_REF] Jin | Behavioural portfolio selection in continuous time[END_REF] and [START_REF] Carlier | Optimal demand for contingent claims when agents have law invariant utilities[END_REF] heavily rely on completeness and there has not yet been any treatment of discrete-time multiperiod models in the literature. In the present article we make a substantial step ahead by studying this problem in a multiperiod, generically incomplete market model. We allow for a possibly stochastic reference point B and need no concavity assumptions on u + , u -: only their behavior at infinity matters.

The issue of well-posedness is a recurrent theme in this literature (see [START_REF] Bernard | Static portfolio choice under cumulative prospect theory[END_REF], He and Zhou). As far as we know, our Theorem 4.3 below is the first result of its kind for discrete-time multiperiod models.

The conditions in [START_REF] Jin | Behavioural portfolio selection in continuous time[END_REF] are not easily checked and have no obvious interpretations. Here we manage to provide intuitive and easily verifiable conditions which apply in the case where u + , u -and the probability distortions are all "power-like" functions satisfying certain parameter constraints and where appropriate moment conditions hold for the price process. We also give examples providing parameter restrictions which are necessary for well-posedness.

Existence of optimal strategies is fairly subtle in this setting. Assuming a certain structure for the information flow (see Assumption 5.1) we manage to establish an existence result in Theorem 5.6 below. Finally, we exhibit examples showing that our assumptions are satisfied in a reasonably large class of models.

The paper is organized as follows: in section 2 we introduce our models; section 3 presents examples pertinent to the well-posedness of the problem; section 4 provides a sufficient condition for well-posedness in a multiperiod market; section 5 proves the existence of optimizers under appropriate conditions, section 6 gives examples while section 7 contains technical material.

Market model description

Let (Ω, F, (F t ) 0≤t≤T , P ) be a discrete-time filtered probability space with time horizon T ∈ N. The set of m-dimensional F t -measurable random variables will be denoted by Ξ m t . Let {S t , 0 ≤ t ≤ T } be a d-dimensional adapted process representing the (discounted) price of d securities in the financial market in consideration. The notation ∆S t := S t -S t-1 will often be used. Trading strategies are given by d-dimensional processes {θ t , 1 ≤ t ≤ T } which are supposed to be predictable (i.e. θ t is F t-1 -measurable). The class of all such strategies is denoted by Φ.

Trading is assumed to be self-financing, so the value process of a strategy θ ∈ Φ is

X z,θ t := z + t j=1 θ j ∆S j ,
where z is the initial capital of the agent in consideration and the concatenation xy of elements x, y ∈ R d means that we take their scalar product. Consider the following technical condition (R). It says, roughly speaking, that there are no redundant assets, even conditionally, see also Remark 9.1 of [START_REF] Föllmer | Stochastic Finance: An Introduction in Discrete Time[END_REF].

(R) The support of the (regular) conditional distribution of ∆S t with respect to F t-1 is not contained in any proper affine subspace of R d , almost surely, for all 1 ≤ t ≤ T .

The following absence of arbitrage condition is standard, it is equivalent to the existence of a risk-neutral measure in discrete time markets with finite horizon, see e.g. [START_REF] Dalang | Equivalent martingale measures and no-arbitrage in stochastic securities market models[END_REF].

(NA) If X 0,θ T ≥ 0 a.s. for some θ ∈ Φ then X 0,θ T = 0 a.s. The next proposition is a trivial reformulation of Proposition 1.1 in [START_REF] Carassus | Optimal strategies and utility-based prices converge when agents' preferences do[END_REF].

Proposition 2.1. The condition (R) + (NA) is equivalent to the existence of F t -measurable positive random variables κ t , π t , 0 ≤ t ≤ T -1 such that ess. inf

ξ∈Ξ d t P (ξ∆S t+1 ≤ -κ t |ξ| /F t ) ≥ π t a.s.
Let W denote the set of R-valued random variables Y such that E|Y | p < ∞ for all p > 0. This family is clearly closed under addition, multiplication and taking conditional expectation. The family of nonnegative elements in W is denoted by W + . With a slight abuse of notation, for a d-dimensional random variable Y , we write Y ∈ W when we indeed mean |Y | ∈ W. We will also need W + t := W + ∩ Ξ 1 t . We now present the hypotheses on the market model that will be needed for our main results in the sequel.

Assumption 2.2. For all t ≥ 1, ∆S t ∈ W. Furthermore, for 0 ≤ t ≤ T -1, there exist κ t , π t ∈ Ξ 1 t satisfying 1/κ t , 1/π t ∈ W + t such that ess. inf ξ∈Ξ d t P (ξ∆S t+1 ≤ -κ t |ξ|/F t ) ≥ π t a.s. (1)
The first item in the above assumption could be weakened to the existence of the N th moment for N large enough but this would lead to complications with no essential gain in generality, which we prefer to avoid. In the light of Proposition 2.1, (1) is a certain strong form of no-arbitrage. Section 6 exhibits concrete examples showing that Assumption 2.2 holds in a broad class of market models. We note that, by Proposition 2.1, Assumption 2.2 implies both (NA) and (R) above. Now we turn to investors' behavior, as modeled by cumulative prospect theory, see [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF], [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF]. Agents' attitude towards gains and losses will be expressed by the functions u + , u -. Agents are assumed to have a (possibly stochastic) reference point B and a distorted view of reality expressed by the probability distortion functions w + , w -.

Formally, we assume that u + , u -: R + → R + are nondecreasing, continuous functions with u + (0) = u -(0) = 0. We fix B, a scalar-valued random variable. The functions w + , w -: [0, 1] → [0, 1] are nondecreasing and continuous with w + (0) = w -(0) = 0 and w + (1) = w -(1) = 1.

Remark 2.3. Actually, the main results of the present article need less about u ± , w ± than stipulated here, in particular, about continuity and monotonicity. But as these are fairly natural requirements for agents' preferences, we make these assumptions throughout the paper.

Example 2.4. A typical choice is taking

u + (x) = x α , u -(x) = kx β
for some k > 0 and setting [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF], the following choice was made: α = β = 0.88, k = 2.25, γ = 0.61 and δ = 0.69. We define, for X 0 ∈ Ξ 1 0 and θ ∈ Φ,

w + (t) = t γ (t γ + (1 -t) γ ) 1/γ , w -(t) = t δ (t δ + (1 -t) δ ) 1/δ , with constants 0 < α, β, γ, δ ≤ 1. In
V + (X 0 ; θ 1 , . . . , θ T ) := ∞ 0 w + P u + X X0,θ T -B + ≥ y dy, and
V -(X 0 ; θ 1 , . . . , θ T ) := ∞ 0 w -P u -X X0,θ T -B - ≥ y dy,
and whenever V -(X 0 ; θ 1 , . . . , θ T ) < ∞ we set

V (X 0 ; θ 1 , . . . , θ T ) := V + (X 0 ; θ 1 , . . . , θ T ) -V -(X 0 ; θ 1 , . . . , θ T ).
We denote by A(X 0 ) the set of strategies θ ∈ Φ such that

V -(X 0 ; θ 1 , . . . , θ T ) < ∞
and we call them admissible (with respect to X 0 ).

Remark 2.5. If there were no probability distortions (i.e. w + (t) = w -(t) = t) then we would sim-

ply get V + (X 0 ; θ 1 , . . . , θ T ) = Eu + X X0,θ T -B + and V -(X 0 ; θ 1 , . . . , θ T ) = Eu -X X0,θ T -B - ,
i.e. the expected "utility" of gains (resp. losses) with respect to the given reference point B.

We refer to [START_REF] Carassus | Portfolio optimization for nonconvex criteria functions[END_REF] for the explicit treatment of this problem in a continuous time, complete case under the assumptions that u + is concave, u -is convex and B is deterministic.

The present paper is concerned with maximizing V (X 0 ; θ 1 , . . . , θ T ) over θ ∈ A(X 0 ). We seek to find conditions ensuring well-posedness, i.e.

sup θ∈A(X0) V (X 0 ; θ 1 , . . . , θ T ) < ∞,
and the existence of θ * ∈ A(X 0 ) attaining this supremum.

A first look at well-posedness

For simplicity we assume that u + (x) = x α and u -(x) = x β for some 0 < α, β ≤ 1; the distortion functions are w + (t) = t γ , w -(t) = t δ for some 0 < γ, δ ≤ 1. The example given below applies also to w + , w -with a power-like behavior near 0 such as those in Example 2.4 above.

Let us consider a two-step market model with S 0 = 0, ∆S 1 uniform on [-1, 1], P (∆S 2 = ±1) = 1/2 and ∆S 2 is independent of ∆S 1 . This choice assures that there is absence of arbitrage. Let F 0 , F 1 , F 2 be the natural filtration of S 0 , S 1 , S 2 .

Let us choose initial capital X 0 = 0 and reference point B = 0. We consider the strategy given by θ 1 = 0 and θ 2 = g(∆S 1 ) for some measurable g : R → R such that the distribution function of θ 2 is

F (y) = 0, y < 1, F (y) = 1 - 1 y , y ≥ 1,
where > 0 will be chosen later; such a g exists as ∆S 1 has uniform law. It follows that

V + (X 0 ; θ 1 , θ 2 ) = ∞ 0 P γ ((θ 2 ∆S 2 ) α + > y) = ∞ 1 1 2 γ 1 y γ/α dy, and V -(X 0 ; θ 1 , θ 2 ) = ∞ 0 P δ ((θ 2 ∆S 2 ) β -> y) = ∞ 1 1 2 δ 1 y δ/β dy. If we have α/γ > β/δ then there is > 0 such that γ α < 1 < δ β , which entails V -(X 0 ; θ 1 , θ 2 ) < ∞ and V + (X 0 ; θ 1 , θ 2 ) = ∞ so the optimization problem becomes ill-posed.
One may wonder whether this phenomenon could be ruled out by restricting the set of strategies e.g. to bounded ones. The answer is no. Considering θ 1 (n

) := 0, θ 2 (n) := min{θ 2 , n} for n ∈ N we obtain that V + (X 0 ; θ 1 (n), θ 2 (n)) → ∞, V -(X 0 ; θ 1 (n), θ 2 (n)) → V -(X 0 ; θ 1 , θ 2 ) < ∞ by monotone convergence, which shows that we still have sup ψ V (X 0 ; ψ 1 , ψ 2 ) = ∞,
where ψ ranges over the family of bounded strategies.

This shows that the ill-posedness phenomenon is not just a pathology but comes from the fact that one may use the information available at time 1 when choosing the investment strategy θ 2 .

We mention another case of ill-posedness which is present already in one-step models, as noticed in He and Zhou and Bernard and Ghossoub. [2010]. We generalize a bit the previous setting, u + , u -, ∆S 1 remain unchanged but we allow general distortions, assuming only that w + (t), w -(t) > 0 for t > 0. The market is defined by S 0 = 0, ∆S 2 = ±1 with probabilities p, 1 -p for some 0 < p < 1 and ∆S 2 is independent of ∆S 1 . Take

θ 1 (n) = 0 = X 0 = B and θ 2 (n) := n, n ∈ N, then V + (X 0 ; θ 1 (n), θ 2 (n)) = n α w + (p) and V -(X 0 ; θ 1 (n), θ 2 (n)) = n β w -(p). If α > β then, whatever w + , w -are, we have V (X 0 ; θ 1 (n), θ 2 (n)) → ∞.
Hence, in order to get a well-posed problem one needs to have α ≤ β, as already observed in [START_REF] Bernard | Static portfolio choice under cumulative prospect theory[END_REF] and He and Zhou.

We add a comment on the case α = β (the choice of [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF]): whatever w + , w -are, we may easily choose p such that the problem becomes ill-posed.

Since it would be difficult to exclude such a simple types of probability laws for ∆S 2 on economic grounds we are led to the conclusion that in order to get a mathematically meaningful optimization problem for a reasonably wide range of price processes one needs to assume both α < β and α/γ ≤ β/δ.

(2)

In the following section we propose an easily verifiable sufficient condition for the well-posedness of this problem in multiperiod discrete-time market models. The decisive condition we require is α/γ < β, see ( 7) below. This is stronger than (2) but still reasonably general. If w -(t) = t (i.e. δ = 1, no distortion on loss probabilities) then ( 7) is essentially sharp, as the present section highlights.

4 Well-posedness in the multiperiod case Assumptions 2.2, 4.1 and 4.2 will be in force throughout this section. We first present the conditions we need on u ± , w ± . Basically, we require that u ± behave in a power-like way at infinity and w ± do likewise in the neighborhood of 0. Condition (7) has already been mentioned in the previous section. It has a rather straightforward interpretation: the investor takes losses more seriously than gains and the distortion on gains is not too strong so that it is still outbalanced by loss aversion (as represented by parameter β). We stress that no concavity assumption is made on u + , u -, unlike in all related papers. Assumption 4.1. We assume that

u + (x) ≤ k + x α + k+ , (3)
k -x β -k-≤ u -(x), (4)
w + (x) ≤ gx γ , (5)
w -(x) ≥ f x, , (6)
with 0 < α, β, γ ≤ 1, k ± , k± , g, f > 0 fixed constants and α γ < β. ( 7
)
This allows us to fix λ such that λγ > 1 and λα < β.

We remark that the functions in Example 2.4 satisfy Assumption 4.1 whenever (7) holds.

The assumption below requires that the reference point B should be comparable to the market performance in the sense that it can be sub-hedged by some portfolio strategy φ ∈ Φ. Assumption 4.2. We fix a scalar random variable B such that, for some strategy φ ∈ Φ and for some b ∈ R, we have

X b,φ T = b + T t=1 φ t ∆S t ≤ B. (8)
The main result of the present section is the following.

Theorem 4.3. Under Assumptions 2.2, 4.1 and 4.2,

sup θ∈A(X0) V (X 0 ; θ 1 , . . . , θ T ) < ∞, whenever X 0 ∈ Ξ 1 0 with E|X 0 | β < ∞.
In particular, the result applies for X 0 a deterministic constant. We need to do some preparatory work before proving Theorem 4.3.

Lemma 4.4. There exist constants k, , ˜ > 0, such that

V + (X 0 ; θ 1 , . . . , θ T ) ≤ kE 1 + |X t + T n=1 (θ n -φ n )∆S n | αλ (9) := Ṽ + (X 0 ; θ 1 , . . . , θ T ), V -(X 0 ; θ 1 , . . . , θ T ) ≥ E [X 0 + T n=1 (θ n -φ n )∆S n -b] β - -˜ (10)
:= Ṽ -(X 0 ; θ 1 , . . . , θ T ).

Proof. We will use the facts that for 0 < η ≤ 1 one has

|x + y| η ≤ |x| η + |y| η (11)
for all x, y ∈ R and for 1 ≤ η we have |x + y| η ≤ 2 η-1 (|x| η + |y| η ). We get, using (5) and Chebishev's inequality:

V + (X 0 ; θ 1 , . . . , θ T ) ≤ 1 + g ∞ 1 E γ u λ + ([X 0 + T n=1 θ n ∆S n -B] + ) y λγ dy (12)
Evaluating the integral and using (3) we continue the estimation as

V + (X 0 ; θ 1 , . . . , θ T ) ≤ 1 + g λγ -1 E γ 2 λ-1 k λ + [X 0 + T n=1 θ n ∆S n -B] αλ + + 2 λ-1 kλ + ≤ 1 + g λγ -1 k λ + 2 λ-1 E(|X 0 + T n=1 (θ n -φ n )∆S n | αλ ) + |b| αλ + kλ + 2 λ-1 + 1 , using the rough estimate x γ ≤ x + 1, x ≥ 0, Assumption 4.2 and the fact that C 1 ≥ C 2 implies that (Y -C 1 ) + ≤ (Y -C 2 ) + .
This gives the first statement. For the second inequality note that, by ( 6) and Assumption 4.2,

V -(X 0 ; θ 1 , . . . , θ T ) ≥ f ∞ 0 P u -([X 0 + T n=1 (θ n -φ n )∆S n -b] -) ≥ y dy = f Eu -[X 0 + T n=1 (θ n -φ n )∆S n -b] - ≥ f k -E [X 0 + T n=1 (θ n -φ n )∆S n -b] β - -f k-. Whenever Ṽ -(X 0 ; θ 1 , . . . , θ T ) < ∞, we set Ṽ (X 0 ; θ 1 , . . . , θ T ) := Ṽ + (X 0 ; θ 1 , . . . , θ T ) -Ṽ -(X 0 ; θ 1 , . . . , θ T ).
For X 0 ∈ Ξ 1 0 we also introduce Ã(X 0 ) as the set of θ ∈ Φ such that Ṽ -(X 0 ; θ 1 , . . . , θ T ) < ∞. For proving Theorem 4.3, we will make use of an auxiliary optimization problem with objective function Ṽ (X 0 ; θ 1 , . . . , θ T ). As no probability distortions are involved this time, we can perform a kind of dynamic programming on this auxiliary problem. To this end we introduce, for all t = 0, . . . , T ,

X t ∈ Ξ 1 t and θ n ∈ Ξ d n-1 , n ≥ t + 1 the quantities Ṽ + t (X t ; θ t+1 , . . . , θ T ) := kE 1 + |X t + T n=t+1 (θ n -φ n )∆S n | αλ /F t , Ṽ - t (X t ; θ t+1 , . . . , θ T ) := E [X t + T n=t+1 (θ n -φ n )∆S n -b] β -/F t -˜ .
Whenever Ṽt (X t ; θ t+1 , . . . , θ T ) < ∞ a.s., we also define Ṽt (X t ; θ t+1 , . . . , θ T ) := Ṽ + t (X t ; θ t+1 , . . . , θ T ) -Ṽt (X t ; θ t+1 , . . . , θ T ).

We denote by Ãt (X t ) the set of (θ t+1 , . . . , θ T ) such that Ṽt (X t ; θ t+1 , . . . , θ T ) < ∞ a.s.

Remark 4.5. Clearly, (θ t+1 , . . . , θ T ) ∈ Ãt (X t ) implies (θ t+m+1 , . . . , θ T ) ∈ Ãt+m (X t + t+m n=t+1 (θ nφ n )∆S n ), for m ≥ 0; this follows from the tower law for conditional expectations and the fact that a bounded from below and integrable random variable is almost surely finite. For the same reason, Ã(X 0 ) ⊂ Ã0 (X 0 ) and, by Lemma 4.4, A(X 0 ) ⊂ Ã(X 0 ).

The crux of our arguments is contained in the next result.

Lemma 4.6. For each 0 ≤ t ≤ T , there exist

C t n ∈ W + n , n = t, . . . , T -1 such that, for (θ t+1 , . . . , θ T ) ∈ Ãt (X t ), Ṽt (X t ; θ t+1 , . . . , θ T ) ≤ Ṽt (X t ; θt+1 , . . . , θT ), where ( θt+1 , . . . , θT ) ∈ Ãt (X t ) satisfies | θn -φ n | ≤ C t n-1 [|X t | + 1], (13)
for n = t + 1, . . . , T , whenever X t ∈ Ξ 1 t .

Proof. Notice that for t = T the statement of the Lemma is trivial as there are no strategies involved. Let us assume that the Lemma is true for t + 1, we will deduce that it holds true for t, too. Let X t ∈ Ξ 1 t and (θ t+1 , . . . , θ T ) ∈ Ãt (X t ). Let X t+1 = X t + (θ t+1 -φ t+1 )∆S t+1 , then X t+1 ∈ Ξ 1 t+1 and by Remark 4.5, (θ t+2 , . . . , θ T ) ∈ Ãt+1 (X t+1 ). By induction hypothesis take ( θt+2 , . . . , θT ) ∈ Ãt+1 (X t+1 ) such that

| θn -φ n | ≤ C t+1 n-1 [|X t + (θ t+1 -φ t+1 )∆S t+1 | + 1] (14)
and Ṽt+1 (X t+1 ; θ t+2 , . . . , θ T ) ≤ Ṽt+1 (X t+1 ; θt+2 , . . . , θT ). It is clear from ( 14) that

T n=t+2 ( θn -φ n )∆S n ≤ H (|X t + (θ t+1 -φ t+1 )∆S t+1 | + 1) for H = T n=t+2 C t+1 n-1 |∆S n | ∈ W + . We have Ṽt (X t ; θ t+1 , . . . , θ T ) = E( Ṽt+1 (X t + (θ t+1 -φ t+1 )∆S t+1 ; θ t+2 , . . . , θ T )/F t ) ≤ E( Ṽt+1 (X t + (θ t+1 -φ t+1 )∆S t+1 ; θt+2 , . . . , θT )/F t ) = Ṽt (X t ; θ t+1 , θt+2 , . . . , θT ). ( 15
)
Fix some αλ < χ < β, we continue the estimation of Ṽ + t = Ṽ + t (X t ; θ t+1 , θt+2 , . . . , θT ) using the (conditional) Hölder inequality for q = χ/(αλ) and 1/p + 1/q = 1.

Ṽ + t ≤ k 1 + E(|X t + (θ t+1 -φ t+1 )∆S t+1 | αλ /F t )+ E(H αλ |X t + (θ t+1 -φ t+1 )∆S t+1 | αλ + H αλ /F t ) ≤ k 1 + |X t | αλ + |θ t+1 -φ t+1 | αλ E(|∆S t+1 | αλ /F t ) + E 1/p (H αλp /F t ) ( E 1/q (|X t | χ /F t ) + E 1/q (|θ t+1 -φ t+1 | χ |∆S t+1 | χ /F t ) + E(H αλ /F t ) ≤ k 1 + |X t | αλ + |θ t+1 -φ t+1 | αλ E(|∆S t+1 | αλ /F t ) + E 1/p (H αλp /F t ) |X t | αλ +|θ t+1 -φ t+1 | αλ E 1/q (|∆S t+1 | χ /F t ) + E(H αλ /F t ) .

It follows that, for an appropriate

H t in W + t , Ṽt (X t ; θ t+1 , . . . , θ T ) ≤ ˜ + H t 1 + |X t | αλ + |θ t+1 -φ t+1 | αλ - E [X t + (θ t+1 -φ t+1 )∆S t+1 + T n=t+2 ( θn -φ n )∆S n -b] β -/F t . (16)
By Lemma 4.7, the event

A := {( θn -φ n )∆S n ≤ 0, n ≥ t + 2; (θ t+1 -φ t+1 )∆S t+1 ≤ -κ t |θ t+1 -φ t+1 |} satisfies P (A|F t ) ≥ πt with 1/π t ∈ W + t , hence considering F := |θ t+1 -φ t+1 |κ t 2 ≥ |X t | + |b| (17) we have (recall that X t+1 = X t + (θ t+1 -φ t+1 )∆S t+1 ), 1 F E [X t+1 + T n=t+2 ( θn -φ n )∆S n -b] β -/F t ≥ 1 F E 1 A |θ t+1 -φ t+1 |κ t 2 β /F t ≥ |θ t+1 -φ t+1 |κ t 2 β πt 1 F . ( 18
)
As a little digression we estimate Ṽt (X t ; φ t+1 , . . . , φ

T ) = E k(1 + |X t | αλ ) -[X t -b] β -/F t + ˜ ≥ -|X t | β -|b| β . ( 19
)
Let us now choose the F t -measurable random variable C t t so large that on the event

F := {|θ t+1 -φ t+1 | > C t t [|X t | + 1]} we have |θ t+1 -φ t+1 |κ t 2 ≥ |X t | + |b| (that is, F ⊂ F holds) 3 |θ t+1 -φ t+1 |κ t 2 β πt ≥ ( + H t )|X t | β , 3 |θ t+1 -φ t+1 |κ t 2 β πt ≥ 2H t + |b| β + ˜ , 3 |θ t+1 -φ t+1 |κ t 2 β πt ≥ H t |θ t+1 -φ t+1 | αλ .
One can easily check that such a C t t exists and is in W + t . On F we have, using |X t | αλ ≤ |X t | β + 1 and thanks to ( 16), ( 18) and ( 19):

Ṽt (X t ; θ t+1 , . . . , θ T ) -Ṽt (X t ; φ t+1 , . . . , φ T ) ≤ ˜ + H t 2 + |X t | β + |θ t+1 -φ t+1 | αλ - |θ t+1 -φ t+1 |κ t 2 β πt + |X t | β + |b| β ≤ ( + H t )|X t | β + H t |θ t+1 -φ t+1 | αλ + 2H t + |b| β - |θ t+1 -φ t+1 |κ t 2 β πt + ˜ ≤ 0. ( 20
)
Consequently, defining

θt+1 := φ t+1 1 F + θ t+1 1 F c , θn := φ n 1 F + θn 1 F c , n = t + 2, . . . , T,
we have, using ( 15) and (20), Ṽt (X t ; θ t+1 , . . . , θ T ) ≤ Ṽt (X t ; θt+1 , . . . , θT ) a.s..

By construction,

| θt+1 -φ t+1 | ≤ C t t [|X t | + 1], and, for n ≥ t + 2, | θn -φ n | ≤ C t+1 n-1 [|X t + ( θt+1 -φ t+1 )∆S t+1 | + 1] ≤ C t+1 n-1 [|X t | + C t t (|X t | + 1)|∆S t+1 | + 1], hence we may set C t n-1 := C t+1 n-1 (C t t |∆S t+1 | + 1) for n ≥ t+2. Clearly, C t n-1 ∈ W + n-1 .
To conclude the proof it remains to check that ( θt+1 , . . . , θT ) ∈ Ãt (X t ). As by hypothesis (θ t+1 , . . . , θ T ) ∈ Ãt (X t ), we get from (15) that Ṽt (X t ; θ t , θt+2 , . . . , θT ) < ∞. Finally,

Ṽ - t (X t ; θt+1 , . . . , θT ) = 1 F (X t -b) β --˜ + 1 F c Ṽ - t (X t ; θ t+1 , θt+2 , . . . , θT ) < ∞ a.s.
Lemma 4.7. There exists πt with 1/π t ∈ W + t such that

P ((θ t+1 -φ t+1 )∆S t+1 ≤ -κ t |θ t+1 -φ t+1 |, ( θn -φ n )∆S n ≤ 0, n = t + 2, . . . , T /F t ) ≥ πt .
Proof. Define the events

A t+1 := {(θ t+1 -φ t+1 )∆S t+1 ≤ -κ t |θ t+1 -φ t+1 |}, A n := {( θn -φ n )∆S n ≤ 0}, t + 2 ≤ n ≤ T.
We prove, by induction, that for m ≥ t + 1,

E(1 At+1 . . . 1 Am /F t ) ≥ πt (m) (21)
for some πt (m) with 1/π t (m) ∈ W + t . For m = t + 1 this is just (1). Let us assume that ( 21) has been shown for m -1, we will establish it for m.

E(1 Am . . . 1 At+1 /F t ) = E(E(1 Am /F m-1 )1 Am-1 . . . 1 At+1 /F t ) ≥ E(π m-1 1 Am-1 . . . 1 At+1 /F t ) ≥ E 2 (1 Am-1 . . . 1 At+1 /F t ) E(1/π m-1 /F t ) ≥ π2 t (m -1) E(1/π m-1 /F t )
by the (conditional) Cauchy inequality. Here 1/π t (m -1) ∈ W + t by the induction hypothesis,

E(1/π m-1 |F t ) ∈ W + t (since 1/π m-1 ∈ W +
) and the statement follows.

Proof of Theorem 4.3. If A(X 0 ) is empty, there is nothing to prove. Otherwise, by Remark 4.5 and Lemma 4.6, for all n ∈ {1, . . . , T }, there is

C 0 n ∈ W + n such that for all θ ∈ A(X 0 ) there exists θ ∈ Ã0 (X 0 ) satisfying | θn -φ n | ≤ C 0 n-1 [|X 0 | + 1], 1 ≤ n ≤ T and Ṽ0 (X 0 ; θ 1 , . . . , θ T ) ≤ Ṽ0 (X 0 ; θ1 , . . . , θT ).
As θ ∈ A(X 0 ), by Lemma 4.4 V (X 0 ; θ 1 , . . . , θ T ) ≤ Ṽ (X 0 ; θ 1 , . . . , θ T ) = E Ṽ0 (X 0 ; θ 1 , . . . , θ T ) ≤ E Ṽ0 (X 0 ; θ1 , . . . , θT )

≤ E Ṽ + 0 (X 0 ; θ1 , . . . , θT ) ≤ kE 1 + |X 0 | αλ + T n=1 | θn -φ n | αλ E(|∆S n | αλ /F n-1 ) ≤ kE 1 + |X 0 | αλ 1 + T n=1 (C 0 n-1 ) αλ E(|∆S n | αλ /F n-1 ) ≤ k2 p-1 p E 1/p (1 + |X 0 | β )E 1/q 1 + T n=1 (C 0 n-1 ) αλ E(|∆S n | αλ /F n-1 ) q , ( 22
)
using Hölder's inequality with p = β/(αλ) and its conjugate number q. As E|X 0 | β < ∞, we get that this expression is finite, showing the Theorem.

Remark 4.8. It is worth contrasting Theorem 4.3 with Theorem 3.2 of [START_REF] Jin | Behavioural portfolio selection in continuous time[END_REF]. The latter states, in a continuous-time context, that in a typical (complete) Brownian market model our optimization problem is ill-posed whenever u + is unbounded and w -(t) = t (i.e. no distortion on losses). This phenomenon stems from the particularity of continuous-time models where the richness of attainable payoffs leads to ill-posedness. However, in our discrete-time models the family of replicable claims is relatively small hence ill-posedness does not occur even if w -is the identity (as long as the other assumptions of Theorem 4.3 hold).

Lemma 5.7. Fix c ∈ R and β , β satisfying χ < β < β < β. Assume that E Ṽt (X t ; θ t+1 , . . . , θ T ) ≥ c for some F t -measurable X t with E|X t | β < ∞ and for some (θ t+1 , . . . , θ T ) ∈ Ãt (X t ). Then there exists K t such that

E|θ t+1 -φ t+1 | β ≤ K t [E|X t | β + 1],
where K t does not depend on either X t or θ.

Proof. Let X t+1 := X t + (θ t+1 -φ t+1 )∆S t+1 and ( θt+2 , . . . , θT ) ∈ Ãt+1 (X t+1 ) such that

| θn -φ n | ≤ C t+1 n-1 [|X t+1 | + 1],
for n = t + 2, . . . , T and Ṽt+1 (X t+1 ; θ t+2 , . . . , θ T ) ≤ Ṽt+1 (X t+1 ; θt+2 , . . . , θT ), see Lemma 4.6. We can obtain equations ( 15) and ( 16) just like in the proof of Lemma 4.6. Furthermore, using ( 18), we get (recall ( 17) for the definition of F ) :

E Ṽt (X t ; θ t+1 , . . . , θ T ) ≤ E(H t (1 + |X t | αλ + |θ t+1 -φ t+1 | αλ )) -E 1 F |θ t+1 -φ t+1 |κ t 2 β πt + ˜ . ( 23
)
We now push further estimations in this last equation. We may estimate, using the Hölder inequality for p = β/β and its conjugate q,

E 1 F |θ t+1 -φ t+1 |κ t 2 β πt ≥ E p 1 F |θt+1-φt+1|κt 2 β π1/p t 1 π1/p t E p/q 1 πq/p t .
The denominator here will be denoted C in the sequel. By Lemma 4.7, C < ∞. Now let us note the trivial fact that for random variables X, Y ≥ 0 such that

EY β ≥ 2EX β one has E[1 {Y ≥X} Y β ] ≥ 1 2 EY β . It follows that if E |θ t+1 -φ t+1 |κ t 2 β ≥ 2E(|X t | + |b|) β (24)
holds true then, applying the trivial x ≤ x p + 1, x ≥ 0,

E p 1 F |θt+1-φt+1|κt 2 β C ≥ E p |θt+1-φt+1|κt 2 β 2 p C ≥ E |θt+1-φt+1|κt 2 β -1 2 p C = c 1 E(|θ t+1 -φ t+1 |κ t ) β -c 2 with suitable c 1 , c 2 > 0.
Using again Hölder's inequality with p = β /β and its conjugate q,

E(|θ t+1 -φ t+1 |κ t ) β ≥ E p |θ t+1 -φ t+1 | β E p/q 1 κ β q t ≥ E|θ t+1 -φ t+1 | β -1 E p/q 1 κ β q t . (25)
With suitable c 1 , c 2 > 0, we get, whenever (24) holds, that

E 1 F |θ t+1 -φ t+1 |κ t 2 β πt ≥ c 1 E|θ t+1 -φ t+1 | β -c 2 . (26)
Estimate also, with p := χ/(αλ),

E H t (1 + |X t | λα + |θ t+1 -φ t+1 | λα ) ≤ E 1/q [H q t ][1 + E 1/p |X t | χ + E 1/p |θ t+1 -φ t+1 | χ ] ≤ E 1/q [H q t ][3 + E|X t | χ + E|θ t+1 -φ t+1 | χ ] ≤ c[1 + E|X t | β + E|θ t+1 -φ t+1 | χ ], (27)
with some c > 0, using that x χ ≤ x β + 1, x 1/p ≤ x + 1, for x ≥ 0. Furthermore, Hölder's inequality with p = β /χ gives

E|θ t+1 -φ t+1 | χ ≤ E χ/β |θ t+1 -φ t+1 | β .
It follows that whenever

E|θ t+1 -φ t+1 | β 1-χ/β ≥ 2c c 1 , (28)
one also has

cE|θ t+1 -φ t+1 | χ ≤ c 1 2 E|θ t+1 -φ t+1 | β . (29)
Finally consider the condition

c 1 2 E|θ t+1 -φ t+1 | β ≥ c[1 + E|X t | β ] + (c 2 -c + 1) + ˜ . (30)
It is easy to see that we can find some K t , large enough, such that

E|θ t+1 -φ t+1 | β ≥ K t [E|X t | β +1]
implies that (24) (recall (25)), ( 28), (30) all hold true. So in this case we have, from ( 23), ( 27), ( 29), ( 26) and ( 30),

E Ṽt (X t ; θ t+1 , . . . , θ T ) ≤ c[1 + E|X t | β ] + c 1 2 E|θ t+1 -φ t+1 | β -c 1 E|θ t+1 -φ t+1 | β + c 2 + ˜ ≤ -(c 2 -c + 1) + c 2 < c.
From this the statement of the lemma follows.

Corollary 5.8. Fix c ∈ R and assume that

Ṽ (X 0 ; θ 1 , . . . , θ T ) = E Ṽ0 (X 0 ; θ 1 , . . . , θ T ) ≥ c for some X 0 ∈ Ξ 1 0 with E|X 0 | β < ∞ and some θ ∈ Ã(X 0 ). Fix β T with χ < β T < β. There exist constants G t , t = 0, . . . , T -1 such that E|θ t+1 -φ t+1 | β T ≤ G t [E|X 0 | β + 1] for t = 0, . . . , T -1, and G t do not depend on X 0 or on θ. Proof. Take β T < β T -1 < . . . < β 1 < β 0 := β. We first prove, by induction on t, that X t := X 0 + t j=1 (θ j -φ j )∆S j , t ≥ 0 satisfy E|X t | βt ≤ C t [E|X 0 | β + 1],
for suitable C t > 0. For t = 0 this is trivial. Assuming it for t we will show it for t + 1. We first remark that E Ṽt (X t ; θ t+1 , . . . , θ T ) = E Ṽ0 (X 0 ; θ 1 , . . . , θ T ) ≥ c and that by the induction hypothesis E|X t | βt < ∞ holds. As θ ∈ Ã(X 0 ) ⊂ Ã0 (X 0 ), (θ t+1 , . . . , θ T ) ∈ Ãt (X t ) (see Remark 4.5). Thus Lemma 5.7 applies with the choice β := (β t+1 +β t )/2 and β := β t , and we can estimate, using Hölder's inequality with p := β /β t+1 (and its conjugate number q), plugging in the induction hypothesis:

E|X t+1 | βt+1 = E|X t + (θ t+1 -φ t+1 )∆S t+1 | βt+1 ≤ E|X t | βt+1 + E|(θ t+1 -φ t+1 )∆S t+1 | βt+1 ≤ E|X t | βt + 1 + E 1/p |(θ t+1 -φ t+1 )| β E 1/q |∆S t+1 | βt+1q ≤ E|X t | βt + 1 + C E|(θ t+1 -φ t+1 )| β + 1 ≤ E|X t | βt + 1 + C K t (E|X t | β + 1) + 1 = E|X t | βt + 1 + C K t (E|X t | βt + 1) + 1 ≤ (1 + CK t )C t E|X 0 | β + 1 + 1 + C + CK t with C := E 1/q |∆S t+1 | qβt+1
, this proves the induction hypothesis for t + 1. Now let us observe that, by Lemma 5.7 (with β = β t+1 , β = β t ),

E|θ t+1 -φ t+1 | β T ≤ E|θ t+1 -φ t+1 | βt+1 + 1 ≤ K t [E|X t | βt + 1] + 1 ≤ K t [C t (E|X 0 | β + 1) + 1] + 1,
concluding the proof.

Proof of Theorem 5.6. Lemma 7.2 with the choice E := ε, l = 2 gives us ε, ε independent, uniformly distributed on [0, 1] and F 0 -measurable. Introduce

A (X 0 ) := {θ ∈ A(X 0 ) : θ t is F t-1 -measurable for all t = 1, . . . , T },
where F t := G t ∨ σ(ε ). Note that if θ ∈ A(X 0 ) then there exists θ ∈ A (X 0 ) such that the law of (θ, ∆S) equals that of (θ , ∆S) (since the law of ε equals that of ε and both are independent of ∆S). It follows that for all θ ∈ A(X 0 ) there is θ ∈ A(X 0 ) with V (X 0 ; θ 1 , . . . , θ T ) = V (X 0 ; θ 1 , . . . , θ T ).

Take θ(j) ∈ A(X 0 ), j ∈ N such that V (X 0 ; θ 1 (j), . . . , θ T (j)) → sup θ∈A(X0)

V (X 0 ; θ 1 , . . . , θ T ), j → ∞.

By Assumption 5.3 and Theorem 4.3, the supremum is finite and we can fix c such that -∞ < c < inf j V (X 0 ; θ 1 (j), . . . , θ T (j)). By Lemma 4.4 it implies that for all j, Ṽ (X 0 ; θ 1 (j), . . . , θ T (j)) > c.

By the discussions above we may and will assume θ(j) ∈ A (X 0 ), j ∈ N. Apply Corollary 5.8 for some β T such that χ < β T < β to get sup

j,t E|θ t (j) -φ t | β T < ∞.
It follows that the sequence of T (d + N ) + 1-dimensional random variables Ỹj := (ε , θ 1 (j) -φ 1 , . . . , θ T (j) -φ T , Z 1 , . . . , Z T ) are bounded in L β T (recall Remark 5.5) and hence

P (| Ỹj | > N ) ≤ E| Ỹj | β T N β T ≤ C N β T , continuity of u + , u -also u ± ([H j ] ± ) tends to u ± ([H] ± ) in law which entails that P (u ± ([H j ] ± ) ≥ y) → P (u ± ([H] ± ) ≥ y)
for all y outside a countable set (the points of discontinuities of the cumulative distribution functions of u ± ([H] ± )).

It suffices thus to find a measurable function h(y) with w + (P (u + [H j ] + ≥ y)) ≤ h(y), j ≥ 1 and ∞ 0 h(y)dy < ∞ and then Fatou's lemma will imply (31). We get, just like in Lemma 4.4, using Chebishev's inequality, (3) and ( 5), for y ≥ 1:

w + (P (u + [H j ] + ≥ y)) ≤ C 1 + |X 0 | αλ + T t=1 E |θ t (j) -φ t | αλ |∆S t | αλ y λγ ≤ C y λγ 1 + |X 0 | αλ + T t=1 E 1/p |θ t (j) -φ t | β T E 1/q W q t ,
for some constant C > 0 and W t ∈ W + , t = 1, . . . , T , using Hölder's inequality with p := β T /(αλ) and its conjugate q (recall that ∆S t ∈ W). We know from the construction that sup j,t E|θ t (j) -

φ t | β T < ∞.
Y t+1 -Y t = µ(Y t ) + σ(Y t ) t+1 ,
where µ : R N → R N is bounded and continuous, σ : R N → R N ×N is bounded and continuous. We assume that there is h > 0 such that inf

|v|=1 v T σ(x)σ T (x)v ≥ h for all x ∈ R N ; t ∈ W, t = 1, . . . , T are independent with supp Law t = R N .
Thus Y t follows a discretized dynamics of a non-degenerate diffusion process. We may think that Y t represent the evolution of N economic factors. Take G 0 trivial and G t := σ( j , j ≤ t), t ≥ 1.

We claim that Y t satisfies Assumption 2.2 with respect to G t . Indeed, Y t ∈ W is trivial and we will show that (1) holds with κ t , π t constants.

Take v ∈ R N . Obviously,

P (v(Y t+1 -Y t ) ≤ -|v|/G t ) = P (v(Y t+1 -Y t ) ≤ -|v|/Y t ).
It is thus enough to show that there is c > 0 such that for each unit vector v and each

x ∈ R N P (v(µ(x) + σ(x) t ) ≤ -1) ≥ c.
Denoting by m an upper bound for |µ(x)|, x ∈ R N , we may write

P (v(µ(x) + σ(x) t ) ≤ -1) ≥ P (vσ(x) t ≤ -(m + 1)).
Here y = v T σ(x) is a vector of length at least √ h, hence the absolute value of one of its components is at least h/N . Thus we have

P (v T σ(x) t ≤ -(m + 1)) ≥ min min i,ki P ( h/N i t ≤ -(m + 1), k i (j) j t ≤ 0, j = i); min i P ( h/N i t ≥ (m + 1), k i (j) j t ≤ 0, j = i)
where i ranges over 1, . . . , N and k i ranges over the (finite) set of all functions from {1, 2, . . . , i -1, i + 1, . . . , N } to {1, -1} (representing all the possible configurations for the signs of y j , j = i). This minimum is positive by our assumption on the support of t . Now we can take S i t := Y i t , i = 1, . . . , d for some d ≤ N . When N > d, we may think that the Y j , d < j ≤ N are not prices of some traded assets but other relevant economic variables that influence the market. It is easy to check that Assumption 2.2 holds for S t , too, with respect to G t .

Enlarging each G t by ε, independent of 1 , . . . , T we get F t as in Assumption 5.1. Clearly, Assumption 2.2 continues to hold for S t with respect to F t and Assumption 5.1 is then also true as S t is a continuous function of 1 , . . . , t .

Example 6.3. Take Y t as in the above example. For simplicity, we assume d = N = 1. Furthermore, let t , t = 1, . . . , T be such that for all ζ > 0, Ee ζ| t| < ∞.

Set S t := exp(Y t ) this time. We claim that Assumption 2.2 holds true for S t with respect to the filtration G t . Obviously, ∆S t ∈ W, t ≥ 1.

We choose κ t := S t /2. Clearly, 1/κ t ∈ W. It suffices to prove that 1/P (S t+1 -S t ≤ -S t /2/G t ) and 1/P (S t+1 -S t ≥ S t /2/G t ) belong to W. We will show only the second containment, the first one being similar. This amounts to checking

1/P (exp{Y t+1 -Y t } ≥ 3/2/Y t ) ∈ W.
Let us notice that

P (exp{Y t+1 -Y t } ≥ 3/2/Y t ) = P (µ(Y t ) + σ(Y t ) t+1 ≥ ln(3/2)/Y t ) = P ( t+1 ≥ ln(3/2) -µ(Y t ) σ(Y t ) /Y t ) ≥ P ( t+1 ≥ ln(3/2) -m √ h ),
which is a deterministic positive constant, by the assumption on the support of t+1 . Defining the enlarged F t , Assumptions 2.2 and 5.1 hold for S t . Examples 6.2 and 6.3 are pertinent, in particular, when the t are Gaussian.

Auxiliary results

We start with a simple observation. Proof. As by Chebishev's inequality and Y ∈ W + , P (Y ≥ y) ≤ M (N )y -N , y > 0, for all N > 0, for a constant M (N ) := EY N , we can choose N so large to have N δ > 1, showing that the integral in question is finite.

The following Lemmata are fairly standard but we include their proofs in want of an appropriate, precise reference.

Lemma 7.2. Let E be uniformly distributed on [0, 1]. Then for each l ≥ 1 there are measurable f 1 , . . . , f l : [0, 1] → [0, 1] such that f 1 (E), . . . , f l (E) are independent and uniform on [0, 1].

Proof. We first recall that if Y 1 , Y 2 are uncountable Polish spaces then they are Borel isomorphic, i.e. there is a bijection ψ : Y 1 → Y 2 such that ψ, ψ -1 are measurable (with respect to the respective Borel fields); see e.g. page 159 of [START_REF] Dellacherie | Probability and potential[END_REF].

Fix a Borel-isomorphism ψ : R → [0, 1] l and define the probability κ(A) := λ l (ψ(A)), A ∈ B(R), where λ l is the l-dimensional Lebesgue-measure restricted to [0, 1] l . Denote by F (x) := κ((-∞, x]), x ∈ R the cumulative distribution function (c.d.f.) corresponding to κ and set F -(u) := inf{q ∈ Q : F (q) ≥ u}, u ∈ (0, 1). This function is measurable and it is well-known that F -(E) has law κ. Now clearly (f 1 (u), . . . , f l (u)) := ψ(F -(u)) is such that (f 1 (E), . . . , f l (E)) has law λ l and the f i are measurable.

Lemma 7.3. Let µ(dy, dz) = ν(y, dz)δ(dy) be a probability on R N2 × R N1 such that δ(dy) is a probability on R N2 and ν(y, dz) is a probabilistic kernel. Assume that Y has law δ(dy) and E is independent of Y and uniformly distributed on [0, 1]. Then there is a measurable function G : R N2 × [0, 1] → R N1 such that (Y, G(Y, E)) has law µ(dy, dz).

Proof. Just like in the previous proof, fix a Borel isomorphism ψ : R → R N1 . Consider the measure on R×R N2 defined by μ(A×B) := A ν(y, ψ(B))δ(dy), A ∈ B(R N2 ), B ∈ B(R). For δ-almost every y, ν(y, ψ(•)) is a probability measure on R. Let F (y, z) := ν(y, ψ((-∞, z]))) denote its cumulative distribution function and define F -(y, u) := inf{q ∈ Q : F (y, q) ≥ u}, u ∈ (0, 1), this is easily seen to be B(R N2 ) ⊗ B([0, 1])-measurable. Then, for δ-almost every y, F -(y, E) has law ν(y, ψ(•)). Hence (Y, F -(Y, E)) has law μ. Consequently, (Y, ψ(F -(Y, E))) has law µ and we may conclude setting G(y, u) := ψ(F -(y, u)). The technique of this proof is well-known, see e.g. page 228 of [START_REF] Bhattacharya | Stochastic Processes with Applications[END_REF].

Lemma 7.4. Let ( Z, W ) be a R × R k -valued random variable with continuous everywhere positive density f (x 1 , . . . , x k+1 ) (with respect to the k + 1-dimensional Lebesgue measure) such that the function x 1 → sup

x 2 ,...,x k f (x 1 , . . . , x k+1 ) (32)

is integrable on R. Then there is a homeomorphism H : R k+1 → [0, 1]×R k such that H i (x 1 , . . . , x k+1 ) = x i for i = 2, . . . , k + 1 and Z := H 1 ( Z, W ) is uniform on [0, 1], independent of W .

Lemma 7. 1 .

 1 If Y ∈ W + then ∞ 0 P δ (Y ≥ y)dy < ∞,for all δ > 0.

  Thus we can find some constant C > 0 such that w + (P (u + [H j ] + ≥ y)) ≤ C /y λγ , for all j. Now trivially w + (P (u + [H j ] + ≥ y)) ≤ w + (1) = 1 for 0 ≤ y ≤ 1. Setting h(y) := 1 for 0 ≤ y ≤ 1 and h(y) := C /y λγ for y > 1, we conclude since λγ > 1 and thus 1/y λγ is integrable on [1, ∞). Let S 0 be constant and ∆S t ∈ W independent t = 1, . . . , T . Take Z i := ∆S i , define G 0 := {∅, Ω} and G t := σ(Z 1 , . . . , Z

	6 Examples
	Example 6.1.

t ), T ≥ 1. Assume that S t satisfies (NA) + (R) w.r.t. G t . Then this continues to hold for the enlargement F t defined in Assumption 5.1. So Assumptions 2.2 and 5.1 hold with κ t , π t almost surely constants since the conditional law of ∆S t w.r.t. F t-1 is a.s. equal to its actual law.

Example 6.2. Fix d ≤ N . Take Y 0 ∈ R N constant and define Y t by the difference equation

Existence of optimal strategies

Throughout this section, Assumptions 2.2, 4.1, 4.2, 5.1 and 5.3 will be in force.

Assumption 5.1. Let G 0 = {∅, Ω}, G t = σ(Z 1 , . . . , Z t ) for 1 ≤ t ≤ T , where the Z i , i = 1, . . . , T are R N -valued independent random variables. S 0 is constant and ∆S t is a continuous function of (Z 1 , . . . , Z t ), for all t ≥ 1 (hence S t is G t -adapted).

Furthermore, F t = G t ∨ F 0 , t ≥ 0, where F 0 = σ(ε) with ε uniformly distributed on [0, 1] and independent of (Z 1 , . . . , Z T ).

We may think that G t contains the information available at time t (given by the observable stochastic factors Z i , i = 1, . . . , t) and F 0 provides some independent random source that we use to randomize our trading strategies (in practice, one may always generate ε e.g. using a computer). The random variables Z i represent the "innovation": the information surplus of F i with respect to F i-1 , in an independent way. For the construction of the optimal strategies we use weak convergence techniques which necessitate the additional randomness provided by ε (the situation is somewhat analogous to the construction of a weak solution for a stochastic differential equation). Assumption 5.1 holds in many cases, see section 6 and Remark 7.8 below.

Remark 5.2. Let us take S 0 = 0, P (∆S 1 = 2) = 3/4, P (∆S 1 = -1) = 1/4 and G i , i = 0, 1 the natural filtration of S. Assume, moreover, that F 0 = σ(ε) where

We thank Andrea Meireles for numerically checking that, somewhat surprisingly,

V (0; θ 1 ).

This simple example shows that additional randomness may increase the satisfaction of the agent, hence using the randomization ε appearing in Assumption 5.1 is, at least, reasonable. This is in stark contrast with the case where there is no distortion present. The tower law for conditional expectations shows that in this case adding an independent random variable to G 0 cannot increase the agent's attainable level of satisfaction.

Assumption 5.3. The random variable B is a continuous function of (Z 1 , . . . , Z T ), X 0 is deterministic and A(X 0 ) is not empty.

Remark 5.4. If B is a continuous function of (S 0 , . . . , S T ) then Assumption 5.1 clearly implies the first part of Assumption 5.3. If u -(x) ≤ c(1 + x η ) for some c ≥ k -and η ≥ β, X 0 , B ∈ W and w -(t) ≤ Ct δ for some δ ≤ 1 and C ≥ 1 then Lemma 7.1 implies that the strategy θ t = 0, t = 1, . . . , T is in A(X 0 ), in particular, the latter set is non-empty. Actually, θ ∈ A(X 0 ) whenever θ t ∈ W, t = 1, . . . , T . This applies, in particular, to Example 2.4.

Remark 5.5. We may and will suppose that Z i are bounded. This can always be achieved by replacing each coordinate Z j i of Z i with arctan Z j i for j = 1, . . . , N , i = 1, . . . , T .

We now present our main result on the existence of an optimal strategy.

Theorem 5.6. Let Assumptions 2.2, 4.1, 4.2, 5.1 and 5.3 hold. Then there is θ

Before proving Theorem 5.6, we need to extend certain arguments of Lemma 4.6 above. We fix some αλ < χ < β for what follows.

for some fixed C > 0, showing that the sequence of the laws of Ỹj is tight. Then, clearly, the sequence of laws of Y j := (ε , θ 1 (j), . . . , θ T (j), Z 1 , . . . , Z T ), is also tight and hence admits a subsequence (which we continue to denote by j) weakly convergent to some probability law µ on B(R T (d+N )+1 ). We will construct, inductively, θ * t , t = 1, . . . , T such that (ε , θ * 1 , . . . , θ * T , Z 1 , . . . , Z T ) has law µ and θ * is F-predictable. Let M be a T (d + N ) + 1-dimensional random variable with law µ. First note that (M 1+T d+1 , . . . , M 1+T d+N ) has the same law as Z 1 , . . . , (M 1+T d+(T -1)N +1 , . . . , M 1+T d+T N ) has the same law as Z T .

Now let µ k be the law of (M 1 , . . . , M 1+kd , M 1+dT +1 , . . . , M 1+dT +N T ) on R kd+N T +1 (which represents the marginal of µ with respect to its first 1 + kd and last N T coordinates), k ≥ 0.

As a first step, we apply Lemma 7.2 with E := ε, l := T to get σ(ε)-measurable random variables ε 1 , . . . , ε T that are independent with uniform law on [0, 1].

Applying Lemma 7.3 with the choice

) has the same law as the marginal of µ 1 with respect to its first 1 + d coordinates. Let Q, Q , U , U random variables such that Q and Q have same law and U and U

. . , M 1+dT +dN ) and U = (Z 1 , . . . , Z T ). As (ε 1 , ε , Z 1 , . . . , Z T ) are independent, we get that Q is independent of U . Now remark that weak convergence preserves independence, so as (ε , θ 1 (j)) and U are independent for all j, we get that Q is independent of U . So we conclude that (ε , G(ε , ε 1 ), Z 1 , . . . , Z T ) has law µ 1 . Define θ * 1 := G(ε , ε 1 ), this is clearly F 0 -measurable. Carrying on, let us assume that we have found θ * j , j = 1, . . . , k such that (ε , θ * 1 , . . . , θ * k , Z 1 , . . . Z T ) has law µ k and θ * j is a function of ε , Z 1 , . . . , Z j-1 , ε 1 , . . . , ε j only (and is thus F j-1 -measurable). We apply Lemma 7.3 with

to get G such that (Y, G(Y, ε k+1 )) has the same law as (M 1 , . . . , M 1+kd , M 1+T d+1 , . . . , M 1+T d+kN , M 1+kd+1 , . . . , M 1+(k+1)d ). Thus

has the same law as Q = (M 1 , . . . , M 1+(k+1)d , M 1+T d+1 , . . . , M 1+T d+kN ). Now choose U = (M 1+dT +kN +1 , . . . , M 1+dT +dN ) and U = (Z k+1 , . . . , Z T ). As Q depends only on (ε 1 , . . . , ε k+1 , ε , Z 1 , . . . , Z k ), which is independent from (Z k+1 , . . . , Z T ), Q is independent of U . Moreover, (ε , θ 1 (j), . . . , θ k+1 (j), Z 1 , . . . , Z k ) and (Z k+1 , . . . , Z T ) are independent for all j and weak convergence preserves independence, so Q is independent of U . This entails that

has law µ k+1 and setting θ * k+1 := G(Y, ε k+1 ) we make sure that θ * k+1 is a function of ε , Z 1 , . . . , Z k , ε 1 , . . . , ε k+1 only, a fortiori, it is F k -measurable. We finally get all the θ * j , j = 1, . . . , T such that the law of (ε , θ * 1 , . . . , θ * T , Z 1 , . . . , Z T ) equals µ = µ T . We will now show that

which will conclude the proof. Indeed, H j := X 0 + T t=1 θ t (j)∆S t -B clearly converges in law to H := X 0 + T t=1 θ * t ∆S t -B, j → ∞ (note that ∆S t and B are continuous functions of the Z t and X 0 is deterministic). By Proof. The conditional distribution function of Z knowing W = (x 2 , . . . , x k+1 ),

, is continuous (due to the integrability of (32) and Lebesgue's theorem). By everywhere positivity of f , F is also strictly increasing in x 1 . It follows that the function H : (x 1 , . . . , x k+1 ) → (F (x 1 , . . . , x k+1 ), x 2 , . . . , x k+1 ) is a bijection and hence a homeomorhpism by Theorem 4.3 in [START_REF] Deimling | Nonlinear functional analysis[END_REF]. The conditional law P (H 1 ( Z, W ) ∈ • |W = (x 2 , . . . , x k+1 )) is uniform on [0, 1] for Lebesgue-almost all (x 2 , . . . , x k+1 ), which shows that H 1 ( Z, W ) is independent of W with uniform law on [0, 1].

Corollary 7.5. Let ( Z1 , . . . , Zk ) be an R k -valued random variable with continuous and everywhere positive density (w.r.t. the k-dimensional Lebesgue measure) such that for all i = 1, . . . , k, the function z → sup

is integrable on R. There are independent random variables Z 1 , . . . , Z k and homeomorphisms

Proof. The case k = 1 is vacuous and k = 2 follows by Lemma 7.4. Assume that the statement is true for k, let us prove it for k + 1. We may set g l (k + 1) := g l (k), 1 ≤ l ≤ k, it remains to construct g k+1 (k + 1) and Z k+1 . Lemma 7.4 provides a homeomorphism s : R k+1 → R k+1 such that s m (x 1 , . . . , x k+1 ) = x m , 1 ≤ m ≤ k and Z k+1 := s k+1 ( Z1 , . . . , Zk+1 ) is independent of ( Z1 , . . . , Zk ) (and hence of (Z 1 , . . . , Z k ) = g k (k) -1 ( Z1 , . . . , Zk )). Define a : R k+1 → R k+1 by a(x 1 , . . . , x k+1 ) := (g k (k) -1 (x 1 , . . . , x k ), s k+1 (x 1 , . . . , x k+1 )) = s(g k (k) -1 (x 1 , . . . , x k ), x k+1 ), a is a homeomorphism since it is the composition of two homeomorphisms. As then a( Z1 , . . . , Zk+1 ) = (Z 1 , . . . , Z k+1 ), the result is shown setting g k+1 (k + 1) := a -1 . Remark 7.6. Condition (33) is quite weak, it holds, for example, when there is C > 0 such that

Assumption 7.7. Let G 0 = {∅, Ω}, G t = σ( Z1 , . . . , Zt ) for 1 ≤ t ≤ T , where the Zi , i = 1, . . . , T are R N -valued random variables with a continuous and everywhere positive joint density f on R T N such that (33) holds with k = T N . S 0 is constant and ∆S t is a continuous function of ( Z1 , . . . , Zt ), for all t ≥ 1.

Furthermore, F t = G t ∨ F 0 , t ≥ 0, where F 0 = σ(ε) with ε uniformly distributed on [0, 1] and independent of ( Z1 , . . . , ZT ).

Remark 7.8. If Assumption 7.7 above holds true then so does Assumption 5.1, in virtue of Corollary 7.5. Assumption 7.7 may be easier to verify in certain model classes.