Ying Hu
email: ying.hu@univ-rennes1.fr.

Hanqing Jin
email: jinh@maths.ox.ac.uk.

Xun Yu Zhou

Consistent Investment of Sophisticated Rank-Dependent Utility Agents in Continuous Time

Keywords: Rank-dependent utility, probability weighting, portfolio selection, continuous time, time inconsistency, intra-personal equilibrium strategy, market price of risk

We study portfolio selection in a complete continuous-time market where the preference is dictated by the rank-dependent utility. As such a model is inherently time inconsistent due to the underlying probability weighting, we study the investment behavior of sophisticated consistent planners who seek (subgame perfect) intra-personal equilibrium strategies. We provide sufficient conditions under which an equilibrium strategy is a replicating portfolio of a final wealth. We derive this final wealth profile explicitly, which turns out to be in the same form as in the classical Merton model with the market price of risk process properly scaled by a deterministic function in time. We present this scaling function explicitly through the solution to a highly nonlinear and singular ordinary differential equation, whose existence of solutions is established. Finally, we give a necessary and sufficient condition for the scaling function to be smaller than 1 corresponding to an effective reduction in risk premium due to probability weighting.

Introduction

The classical expected utility theory (EUT) is unable to explain many puzzling phenomena and paradoxes, such as, just to name a few, the Allais paradox, the co-existence of riskaverse and risk-seeking behavior of a same individual, and the disposition effect in financial investment. Rank-dependent utility theory (RDUT), first proposed by [START_REF] Quiggin | A theory of anticipated utility[END_REF], has been developed to address some of these puzzles and has thus far been widely considered as one of the most prominent alternative theories on preferences and choices. In addition to a concave outcome utility function as in EUT, RDUT features a probability weighting (or distortion) function whose slopes give uneven weights on random outcomes when calculating the mean. The presence of probability weighting is supported by a large amount of experimental and empirical studies, many of which find that such a weighting function typically displays an "inverse S-shape" (namely, it is first concave and then convex in its domain); see, e.g. [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF], [START_REF] Wu | Curvature of the probability weighting function[END_REF] and [START_REF] Tanaka | Risk and time preferences: Linking experimental and household survey data from Vietnam[END_REF]. This particular shape captures individuals' tendency to exaggerate the tiny probabilities of both improbable large gains (such as winning a lottery) and improbable large losses (such as encountering a plane crash). In particular, [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF] propose a specific parametric class of inverse S-shaped weighting functions, which will be used as a baseline example to test the assumptions in our paper.

In this paper we study a continuous-time financial portfolio selection model in which an agent pursues the highest rank-dependent utility (RDU) value. In contrast to classical continuous-time portfolio models such as Merton's [START_REF] Merton | Lifetime portfolio selection under uncertainty: The continuous-time case[END_REF], a dynamic RDU model is intrinsically time inconsistent; namely, any "optimal" strategy for today will generally not be optimal for tomorrow. As a result, there is no notion of a dynamically optimal strategy for a time-inconsistent model because any such a strategy, once devised for this moment, will have to be abandoned immediately (and indeed infinitesimally) at the next moment. The time inconsistency of the RDU model emanates from the probability weighting which weights the random outcomes unevenly according to their probabilities of occurrence. Consider as an example a 10-period binomial lattice model with equal probabilities of moving up and down at any given state. Standing at t = 0 the probability of reaching the top most state (TMS) at t = 10 is extremely small (2 -10 ). If the agent has an inverse S-shaped probability weighting then she will greatly inflate this probability. As time goes by and the agent moves along the lattice the probabilities of this same event -eventually reaching the TMS -keep changing, and so do the degrees of probability weighting. Indeed, once at t = 9 the probability of finally reaching the TMS is either 1/2 or 0 -which any reasonably intelligent individual is able to tell -and hence there is no probability weighting at all. We hereby see an inconsistency in the strength of probability weighting over time which is the key reason behind the time inconsistency of an RDU model. Time inconsistency changes fundamentally the way we deal with dynamic optimization in general. Optimization is intimately associated with decision making, and finding an optimal solution is therefore to advise on the best decisions. Now, there is no optimal solutions under time inconsistency: so what is the purpose of studying a time-inconsistent problem?

In his seminal paper, [START_REF] Strotz | Myopia and inconsistency in dynamic utility maximization[END_REF] describes three types of agents when facing time inconsistency. Type 1, a "spendthrift" (or a naiveté as in the more recent literature), does not recognize the time inconsistency and at any given time seeks an optimal solution for that moment only. As a result, his strategies are always myopic and change all the times. The next two types are aware of time inconsistency but act differently. Type 2 is a "precommitter" who solves the optimization problem only once at time 0 and then commits to the resulting strategy throughout, even though she knows that the original solution may no longer be optimal at later times. Type 3 is a "thrift" (or a sophisticated agent) who is unable to precommit and realizes that her future selves will disobey whatever plans she makes now. Her resolution is to compromise and choose consistent planning in the sense that she optimizes taking the future disobedience as a constraint. The Strotzian approach to time inconsistency is, therefore, descriptive, namely to describe what people actually do -the different reactions and behaviors in front of the inconsistency, as opposed to being normative, namely to tell people what to do.

It is both interesting and challenging to formulate mathematical models for each of the three types and solve them. It is interesting because these models are very different from the classical stochastic control based ones and different from each other, and it is challenging because the most powerful tools for tackling dynamic optimization such as dynamic programming and martingale analysis are based on time consistency and hence fail for time-inconsistent problems.

Recently, there is an upsurge of research interest and effort in the fields of stochastic control and mathematical finance/insurance in studying time-inconsistent models, mostly focusing on three different problems: mean-variance portfolio selection, and those involving non-exponential discounting or probability weighting. Earlier works focused on Type 2, precommitted agents (see, e.g., Li The Type 3 problem can be mathematically formulated as a game in the following way.

The sophisticated agent, anticipating the disagreement between her current and future selves, searches for a dynamic strategy that all the future selves have no incentive to deviate from.

The resulting strategy is a (subgame perfect) intra-personal equilibrium that will be carried through. In the continuous-time setting, [START_REF] Ekeland | Being serious about non-commitment: subgame perfect equilibrium in continuous time[END_REF] are the first to give a formal definition of such an equilibrium (albeit for a deterministic Ramsey model with non-exponential discounting), based on a first-order condition of a "spike variation" of the equilibrium strategy. This paper derives the consistent investment strategies of a Type 3 RDU agent in a continuous-time market in which the asset prices are described by stochastic differential equations (SDEs). We make several contributions, both methodologically and economically. First of all, to our best knowledge, this paper is the first to formulate and attack an RDU investment problem in the continuous-time setting. There are substantial difficulties in approaching the problem. Most notably, in deriving an equilibrium strategy, one needs to analyze the effect of the aforementioned spike variation on the RDU objective functional. In the absence of probability weighting, there is a well-developed approach at disposal to do this, based on calculus of variations and sample-path dependence of SDEs on parameters. 1 With probability weighting, however, we need to study the distributional dependence of the SDE solutions on parameters, which is actually a largely unexplored topic to our best knowledge. Starting from scratch, we carry out a delicate analysis to solve the problem thoroughly. The final solution requires the existence of solutions to a highly nonlinear, singular ordinary differential equation (ODE), the validity of an inequality, and the existence of the Lagrange multiplier to a budget constraint. We then provide sufficient conditions on the model primitives to ensure that these three requirements are met.

The equilibrium strategy is a replicating portfolio to a terminal wealth profile we derive explicitly, assuming that the market is complete. Curiously, the terminal wealth is of the same form as that with the classical Merton optimal strategy under EUT, except that the market price of risk or risk premium process needs to be multiplied by a scaling factor, the latter in turn determined by a solution to the aforementioned nonlinear ODE. To put this in a different way, the sophisticated RDU behaves as if she was an EUT agent, albeit in a revised market where the risk premium is properly scaled. The scaling factor depends on the original investment opportunity set and the agent's probability weighting function, but not her outcome utility function. This suggests that the additional constraint arising from the consistent planning due to time inconsistency can be transferred to the market opportunity set. This observation could be a key leading to identifying market equilibria where all the agents are RDU consistent planners.

When the scaling factor is less than 1, the risk premium is reduced and the agent is more risk averse than her EUT counterpart. In this case the probability weighting and consistent planning make the agent to take less risky exposure. We present a necessary and sufficient condition for this to happen.

It should be noted that, in order to derive our main results, we make several assumptions on the model primitives. Some of them may look quite technical especially those on the weighting function. However, we do not impose those assumptions for mathematical convenience; instead we make sure that they are mathematically mild and economically reasonable. In particular, all the assumptions on the weighting function are satisfied by the baseline function proposed by [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF].

The rest of the paper is organized as follows. In Section 2, we state our problem and define the intra-personal equilibrium in the same spirit as in our previous work [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF][START_REF] Hu | Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium[END_REF].

In Section 3, we present sufficient conditions under which the equilibrium terminal wealth can be explicitly derived. In Section 4, we examine these sufficient conditions closely on our model primitives. Section 5 is devoted to a concrete example demonstrating the results of the paper. In Section 6, we present an equivalent condition for an effectively reduced risk premium. Finally, Section 7 concludes. In Appendices, we state related general results on a class of singular ODEs, and verify our assumptions on a family of time-varying Tversky-Kahneman's weighting functions.

Problem Formulation

The market

We consider a continuous-time market in a finite time horizon [0, T ], where there are a riskfree asset and n risky assets being traded frictionlessly with price processes S 0 (•) and S i (•), i = 1, • • • , n, respectively. The risk-free interest rate, without loss of generality, is set to be r(•) ≡ 0, or equivalently, S 0 (t) ≡ S 0 (0). The dynamics of S i (•), i = 1, • • • , n, of the risky assets follow a multi-dimensional geometric Brownian motion

dS i (t) = S i (t) µ i (t)dt + n j=1 σ i,j (t)dW j (t) , i = 1, • • • , n, where µ i (•) and σ i,j (•) are all deterministic functions of time t, W (•) = (W 1 (•), • • • , W n (•)) is
an n-dimensional standard Brownian motion in a filtered probability space (Ω, F, (F t ) t∈[0,T ] , P) with F t = σ(W s : s ≤ t) ∨ N (P). Here denotes the matrix transpose.

Denote µ(t) = (µ 1 (t), • • • , µ n (t)) and σ(t) = (σ i,j (t)) n×n . We assume that σ(•) is invertible and that the market price of risk is θ(t) := σ(t) -1 µ(t); so the market is arbitrage free and complete.

A trading strategy is a self-financing portfolio described by an F t -adapted process π(

•) = (π 1 (•), • • • , π n (•))
, where π i (t) is the dollar amount allocated to asset i at time t. Under such a portfolio π(•), the dynamics of the corresponding wealth process X(•) evolve according to the wealth equation

dX(t) = π(t) µ(t)dt + π(t) σ(t)dW (t). ( 2

Rank-dependent utility

An agent, with an initial endowment x 0 at time t = 0, pursues the highest possible rankdependent utility (RDU) of her wealth at t = T by dynamically trading in the market. At any given time t with the wealth state X(t) = x she takes an admissible portfolio π(•) on [t, T ] leading to the terminal wealth X(T ). The RDU value of this terminal wealth is

J(X(T ); t, x) = +∞ 0 w (t, P t (u(X(T )) > y)) dy + 0 -∞ [w (t, P t (u(X(T )) > y)) -1] dy,
where w(t, •) is the probability weighting applied at time t, u(•) is the (outcome) utility function, and P t := P(•|F t ) denotes the conditional probability given F t , which includes the information X(t) = x. The agent's original objective is to maximize J(X(T ); t, x) by choosing a proper admissible investment strategy.

Here we allow the weighting function w to depend explicitly on time t. This is not just for mathematical generality; the time variation of probability weighting is supported empirically (Dierkes 2013, Cui et al 2020) and argued for on a psychology ground (Cui et al 2020).

Equilibrium strategies

As discussed in Introduction, probability weighting in general causes time-inconsistency. This means an optimal strategy with respect to J(X(T ); 0, X(0)) is not necessarily still optimal with respect to J(X(T ); t, X(t)) for a future time t > 0, where X(•) is the "optimal" wealth process projected at t = 0 for the objective J(X(T ); 0, X(0)). In this paper, we study the behaviors of a sophisticated agent who is aware of the time-inconsistency but lacks commitment, and who instead seeks a consistent investment among all the different selves t ∈ [0, T ] by finding intra-personal equilibrium strategies.

Precisely, given an admissible trading strategy π(•) with the corresponding wealth process X(•) starting from X(0) = x 0 , a time t ∈ [0, T ) and a small number ε ∈ (0, T -t), we define a slightly perturbed strategy which adds $k on top of π(•) over the small time interval [t, t + ε) while keeping π(•) unchanged outside of this interval. Here k is an F t -measurable random vector. This technique is called a spike variation. It follows from the wealth equation (2.1) that the perturbed terminal wealth is X(T ) + k ∆(t, ε), where ∆(t, ε) = t+ε t µ(s)ds + t+ t σ(s)dW (s). Definition 2.1 An admissible strategy π(•) with the wealth process X(•) starting from X(0) = x 0 is called an equilibrium strategy, if for any t ∈ [0, T ) and any F t -measurable random vector k, one has

lim sup ε↓0 J(X(T ) + k ∆(t, ε); t, X(t)) -J(X(T ); t, X(t)) ε ≤ 0. (2.2)
This definition follows most existing works on time-inconsistent optimal controls in continuous time; see e.g. [START_REF] Ekeland | Being serious about non-commitment: subgame perfect equilibrium in continuous time[END_REF], [START_REF] Bjork | A general theory of Markovian time inconsistent stochastic control problems[END_REF], and [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF][START_REF] Hu | Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium[END_REF].

Theoretically, an equilibrium strategy is an "infinitesimally" sub-game perfect equilibrium among all the selves t ∈ [0, T ). 2 The objectives of this paper are to find conditions under which there exists an equilibrium strategy, to derive the terminal wealth under such a strategy, and to draw economic implications and interpretations of the results. Since the market is complete, once the desired terminal wealth profile is identified its replicating portfolio is then the corresponding equilibrium strategy for consistent investment.

Assumptions

In this subsection we collect all the assumptions needed on the market parameters as well as on the preference functions. 3 These assumptions are henceforth in force, without necessarily being mentioned again in all the subsequent statements of results. 2 Definition 2.1 is also in line with the original Strotz's vision of intrapersonal equilibrium [START_REF] Strotz | Myopia and inconsistency in dynamic utility maximization[END_REF], adjusted for the continuous-time setting. More specifically, a Strotzian equilibrium strategy stipulates that for any given self t, all the future selves will commit to the strategy, because any deviation from the strategy will make the deviating self t worse off. In a continuous-time setting, however, any fixed t alone has no influence on the terminal wealth because it has a measure of zero. Therefore, one considers instead a small "alliance" of self t: the interval [t, t + ε). Definition 2.1 posits that a "deviation-in-alliance" from equilibrium fares worse in a first-order sense.

3 Throughout this paper, by an "increasing" function we mean a "non-decreasing" function, namely f is increasing if f (x) ≥ f (y) whenever x > y. We say f is "strictly increasing" if f (x) > f (y) whenever x > y.

Similar conventions are used for "decreasing" and "strictly decreasing" functions.

Assumption 2.3 (i) w(•, •) : [0, T ] × [0, 1] → [0, 1] is measurable. Moreover, for each t ∈ [0, T ], w(t, •) : [0, 1] → [0, 1] is strictly increasing, C 2 , and w(t, 0) = 0, w(t, 1) = 1 ∀t ∈ [0, T ]. (ii) u(•) : R → R is strictly increasing, strictly concave, C 3 , and u(-∞) = -∞, u (-∞) = +∞, u (+∞) = 0.
These two are very weak assumptions, representing some "minimum requirements" for the model primitives.

Define I(x) := (u ) -1 (x) , x > 0, and l(x) := -ln u (x), x ∈ R.

Assumption 2.4 (i)

There exists α > 0 such that lim sup x→+∞

x -α -I (x) < +∞, or equiva- lently, lim sup x→-∞ -u (x) u (x) α < +∞. (ii) There exist a > 0, b > 0 such that l (x) ≤ e a|l(x)|+b , l (x) ≤ e a|l(x)|+b ∀x ∈ R. (iii) u (x) ≥ 0 ∀x ∈ R.
(iv) For any t ∈ [0, T ], there exist c > 0 and m ∈ (-1, 0), both possibly depending on t, such that

w p (t, p) ≤ c[p m + (1 -p) m ] ∀ p ∈ (0, 1).
Assumption 2.4-(i) and -(ii) are very mild conditions satisfied by most commonly used utility functions defined on R (e.g. the exponential utility). Assumption 2.4-(iii) implies the risk prudence of the agent, capturing her tendency to take precautions against future risk.

Many common utility functions are prudent. 4 Assumption 2.4-(iv) is to control the level of probability weighting on very small and very large probabilities. It is satisfied by some wellknown weighting functions, e.g. that of [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF]; see Appendix B.

Assumption 2.5 There exist constants ν > 0, ζ > 0, such that 0 -∞ u (x) -ν dx < +∞, ∞ 0 u (x) ν dx < +∞, +∞ -∞ u (x)e -ζx 2 dx < +∞.
This is also a mild assumption, which is satisfied by, say, the exponential utility function.

A crucial function

The following real-valued function, generated from the weighting function, will play a central role throughout this paper:

h(t, x) := E w p (t, N (ξ))e xξ > 0, t ∈ [0, T ], x ∈ R,
where (and henceforth) w p (t, p) := ∂ ∂p w(t, p), ξ is a standard normal random variable with N being its probability distribution function. Similarly, we denote by h x (t, x) and h x (t, x) the first-and second-order partial derivatives of h in x respectively, and in general h

(n) x (t, x) the n-th order partial derivative of h in x. Lemma 2.6 h(t, x) < +∞ ∀(t, x) ∈ [0, T ] × R. Proof. Fix (t, x) ∈ [0, T ] × R. It follows from Assumption 2.4-(iv) that w p (t, p) ≤ c(p m + (1 - p) m ) ∀ p ∈ (0, 1) for some c > 0 and m ∈ (-1, 0). Denote m := m-1 2 ∈ (-1, m), p := m m > 1, q := p p-1 > 1. Applying Young's inequality a m b ≤ a m p + b q q ∀a ≥ 0, b ≥ 0, we obtain E[(N (ξ)) m e ±xξ ] ≤ E 1 p (N (ξ)) m + 1 q e ±xqξ = 1 ( m + 1)p + 1 q e (qx) 2 /2 . So h(t, x) ≤ c E[N (ξ) m e xξ ] + E[N (-ξ) m e xξ ] = c(E[N (ξ) m e xξ ] + E[N (ξ) m e -xξ ]) ≤ 2c 1 ( m + 1)p + 1 q e (qx) 2 /2 < +∞.
The proof is complete.

The following lemma collects some basic properties of h which will be useful in the sequel.

Lemma 2.7 For any t ∈ [0, T ], h(t, •) has the following properties:

(i) h(t, 0) = 1. (ii) h(t, x) is C ∞ in x ≥ 0, with h (n) x (t, x) = E w p (t, N (ξ))ξ n e xξ , x ≥ 0, ∀n ≥ 0. (iii) h (n)
x (t, x) is convex in x ≥ 0 for all even n ≥ 0, and increasing in x ≥ 0 for all odd n ≥ 1.

(iv) ln h(t, x) is convex in x ≥ 0. Proof. (i) We have h(t, 0) = E[w p (t, N (ξ))] = ∞ -∞ w p (t, N (x))dN (x) = 1 0 w p (t, p)dp = w(t, 1) -w(t, 0) = 1.
(ii) For ε = 0, write

h(t, x + ε) -h(t, x) ε = E w p (t, N (ξ))ξ 1 0 e (x+θε)ξ dθ .
For x ≥ 0, we have

0 < 1 0 e (x+θε)ξ dθ ≤ e (x+|ε|)ξ + e -|ε|ξ .
Fix a < 0. For any integer n ≥ 0, it follows from h(t, a) < +∞ that there exists a constant

C n > 0 such that |ξ n | ≤ C n (e aξ/2 + e ξ ).
Now, for any ε with |ε| < min(-a/2, 1), we have

|w p (t, N (ξ))ξ 1 0 e (x+θε)ξ dθ| ≤ w p (t, N (ξ))|ξ| e (x+|ε|)ξ + e -|ε|ξ ≤ C 1 w p (t, N (ξ)) e (x+|ε|+1)ξ + e (1-|ε|)ξ + e (x+|ε|+a/2)ξ + e (a/2-|ε|)ξ
< C 1 w p (t, N (ξ)) e (x-a/2+1)ξ + e (x+1)ξ + e (1+a/2)ξ + e ξ + e (x+a/2)ξ + e xξ + e aξ + e aξ/2 , where to deduce the last inequality we have repeatedly used the fact that e y < e y 1 + e y 2 for any y 2 > y 1 and y ∈ [y 1 , y 2 ]. From the assumption of the lemma, it follows that the last term of the above is a random variable with a finite mean; hence Lebesgue's dominated convergence theorem yields

h x (t, x) = E[w p (t, N (ξ))ξe xξ ].
Similarly, we can derive the desired expressions of higher-order derivatives.

(iii) The result is straightforward by (ii).

(iv) It follows from (ii) and the Cauchy-Schwarz inequality that, for x ≥ 0,

h x (t, x) = E w p (t, N (ξ))ξe xξ = E w p (t, N (ξ))ξe xξ/2 • w p (t, N (ξ))e xξ/2 ≤ E w p (t, N (ξ))ξ 2 e xξ • E w p (t, N (ξ))e xξ = h x (t, x) h(t, x). Thus ∂ 2 ∂x 2 ln h(t, x) = h x (t, x)h(t, x) -(h x (t, x)) 2 h 2 (t, x) ≥ 0,
establishing the desired convexity.

3 Equilibrium Strategies

An Ansatz and an ODE

Our approach to deriving the equilibrium strategies is inspired by an Ansatz we now make. If the agent was an expected utility maximizer, then her optimal strategy would be to dynamically replicate the terminal wealth I(κρ(T )) where ρ(•) is the state-price density process defined as

ρ(t) := exp - 1 2 t 0 |θ(s)| 2 ds - t 0 θ(s) dW (s) , (3.1)
and κ is the Lagrange multiplier for the budget constraint. We conjecture that in the current RDU setting the terminal wealth from an equilibrium strategy is still of the form I(κρ(T ))

with a revised state-price density process determined by multiplying the market price of risk function θ(•) by a scaling function λ(•):

ρ(t) = exp - 1 2 t 0 |λ(s)θ(s)| 2 ds - t 0 λ(s)θ(s) dW (s) . (3.2)
Since now we have conjectured a specific form of the desired terminal wealth profile X(T ),

we will be able to calculate its RDU value along with that of a slightly perturbed wealth process in the spirit of Definition 2.1. Then, the equilibrium condition (2.2) will lead to an equation that can be used to identify λ(•) as well as to other conditions.

It turns out that the equation to derive λ(•) is an ODE, explicitly expressed in the following form:

     Λ (t) = -|θ(t)| 2 h(t, √ Λ(t)) h x (t, √ Λ(t)) 2 Λ(t), t ∈ [0, T ), Λ(T ) = 0. (3.3)
This is a highly nonlinear ODE that is singular at T . The existence of its positive solutions will be established in Subsection 4.2.

The scaling function λ(•) in determining (3.2) is then given by

λ(t) := -Λ (t)/|θ(t)| 2 > 0, t ∈ [0, T ), (3.4)
where Λ(•) is a positive solution of (3.3).

Terminal wealth of an equilibrium strategy

The following result gives a complete solution to our problem by presenting the explicit terminal wealth profile of an equilibrium strategy. ), and that the following inequality holds for any c ∈ R:

Theorem 3.1 Assume that equation (3.3) admits a solution Λ(•) ∈ C[0, T ] ∩ C 1 [0, T ) with Λ(t) > 0 ∀t ∈ [0, T
+∞ -∞ w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) g (x) + c -g(x) Λ(t) g (x) 2 du(x) ≥ 0, a.e.t ∈ [0, T ). (3.5)
Moreover, assume there is κ > 0 such that the following holds

E [ρ(T )I(κρ(T ))] = x 0 (3.6)
where x 0 > 0 is the initial endowment of the agent at t = 0. Then the portfolio replicating the terminal wealth

X(T ) := I (κρ(T )) (3.7)
where ρ(T ) is determined through (3.2) -(3.4), is an equilibrium strategy.

Proof. Denote

E s,t := t s λ(v)θ(v) dW (v) ∀0 ≤ s ≤ t ≤ T, f (x) := I(κe -1 2 Λ(0) e -x ) =: g -1 (x), v(x) := u -1 (x), x ∈ R.
It is easy to see that, for any s ∈ [0, T ) and conditional on F s , E s,t is normal, i.e., E s,T |F s ∼ N (0, Λ(s)), and (3.7) can be rewritten as X(T ) = f (E 0,T ).

Let π(•) be the replicating strategy of X(T ), which exists by the market completeness.

Moreover, the budget constraint (3.6) ensures that π(•) is an admissible portfolio starting from the initial wealth x 0 . The goal is to prove that π(•) is an equilibrium strategy.

Fix t ∈ [0, T ). Consider the perturbed strategy described in Subsection 2.3 with the perturbed final wealth X(T ) + k ∆(t, ε), where ε ∈ (0, T -t). To compute the RDU value of this perturbed strategy, we first calculate, for any y ∈ R:

P t u(X(T ) + k ∆(t, ε)) > y = P t X(T ) > v(y) -k ∆(t, ε) = P t E t+ε,T > g(v(y) -k ∆(t, ε)) -E 0,t+ε = E t P t+ε E t+ε,T > g(v(y) -k ∆(t, ε) -E 0,t+ε ) = E t N E 0,t+ε -g(v(y) -k ∆(t, ε)) Λ(t + ε) ,
where

P t = P(•|F t ) and E t = E[•|F t ]. Denote m(s, y) = v(y) -k ∆(t, s) and Y (s, y) = E 0,t+s -g(m(s,y)) √ Λ(t+s) , s ∈ [0, ε), y ∈ R. Applying Itô's formula and noting that Λ (t + s) = -λ(t + s) 2 |θ(t + s)| 2 , we derive 5 dY (s) = Y (s) 2Λ(t + s) λ(t + s) 2 |θ(t + s)| 2 + g (m(s))k µ(t + s) Λ(t + s) - g (m(s))|k σ(t + s)| 2 2 Λ(t + s) ds + λ(t + s)θ(t + s) + g (m(s))k σ(t + s) Λ(t + s) dW (t + s), s ∈ [0, ε).
Applying Itô's formula again yields

dN (Y (s)) = N (Y (s))dY (s) + 1 2Λ(t + s) N (Y (s))|λ(t + s)θ(t + s) + g (m(s))k σ(t + s)| 2 ds = N (Y (s))dY (s) - 1 2Λ(t + s) N (Y (s))Y (s)|λ(t + s)θ(t + s) + g (m(s))k σ(t + s)| 2 ds = A(s)ds + B(s)dW (t + s), s ∈ [0, ε),
where

A(s) := N (Y (s)) √ Λ(t+s) g (m(s))k µ(t + s) -g (m(s)) 2 |σ(t + s) k| 2 -Y (s)g (m(s)) 2 2 √ Λ(t+s) |σ(t + s) k| 2 -Y (s)g (m(s)) √ Λ(t+s) k σ(t + s)λ(t + s)θ(t + s) = N (Y (s)) √ Λ(t+s) -1 2 g (m(s)) + g (m(s)) 2 Y (s) √ Λ(t+s) |σ(t + s) k| 2 +g (m(s)) 1 -Y (s)λ(t+s) √ Λ(t+s) θ(t + s) σ(t + s) k , B(s) := N (Y (s)) √ Λ(t+s) λ(t + s)θ(t + s) + g (m(s))k σ(t + s) . (3.8)
Integrating from s = 0 to s = δ, where δ ∈ (0, ε], and then taking conditional expectations on the above, we obtain

E t [N (Y (δ))] = N (Y (0)) + δ 0 E t [A(s)]ds, (3.9)
where we have used the fact that

• 0 B(s)dW (t + s) is a martingale on [0, ε]
for sufficiently small ε > 0 and hence

E t δ 0 B(s)dW (t + s) = 0, δ ∈ (0, ε].
(3.10)

A proof of this martingality will be delayed to Subsection 3.3. Now, we have lim sup

↓0 w t, P t (u(X(T ) + k ∆(t, ε)) > y) -w (t, P t (u(X(T )) > y)) ε = lim sup ε↓0 w(t, E t [N (Y (ε))]) -w(t, N (Y (0))) ε = lim sup ε↓0 w t, N (Y (0)) + ε 0 E t [A(s)]ds -w(t, N (Y (0))) ε = w p (t, N (Y (0))) • lim sup ε↓0 1 ε ε 0 E t [A(s)]ds = w p (t, N (Y (0)))A(0).
Hence (from this point on we will write back the variable y)

lim sup ε↓0 J(X(T )+k ∆(t,ε);t,X(t))-J(X(T );t,X(t)) ε = lim sup ε↓0 +∞ -∞ w(t,Et[N (Y (ε,y))])-w(t,N (Y (0,y))) ε dy ≤ +∞ -∞ lim sup ε↓0 w(t,Et[N (Y (ε,y))])-w(t,N (Y (0,y))) ε dy = +∞ -∞ w p (t, N (Y (0, y)))A(0, y)dy = -|σ(t) k| 2 1 2 √ Λ(t) +∞ -∞ w p (t, N (Y (0, y)))N (Y (0, y)) g (m(0, y)) + g (m(0, y)) 2 Y (0,y) √ Λ(t) dy +θ(t) σ(t) k 1 √ Λ(t) +∞ -∞ w p (t, N (Y (0, y)))N (Y (0, y))g (m(0, y)) 1 -Y (0,y) √ Λ(t)
λ(t) dy.

(3.11)

In the above, the first inequality

lim sup ε↓0 +∞ -∞ w(t,Et[N (Y (ε,y))])-w(t,N (Y (0,y))) ε dy ≤ +∞ -∞ lim sup ε↓0 w(t,Et[N (Y (ε,y))])-w(t,N (Y (0,y))) ε dy (3.12)
will be proved in Subsection 3.4.

By Definition 2.2, the underlying strategy is an equilibrium if the right hand side of (3.11) is non-positive for any k ∈ R n . 6 However, the right hand side, being quadratic in σ(t) k while

σ(t) is invertible, is non-positive for any k if and only if (noting θ(t) = 0) +∞ -∞ w p (t, N (Y (0, y)))N (Y (0, y))g (m(0, y)) 1 - Y (0, y) Λ(t) λ(t) dy = 0, (3.13)
and

+∞ -∞ w p (t, N (Y (0, y)))N (Y (0, y)) g (m(0, y)) + g (m(0, y)) 2 Y (0, y) Λ(t) dy ≥ 0. (3.14)
Recall that m(0, y) = v(y) and Y (0, y) = E 0,t -g(v(y)) √

Λ(t)

. By changing variables x = v(y) and

z = E 0,t -g(x) √ Λ(t)
, the equation (3.13) becomes

0 = +∞ -∞ w p t, N E 0,t -g(x) Λ(t) N E 0,t -g(x) Λ(t) g (x) 1 - E 0,t -g(x) Λ(t) λ(t) u (x)dx = - +∞ -∞ w p t, N E 0,t -g(x) Λ(t) N E 0,t -g(x) Λ(t) 1 - E 0,t -g(x) Λ(t) λ(t) du (x) = - +∞ -∞ w p (t, N (z))N (z) 1 - z Λ(t) λ(t) de √ Λ(t)z-E 0,t = -e -E 0,t +∞ -∞ w p (t, N (z))N (z)( Λ(t) -zλ(t))e √ Λ(t)z dz = -e -E 0,t E w p (t, N (ξ)) Λ(t) -ξλ(t) e √ Λ(t)ξ ,
where ξ ∼ N (0, 1). This equation is further equivalent to

Λ(t)E w p (t, N (ξ))e √ Λ(t)ξ = λ(t)E w p (t, N (ξ))ξe √ Λ(t)
w p t, N E 0,t -g(x) Λ(t) N E 0,t -g(x) Λ(t) g (x) + g (x) 2 E 0,t -g(x) Λ(t) du(x) ≥ 0, (3.15)
which is satisfied under (3.5).

In the above proof, there are two technical results, the martingale condition (3.10) and the inequality (3.12), left unproved. We provide proofs in the next two subsections. Recalling that m(s) = f (E 0,t+s -Λ(t + s)Y (s)) and f (x) = -κI (κe -x )e -x where κ := κe -1 2 Λ(0) , we have

g (m(s)) = g (f (E 0,t+s -Λ(t + s)Y (s))) = 1 f (E 0,t+s -Λ(t + s)Y (s)) = 1
-κI (κe

√ Λ(t+s)Y (s)-E 0,t+s ) e E 0,t+s - √ Λ(t+s)Y (s) ≤ K κ κα e (α-1) √ Λ(t+s)Y (s)-(α-1)E 0,t+s + e E 0,t+s - √ Λ(t+s)Y (s) ,
where the last inequality is due to (3.16). However, N (x)e γx is bounded in x ∈ R for any given γ ∈ R; hence 0 < N (Y (s))g (m(s)) ≤ c 1 e -(α-1)E 0,t+s + c 2 e E 0,t+s .

The right hand side above is locally square integrable as a process in s. The proof is complete.

The inequality (3.12)

We now prove the inequality (3.12). In view of Fatou's lemma, it suffices to show that the integrand on its left hand side is dominated by an integrable function. Throughout this subsection, we keep t ∈ [0, T ) fixed.

Lemma 3.3 For any constant γ ∈ (0, 1] and ε ∈ (0, T -t), there exists a constant c 1 (γ) > 0 such that

|A(δ, y)| N (Y (δ, y)) 1-γ = |A(δ, y)| N (-Y (δ, y)) 1-γ ≤ c 1 (γ)e γ 4Λ(t+δ) E 2 0,t+δ ∀δ ∈ (0, ε], y ∈ R.
Proof. As before, we drop the y variable to save space in this proof. Since g(x) = l(x) + ln κ -

|A(δ)| N (Y (δ)) 1-γ ≤ c 2 (γ)N (Y (δ)) γ (e a|g(m(δ))|+b + |Y (δ)|e 2a|g(m(δ))|+2b ) = c 2 (γ) √ 2π e -γ 2 Y (δ) 2 (e a|g(m(δ))|+b + |Y (δ)|e 2a|g(m(δ))|+2b ) ≤ c 3 (γ)e -γ 4 Y (δ) 2 e 2a|g(m(δ))|+2b = c 3 (γ)e -γ 4Λ(t+δ) (E 0,t+δ -g(m(δ))) 2 e 2a|g(m(δ))|+2b ,
where c 2 (γ) > 0 and c 3 (γ) > 0 are suitable constants. Making use of the general inequality

(x + y) 2 ≥ 1 2 x 2 -y 2 , we deduce |A(δ)| N (Y (δ)) 1-γ ≤ c 3 (γ)e -γ 4Λ(t+δ) (E 0,t+δ -g(m(δ))) 2 e 2a|g(m(δ))|+2b ≤ c 3 (γ)e γ 4Λ(t+δ) E 2 0,t+δ e -γ 8Λ(t+δ) g(m(t+δ)) 2 +2a|g(m(t+δ))|+2b ≤ c 3 (γ)e n 1 (γ) e γ 4Λ(t+δ) E 2 0,t+δ , (3.17)
where n 1 (γ) := max x∈R {-γ 8Λ t+δ x 2 + 2a|x| + 2b} < +∞. This completes the proof.

Lemma 3.4 For any γ ∈ (0, 1), there exists ε ∈ (0, T -t) and a constant

M 1 (γ) < +∞ such that E t |A(δ, y)| [E t N (Y (δ, y))] γ ≤ M 1 (γ) ∀ δ ∈ (0, ], y ∈ R.
Proof. Again we omit to write out y. Denote m = 1/γ and n = m m-1 . Then Cauchy-Schwarz's inequality yields

E t |A(δ)| [E t N (Y (δ))] γ ≡ E t |A(δ)| [E t N (Y (δ))] 1/m ≤ E t |A(δ)| n N (Y (δ)) -n/m 1/n . Write |A(δ)| n N (Y (δ)) -n/m = N (Y (δ)) N (Y (δ)) n/m |A(δ)| N (Y (δ)) 1/m n = N (Y (δ)) N (Y (δ)) n/m |A(δ)| N (Y (δ)) (γ+1)/2 n N (Y (δ)) n(1-γ)/2 .
Noting N (x) N (x) ≤ C(|x| + 1) ∀x ∈ R for some C > 0, we conclude that the last term above converges to 0 faster than the first term going to +∞ when Y (δ) goes to +∞; hence we can find a bound M 2 (γ) > 0 such that

|A(δ)| n N (Y (δ)) -n/m ≤ M 2 (γ) |A(δ)| N (Y (δ)) (γ+1)/2 n ≤ M 2 (γ)c 1 ((1 -γ)/2) n e 1 8Λ(t+δ) E 2 0,t+δ ,
where the last inequality is due to Lemma 3.3. When ε > 0 is small enough, there exists (i) For any γ > 0, there are a sufficiently small ε ∈ (0, T -t) and a function H 1 (•; γ) with

M 3 (γ) < ∞ such that E t [e 1 8Λ(t+δ) E 2 0,t+δ ] < M 3 (γ)
y(γ)
-∞ H 1 (y; γ)dy < +∞ for some y(γ) < 0, such that when δ ∈ (0, ε), we have

(E t |A(δ)|) γ ≤ H 1 (y; γ) ∀y ∈ (-∞, y(γ)].
(ii) For any γ > 0, there are a sufficiently small ε ∈ (0, T -t) and a function H 2 (•; γ) with +∞ ȳ(γ) H 2 (y; γ)dy < +∞ for some ȳ(γ) > 0, such that when δ ∈ (0, ε), we have

(E t |A(δ)|) γ ≤ H 2 (y; γ) ∀y ∈ [ȳ(γ), +∞).
Proof. We prove only (i), while (ii) being similar. Applying γ = 1 to the second inequality of (3.17), we get

|A(δ)| ≤ c 3 (1)e 1 4Λ(t+δ) E 2 0,t+δ e -1 8Λ(t+δ) g(m(δ)) 2 +2a|g(m(δ))|+2b .
Define

n 2 := max x∈R {-1 16Λ(t+δ) x 2 + 2a|x| + 2b} < +∞. Then |A(δ)| ≤ c 3 (1)e n 2 e 1 4Λ(t+δ) E 2 0,t+δ e -1 16Λ(t+δ) g(m(δ)) 2
.

So E t |A(δ)| ≤ c 3 (1)e n 2 E t [e 1 4Λ(t+δ) E 2 0,t+δ e -1 16Λ(t+δ) g(m(δ)) 2 ] ≤ c 3 (1)e n 2 E t e 1 2Λ(t+δ) E 2 0,t+δ E t e -1 8Λ(t+δ) g(m(δ)) 2 . Set n 3 := c 3 (1) 2 e 2n 2 E t [e 1 2Λ t+δ E 2 0,
t+δ ] which is a finite constant when ε (and hence δ) is sufficiently small, and

ξ := k ∆(t, δ) ∼ N (0, η 2 ) conditional on F t , where η 2 ≡ η(δ) 2 = Var(ξ) ≤ |k| 2 t+δ t |σ(s)| 2 ds. Then (E t |A(δ)|) 2 ≤ n 3 E t [e -1 8Λ(t+δ) g(m(δ)) 2 ] = n 3 E t [e -1 8Λ(t+δ) ln 2 u (v(y)-ξ) ] = n 3 √ 2π +∞ -∞ e -1 8Λ(t+δ) ln 2 u (v(y)+ηz) e -z 2 /2 dz = n 3 √ 2π +∞ -∞ e -1 8Λ(t+δ) ln 2 u (ηz) e -(z-v(y)/η) 2 /2 dz = n 3 √ 2π v(y)/(2η) -∞ + 0 v(y)/(2η) + +∞ 0 e -1 8Λ(t+δ) ln 2 u (ηz) e -(z-v(y)/η) 2 /2 dz.
We now find an integrable bound (as a function of y) for each of the three integrals in the above.

For the first integral, take y 0 < 0 small enough such that v(y 0 ) < 0 and u (v(y 0 )/2) > 1.

Then v(y) < 0 and u (v(y)/2) ≥ u (v(y 0 )/2) > 1 ∀y ≤ y 0 . Thus

v(y)/(2η) -∞ e -1 8Λ(t+δ) ln 2 u (ηz) e -(z-v(y)/η) 2 /2 dz < √ 2πe -1 8Λ(t+δ) ln 2 u (v(y)/2) ≤ c 4 (a 1 )u (v(y)/2) -a 1 , y ≤ y 0 ,
for any a 1 > 0 and some c 4 (a 1 ) < +∞. 7For the second integral, we have

0 v(y)/(2η)
e -1 8Λ(t+δ) ln 2 u (ηz) e -(z-v(y)/η) 2 /2 dz ≤ e -v(y) 2 /(8η 2 ) |v(y)/(2η)| ≤ c 5 e -v(y) 2 /(9η 2 )

for some constant c 5 > 0.

For the last integral, we get

+∞ 0 e -1 8Λ(t+δ) ln 2 u (ηz) e -(z-v(y)/η) 2 /2 dz = +∞ 0 e -1 8Λ(t+δ) ln 2 u (ηz) e -z 2 /2 e zv(y)/η e -v(y) 2 /(2η 2 ) dz < e -v(y) 2 /(2η 2 ) +∞ 0 e -z 2 /2 dz = π 2 e -v(y) 2 /(2η 2 ) .
Combining the three integrals, we conclude that, for any a 1 > 0, a 2 > 0, when ε ∈ (0, T -t)

is small enough and y 1 < 0 with v(y 1 ) sufficiently negative, there exists constant c 6 > 0 such that for any δ ∈ (0, ε),

(E t |A(δ)|) 2 ≤ c 6 [u (v(y)) -a 1 + e -a 2 v(y) 2 ] := H 1 (y; γ) 2/γ ∀y ≤ y 1 .
Finally, for any γ > 0, we can take suitable a 2 and a 3 (which may depend on γ) such that, in view of Assumption 2.5,

y(γ)
-∞ H 1 (y)dy < +∞, where y(γ) is sufficiently negative.

Theorem 3.6 There exists an integrable function H(•) such that when ε > 0 is sufficiently small, it holds that

w(t, E t [N (Y (ε, y))]) -w(t, N (Y (0, y))) ε ≤ H(y) ∀y ∈ R.
Proof. Fix ε > 0. For any y ∈ R, by the mean-value theorem, there exists δ ∈ [0, ] (which depends on t and y) such that

w(t, E t [N (Y (ε, y))]) -w(t, N (Y (0, y))) ε = w p (t, E t [N (Y (δ, y)])E t [A(δ)] ≤ w p (t, E t [N (Y (δ, y)])E t [|A(δ)|],
where we have used the fact that, by virtue of (3.9),

d dδ E t [N (Y (δ, y))] = E t [A(δ)]
. By Assumption 3.1-(iv), we have

w(t, E t [N (Y (ε, y))]) -w(t, N (Y (0, y))) ε ≤ c E t [|A(δ)|] (E t [N (Y (δ, y)]) -m + c E t [|A δ |] (1 -E t [N (Y (δ, y)]) -m
for some c > 0 and m ∈ (-1, 0).

Recall that E t [N (Y (δ, y)] = P t u(X(T ) + k ∆(t, δ)) > y . Choose y 1 and y 2 such that

; (1 + m)/2)M 1 ((2m)/(m -1)) (1-m)/2 , Ĥ2 (y) := 2cH 2 (y; (1 + m)/2)M 1 ((2m)/(m -1)) (1-m)/2 .
Then:

(i) For any y < y, we deduce

w(t, E t [N (Y (ε, y))]) -w(t, N (Y (0, y))) ε ≤ 2c E t [|A(δ)|] (E t [N (Y (δ, y)]) -m ≤ 2c(E t [|A(δ)|]) (1+m)/2 (E t [|A(δ)|]) (1-m)/2 (E t [N (Y (δ, y)]) -m ≤ 2cH 1 (y; (1 + m)/2)M 1 ((2m)/(m -1)) (1-m)/2 = Ĥ1 (y).
(ii) Similarly, for any y > ȳ, we have w(t,Et[N (Y (ε,y))])-w(t,N (Y (0,y))) ε ≤ Ĥ2 (y).

(iii) Finally, w(t,Et[N (Y ( ,y))])-w(t,N (Y (0,y))) ≤ 2cM 1 (-m) when y ∈ [y, ȳ].

Then H(•), where H(y) = Ĥ1 (y)1 y<y +2cM 1 (-m)+ Ĥ2 (y)1 y≥ȳ , y ∈ R, is the desired function.

Sufficient Conditions

Theorem 3.1 holds under three major assumptions: the existence of a positive constant κ such that the budget constraint (3.6) holds, the existence of a positive solution to the ODE (3.3), and the validity of the inequality (3.5). It is hard to verify these assumptions (and hence hard to tell if they are reasonable) because they are not imposed directly on the model primitives. This section explores equivalent or at least sufficient conditions, that are shown to be reasonable economically, under which these three assumptions are satisfied respectively.

Budget constraint (3.6)

The equation (3.6) is not a "standard" budget constraint appearing in the classical Merton problem because ρ(T ) and ρ(T ) are in general different. However, we have the following result.

E[ρ(T )I(κρ(T ))] = E Q I(κe -1 2 T 0 |λ(s)θ(s)| 2 ds-T 0 λ(s)θ(s) dW (s) ) = E Q I(κe -1 2 T 0 |λ(s)θ(s)| 2 ds+ T 0 λ(s)|θ(s)| 2 ds-T 0 λ(s)θ(s) d W (s) ) = E I(κe -1 2 T 0 |λ(s)θ(s)| 2 ds+ T 0 λ(s)|θ(s)| 2 ds-T 0 λ(s)θ(s) dW (s) ) = E Q I(κe -1 2 T 0 |λ(s)θ(s)| 2 ds+ T 0 λ(s)|θ(s)| 2 ds-T 0 λ(s)θ(s) d W (s) ) = E ρ(T )I(κe -1 2 T 0 |λ(s)θ(s)| 2 ds+ T 0 λ(s)|θ(s)| 2 ds-T 0 λ(s)θ(s) d W (s) ) = E ρ(T )I κe T 0 λ(s)(1-λ(s))|θ(s)| 2 ds ρ(T ) .
This establishes the desired equivalence with κ = κe

Existence of positive solutions to (3.3)

The main result of this paper depends crucially on the existence of a positive solution to the ODE (3.3). Note that this equation is highly nonlinear, and singular at t = T in that the denominator of the right hand side of the equation is 0 at T . In this subsection we provide conditions under which (3.3) admits positive solutions, by applying a general existence result for a class of ODEs with singular initial/terminal values (see Appendix A).

Setting

y(t) = Λ(T -t), (3.3) is equivalent to      y (t) = |θ(T -t)| 2 h(T -t, √ y(t)) h x (T -t, √ y(t)) 2 y(t), t ∈ (0, T ], y(0) = 0. (4.2)
In the rest of this subsection we study equation (4.2) instead of (3.3). The key idea is to first establish the local existence in the right neighborhood of t = 0, and then extend it globally to the whole time interval [0, T ].

We introduce the following assumption on the function h(•, •) (which depends directly on the probability weighting function) and on the market represented by θ(•):

Assumption 4.2 (i) h x (t, 0) ≥ 0 and h x (t, 0) ≥ 0 ∀t ∈ [0, T ]. (ii) lim sup t↑T h x (t,0) √ |θ(t)| 2 (T -t)
< 1 and lim inf t↑T |θ(t)| 2 > 0.

(iii) sup t∈[0,T ] h(t, 1) < +∞ and lim sup t↑T h x (t, 1) < +∞.

(iv) inf t∈[0,T ] h x (t, 0) > 0. (i) h(t, x) and h (t, x) are both increasing in x ≥ 0, and

h x (t, x) is convex in x ≥ 0. (ii) xh x (t, 0) ≤ h x (t, x) ≤ h x (t, 0) + xh x (t, x) ∀x ≥ 0.
Proof. (i) It follows from Lemma 2.7-(iii) that h x (t, x) and h

(3)

x (t, x) are both increasing in x ≥ 0. Assumption 4.2-(i) then leads to the desired results readily.

(ii) Applying the Taylor expansion, we have for any x > 0, there exists ζ ∈ [0, x] such that

h x (t, x) = h x (t, 0) + xh x (t, 0) + 1 2 x 2 h (3) x (t, ζ) ≥ xh x (t, x).
On the other hand, the convexity of h x (t, x) in x ≥ 0 implies that

h x (t, x) ≤ h x (t, 0) + xh x (t, x).
The proof is complete.

Lemma 4.4 Under Assumption 4.2, there exist k 1 > 0 greater than any given number and

δ 1 > 0 such that |θ(T -t)| 2 √ th(T -t, √ k 1 t) h x (T -t, √ k 1 t) 2 < 1 ∀t ∈ (0, δ 1 ]. (4.3)
Moreover, there exist k 2 > 0 less than any given positive number and δ 2 > 0 such that where the finiteness is due to Assumption 4.2-(iii). On the other hand, it follows from Lemma 4.3-(ii) that, for any k > 0,

|θ(T -t)| 2 √ th(T -t, √ k 2 t) h x (T -t, √ k 2 t) 2 > 1 ∀t ∈ (0, δ 2 ]. ( 4
h x (T -t, √ kt) √ t ≥ √ kh x (T -t, 0) ≥ √ k inf s∈[0,T ] h x (s, 0) > 0, (4.6)
where the last inequality is due to Assumption 4.2-(iv). Combining (4.5) and (4.6) and noting the boundedness of θ(•), we conclude that there is k = k 1 greater than any given number such that (4.3) is satisfied.

Next, by Lemma 4.3-(ii), we deduce that for sufficiently small k > 0, lim sup

t↓0 h x (T -t, √ kt) |θ(T -t)| 2 t ≤ lim sup t↓0 h x (T -t, 0) |θ(T -t)| 2 t + √ k h x (T -t, √ kt) |θ(T -t)| 2 ≤ lim sup t↓0 h x (T -t, 0) |θ(T -t)| 2 t + √ k lim sup t↓0 h x (T -t, 1) |θ(T -t)| 2 < 1,
where the last inequality follows from Assumption 4.2-(ii) and -(iii). Since h(T -t, √ kt) ≥ h(T -t, 0) = 1, we conclude that there is k = k 2 less than any given positive number so that (4.4) is satisfied.

The following establishes the local existence of the ODE (4.2). Proof. Fix k 1 > 0 and 0 < k 2 < k 1 as in Lemma 4.4, and take δ = δ 1 ∧ δ 2 . Then it is easy to check that β(t) = k 1 t and α(t) = k 2 t satisfy all the requirements in Theorem A.2 on (0, δ];

hence the result.

Next we extend the local solution y(•) obtained in Proposition 4.5 to the whole time interval

[0, T ]. Denote y(δ) = x 1 > 0.
Without loss of generality, we assume x 1 < 1 (otherwise, we can make δ closer to 0 to reduce x 1 ).

Proposition 4.6 Under Assumption 4.2-(i) and (iv), for any δ > 0 and 0

< x 1 < 1, the equation      y (t) = |θ(T -t)| 2 h(T -t, √ y(t)) h x (T -t, √ y(t)) 2 y(t), t ∈ (δ, T ], y(δ) = x 1 (4.7) has a solution y(•) ∈ C 1 [δ, T ] with y(t) > 0 ∀t ∈ [δ, T ].
Proof. Since ln h(x, t) is convex in x ≥ 0 (by Lemma 2.7-(iv)), we have

h x (t, x) h(t, x) ≥ h x (t, √ x 1 ) h(t, √ x 1 ) ≥ √ x 1 h x (t, 0) h(t, √ x 1 ) ≥ √ x 1 h x (t, 0) h(t, 1) ≥ c 1 > 0 ∀t ∈ [0, T ], x ≥ √ x 1 ,
where the second inequality is due to Lemma 4.3-(ii) and c 1 (which may depend on x 1 ) is a constant arising from Assumption 4.2-(iii) and -(iv). Hence for any y ≥ x 1 , we have

|θ(T -t)| 2 h(T -t, √ y) h x (T -t, √ y) 2 y ≤ c 2 y,
where c 2 > 0 is a constant depending on c 1 and the bound of |θ(•)| 2 .

Denote c 3 = x 1 e c 2 T < +∞, and a truncation function r(y) = (y ∨ x 1 ) ∧ c 3 for y ≥ 0.

Consider the ODE

   y (t) = f (t, y(t)), t ∈ (δ, T ], y(δ) = x 1 (4.8)
where

f (t, y) := |θ(T -t)| 2 h(T -t, r(y)) h x (T -t, r(y)) 2 r(y).
It is easy to show that f (•, •) satisfies the conditions in Theorem A.3 on (δ, T ]; hence (4.8) admits a solution y(•). Moreover, since y (t) ≥ 0 we have

y(t) ≥ x 1 ∀t ∈ [δ, T ].
Take

t 1 := inf{t ∈ (δ, T ] : y(t) ≥ c 3 } ∧ T . If t 1 < T , then y(t 1 ) = c 3 and y(t) < c 3 ∀t < t 1 . Consequently y (t) = |θ(T -t)| 2 h(T -t, √ y(t)) h x (T -t, √ y(t)) 2 y(t) ≤ c 2 y(t), ∀t ∈ (δ,
Λ(•) ∈ C[0, T ] ∩ C 1 [0, T ) satisfying Λ(t) > 0 ∀t ∈ [0, T ).

Inequality (3.5)

We now provide conditions on the model primitives under which the inequality (3.5) holds. We A convex weighting function captures risk aversion in terms of exaggerating the small probability of very "bad" events while downplaying very "good" events; see [START_REF] Yaari | The dual theory of choice under risk[END_REF].

On the other hand, as discussed in Introduction, an inverse S-shaped probability weighting function is more interesting as it reflects the tendency of inflating the small probabilities of both tails which are consistent with the conclusions of many experimental and empirical works.

I (e √ Λ(t)y-c ) = 0 ∀c ∈ R, ∀t ∈ [0, T ). (4.9)
Proof. It follows from (3.16) that 0 ≤ w p (t, N (y))

-I (e

√ Λ(t)y-c ) ≤ Kw p (t, N (y))(e α √ Λ(t)y-αc + 1). However +∞ -∞ w p (t, N (y))(e α √ Λ(t)y-αc + 1)N (y)dy = e -αc h(t, α Λ(t)) + 1 < +∞, implying lim |y|→+∞ w p (t, N (y))(e α √ Λ(t)y-αc + 1)N (y) = 0.
This completes the proof.

We now analyze the integral in (3.5) by decomposing it into M 0 + M , where

M 0 = +∞ -∞ w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) c -g(x) Λ(t) g (x) 2 du(x),
and

M = +∞ -∞ w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) g (x)du(x) = +∞ -∞ w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) u (x)dg (x).
Applying integration by parts and noting that N (x) = -xN (x), we can further decompose

M into -(M 1 + M 2 + M 3 )
where

M 1 = +∞ -∞ g (x)w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) 2 u (x) -g (x) Λ(t) dx M 2 = +∞ -∞ g (x)w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) u (x) -g (x) Λ(t) dx = - +∞ -∞ g (x)w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) u (x) c -g(x) Λ(t) -g (x) Λ(t) dx = +∞ -∞ g (x) 2 w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) u (x) c -g(x) Λ(t) dx ≡ M 0 , M 3 = +∞ -∞ g (x)w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) u (x)dx,
assuming that for any c ∈ R,

w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) u (x)g (x) = 0 when |x| → +∞. (4.10)
Since g (x) = -u (x) u (x) , (4.10) can be written as

w p t, N c -g(x) Λ(t) N c -g(x) Λ(t) u (x) = 0 when |x| → ∞,
which is equivalent to (4.9). Now, denoting y = c-g(x) √ Λ(t)

, we have

M 0 + M = -(M 1 + M 3 ) = - +∞ -∞ g (x)N (y) w p (t, N (y)) N (y)u (x) dy dx + w p (t, N (y))) u (x) dx = - +∞ -∞ g (x)N (y)d w p (t, N (y)) u (x) = - -∞ +∞ g (f (c -Λ(t)y))N (y)d w p (t, N (y)) u (f (c -Λ(t)y)) = +∞ -∞ N (y) f (c -Λ(t)y) d w p (t, N (y)) u (f (c -Λ(t)y)) .
Hence, inequality (3.5) is equivalent to

+∞ -∞ N (y) f (c -Λ(t)y) d w p (t, N (y)) u (f (c -Λ(t)y)) ≥ 0 (4.11)
for any c ∈ R, a.e. t ∈ [0, T ).

An obvious sufficient condition for (4.11) is that

w p (t, N (y)) u (f (c -Λ(t)y)) ≡ w p (t, N (y)) e √ Λ(t)y-c
is increasing in y for any c ∈ R, which holds automatically if w(t, •) is convex. -κI (κe

√ Λ(t)y-c )
dy > 0, and

M 5 = +∞ -∞ ∂w p (t,N (y)) ∂y -κI (κe √ Λ(t)y-c ) N (y)dy = E[F (ξ)G(ξ)] with F (y) = ∂w p (t,N (y)) ∂y and G(y) = 1 -κI (κe √ Λ(t)y-c ) > 0.
Since w(t, •) is inverse S-shaped, there exists q ∈ (0, 1) such that w p (t, p) ≤ 0 on (0, q), w p (t, p) ≥ 0 on (q, 1), and w p (t, q) = 0. Furthermore, for any b > 0, Denote c = N -1 (q). Then F (ξ) is negative on (-∞, c), positive on (c, +∞), and F (c) = 0.

Since u (x) is increasing in x, so is I (x) = 1/u (I(x)); hence G(•) is increasing. Then we have

F (ξ)(G(ξ) -G(c)) ≥ 0. Consequently, 0 ≤ E[F (ξ)(G(ξ) -G(c))] = E[F (ξ)G(ξ)] -G(c)E[F (ξ)] = E[F (ξ)G(ξ)] -G(c)h x (t, 0),
which implies

E[F (ξ)G(ξ)] ≥ G(c)h x (t, 0) ≥ 0.
The proof is complete.

An Example

In this section, we give a concrete example to demonstrate our results. where N is the probability distribution function of a standard normal. So in this example our weighting function is time-invariant, and we will drop t and write w(p) = w(t, p) throughout this section.

It is clear that w(0) = 0, w(1) = 1, w (p) = F (N -1 (p)) N (N -1 (p)) > 0. So w is a probability weighting function satisfying Assumption 2.3-(i). Moreover, we have

w(N (x)) = F (x) = N (x/ √ 2), w (N (x)) = F (x) N (x) = 1 √ 2 N (x/ √ 2) N (x) = 1 √ 2 e x 2 /4 , h(t, x) ≡ E[w (N (ξ))e xξ ] = 1 √ 2 R 1 √ 2π e xz e -z 2 /4 dz = e x 2 .
Now, take any m ∈ (-1, -1/2). Then

lim p↓0 w (p) p m = lim x→-∞ w (N (x)) N (x) m = lim x→-∞ 1 √ 2 e x 2 /4 N (x) m = 1 √ 2 lim x→-∞ e x 2 /(4m) N (x) m = 1 √ 2 lim x→-∞ xe x 2 /(4m) /2m N (x) m = 1 √ 2 1 2m lim x→-∞ xe x 2 ( 1 4m + 1 2 ) m = 0.
Also, we have lim p↑0

w (p) (1-p) m = lim q↓0 w (1-q) q m = lim q↓0 w (q)
q m = 0. So Assumption 2.4-(iv) holds. Since h(t, x) = e x 2 , it is straightforward to verify that Assumption 4.2 is satisfied. Moreover,

w (p) = 1 2 e (N -1 (p)) 2 /4
; so w is concave on [0, 1/2] and convex on [1/2, 1]. It is therefore inverse S-shaped, satisfying Assumption 4.8. We have now checked the validity of all the assumptions on w in this paper.

With the explicit form of h, the ODE (3.3) becomes

Λ (t) = - 1 4 |θ(t)| 2 , t ∈ [0, T ), Λ(T ) = 0, whose solution is Λ(t) = 1 4 T t |θ(s)| 2 ds. Thus, λ(t) = -Λ (t)/|θ(t)| 2 = 1/2.
As discussed at the end of Section 4.1, λ(t) = 1/2 means that the RDU agent in this example acts like an EUT agent with the market price of risk is revised to half of the original one. In other words, because of the probability weighting and the consistent planning (recall that consistent planning amounts to a heavily constrained optimization), the agent will lose half of the risk premium or Sharpe ratio.

To complete this example, we present the equilibrium portfolio π * when the exponential utility function is u(x) = 1 -e -αx , x ∈ R, for some α > 0, which clearly satisfies all the assumptions on the utility function in this paper.

In this case, u (x) = αe -αx and hence I(x) = ln α-ln x α , x > 0. It is easy to check that there exists κ > 0 satisfying the budget constraint (3.6). Hence the desired terminal wealth is

X * (T ) = I(κρ(T )) = c + 1 α T 0 λ(t)θ(t) d W (t),
where W (t) = W (t)+ t 0 θ(s)ds is a standard Brownian motion under the risk-neutral measure, Q, of the original market, and c is a constant dependent of the market parameters.

By the pricing theory, the replicating wealth process X * (•) of X * (T ) is a Q-martingale (recall that the risk-free rate has been assumed to be 0); hence

X * (t) = c + 1 α t 0 λ(s)θ(s) d W (s), t ∈ [0, T ].
Matching the above with the wealth equation dX * (t) = π * (t) σ(t)d W (t), we obtain the equilibrium portfolio

π * (t) = 1 α λ(t)(σ(t) ) -1 θ(t).
Recall that the optimal portfolio of an EUT agent with the same exponential utility is π EU T (t) = 1 α (σ(t) ) -1 θ(t). Hence the risky investment of the RDU agent at t is that of the EUT agent multiplied by λ(t). Note that this result does not depend on the spcific form of the weighting function so long as λ(•) exists. In the special case when w(t, p) = F (N -1 (p)), λ(t) ≡ 1/2; so the risk exposure is reduced by half.

Reduction in Risk Premium

We have seen that the sophisticated RDU agent behaves as if the risk premium is factored by λ(•). In the example with the specific probability weighting function presented in Section 5, λ(t) = 1/2; so the risk premium is reduced and the agent acts more cautiously than her EUT counterpart. In this section, we answer the general question of when there is a reduction in risk premium or, equivalently, when λ(t) < 1.

We assume Λ(•) exists. It follows from (3.3) that

λ(t) = Λ(t)h(t, Λ(t) ) h x (t, Λ(t) ) > 0, t ∈ [0, T ).
For any x ∈ R + , define a probability measure Q x by dQ x dP = e -x 2 2 +xξ , under which ξ -x ∼ N (0, 1). So Proof. This is obvious.

h(t, x) = e x 2 2 E[w p (t, N (ξ))e -x 2 2 +xξ ] = e x 2 2 E Q x [w p (t, N (ξ -x + x))] = e x 2 2 E[w p (t, N (ξ + x))] = e x 2
In the example in Section 5, H(t, x) = e -x 2 /2 h(t, x) = e x 2 /2 , which is strictly increasing in x ∈ R + ; hence Corollary 6.2 applies. On the other hand, as explained earlier, a strict convexity of w(t, p) in p underlines strict overweighting of the left tail (i.e. the bad events) and strict underweighting of the right tail (i.e. the good events); so it enhances the level of risk aversion leading to a smaller risk premium and less risky exposure.

Even if w is more general including being inverse S-shaped, it is still possible that λ(t) < 1, as demonstrated by the example in Section 5. In this case, the equivalent condition H x (t, Λ(t)) > 0 can be used to uncover the balance between the risk-averse component and the risk-seeking component in probability weighting needed to render an overall reduction in risk premium. We leave a detailed study along this direction to interested readers.

Concluding Remarks

A continuous-time RDU portfolio selection problem is inherently time inconsistent. A sophisticated agent, realizing that in the future she herself might disagree to her current planning, resorts to consistent investment by implementing intra-personal equilibrium strategies from which she will have no incentive to deviate at any point in time. We have solved the open problem of deducing such equilibrium strategies, by developing an approach that to our best knowledge is new to the literature. The main technical thrust of our approach is to express the first-order derivative of the small deviation from an equilibrium as a quadratic function of the deviating amount. The definition of the equilibrium requires the quadratic function to have a constant sign whatever the amount might be. This leads to an equality (which in turn leads to an ODE) and an inequality, the two constituting the main sufficient conditions for deriving explicitly the final wealth profile and, hence, the resulting strategy courtesy of the market completeness.

With an intra-personal equilibrium strategy, the agent, at any given time, in effect solves a constrained dynamically optimal RDU model in which the constraint is to honor all her future strategies. One may think this would lead to an extremely complicated terminal wealth profile.

Our result, however, shows that the terminal wealth is surprisingly simple -it resembles that of an optimal Merton portfolio, except that the investment opportunity set needs to be modified properly. In other words, the RDU agent behaves as if she was an EUT agent, only that she is in a fictions market where she blends her probability weighting function into the market price of risk. This observation may in turn shed lights on finding intertemporal market equilibria for markets where all the agents are EUT and/or RDU consistent planners.

We derived our equilibrium strategy based on an Ansatz; as such, our results do not rule out the possibility of having other equilibria beyond our Ansatz. In general, uniqueness of intra-personal equilibrium for time-inconsistent problems remains a very challenging research question. 10 It is particularly the case for the RDU model, or so we believe. 10 To our best knowledge, Hu et al ( 2017) is the only paper that addresses the uniqueness in continuous time.

Denote a function α(p; δ)

:= [p δ + (1 -p) δ ] -1 δ , p ∈ [0, 1], (B.3)
with parameter δ ∈ (0, 1). It is easy to see that 1 ≤ p δ + (1 -p) δ < 2; hence

2 -1 δ < α(p; δ) ≤ 1. (B.4) Moreover, α(p; δ) = α(1 -p; δ), w T K (p; δ) = p δ α(p; δ). Hence α (p; δ) = - 1 δ [p δ + (1 -p) δ ] -1 δ -1 δ[p δ-1 -(1 -p) δ-1 ] = -α(p; δ) 1+δ [p δ-1 -(1 -p) δ-1 ],
w KT (p; δ) = δp δ-1 α(p; δ) + p δ α (p; δ) = δα(p; δ)p δ-1 + α(p; δ) 1+δ p δ (1 -p) δ-1 -α(p; δ) 1+δ p 2δ-1

< p δ-1 + (1 -p) δ-1 ,
where the last inequality is by (B.4). This completes the proof. Proof. The statement is symmetric for x ≤ 0 and x ≥ 0; hence it suffices to prove for the case when x < 0 (the case of x = 0 is trivial). According to Lemma B.4, we know h(t, 2) ≤ K for some constant K independent of t. However, h(t, x) is increasing in x; so sup t∈[0,T ] h(t, 1) ≤ sup t∈[0,T ] h(t, 2) < K.

Next, by the fact that ξ 2 < e ξ + e -ξ we have h x (t, 1) ≤ h(t, 2) + h(t, 0) < K + 1.

Finally, we check Assumption 4.2-(ii). We first need two lemmas.

  and Ng 2000, Zhou and Li 2000, He and Zhou 2011), and later ones gradually shifted to Type 3, consistent planners or sophisticated agents (Ekeland and Lazrak 2006, Bjork and Murgoci 2010, Hu et al 2012, Hu et al 2017, Bjork et al 2014).

Assumption 2 . 2

 22 The functions µ(•), σ(•) and θ(•) are all right continuous and uniformly bounded, and θ(t) = 0 ∀t ∈ [0, T ].

3. 3 1 √

 31 The martingale condition (3.10) Proposition 3.2 The martingale condition (3.10) holds for sufficiently small ε > 0. Proof. As before the variable y is suppressed. It suffices to prove that B(s) is locally square integrable at s = 0+. Note that λ(•), θ(•), σ(•) are all bounded, and Λ(t+s) is locally bounded at s = 0+. It thus follows from the expression of B(•) that we only need to estimate a bound of N (Y (s))g (m(s)). Because lim sup y→+∞ y -α -I (y) < +∞ (Assumption 2.4-(i)) and 1 -I (y) ≡ -u (I(y)) is increasing in y (Assumption 2.4-(iii)), we have that there exists K > 0 such that 1 -I (y) ≤ K(y α + 1) ∀ y > 0. (3.16)

1 2 Λ

 2 (0), Assumption 2.4-(ii) is satisfied with l(•) replaced by g(•) (and the constants a and b properly modified). Hence,

Proposition 3 . 5

 35 for any δ ∈ (0, ]. This leads to the desired inequality. We have the following conclusions:

E

  t [N (Y (δ, y 1 )] < 1/2 and E t [N (Y (-δ, y 2 )] < 1/2 for all δ ∈ (0, ε]. Using notations in Lemma 3.4 and Proposition 3.5, denote y := y((1 + m)/2)) ∧ y 1 , ȳ := ȳ((1 + m)/2) ∨ y 2 , and Ĥ1 (y) := 2cH 1 (y

Theorem 4 . 1 . 1 )

 411 The following two statements are equivalent:(i) There exists κ > 0 such that (3.6) holds.(ii) There exists κ > 0 such thatE[ρ(T )I(κρ(T ))] = x 0 . (4Proof. Let Q and Q be respectively the equivalent martingale measures corresponding to ρ(T ) and ρ(T ); namely, dQ dP = ρ(T ), d Q dP = ρ(T ). By Girsanov's theorem, W (•) and W (•) are respectively Brownian motions under Q and Q, where W (t) := W (t) + t 0 θ(s)ds, W (t) := W (t) + t 0 λ(s)θ(s)ds, t ∈ [0, T ]. Now, we have

T 0 λ

 0 (s)(1-λ(s))|θ(s)| 2 ds . So the assumption in Theorem 3.1 regarding the existence of a positive constant κ satisfying (3.6) boils down to the familiar condition (4.1). The latter condition is standard in the classical Merton problem in which the pricing kernel is ρ(T ) or, equivalently, the market price of risk process is λ(•)θ(•). Note that because the probability weighting function w has been embedded into λ(•), the existence of a positive κ satisfying (4.1) becomes a condition on the utility function u only, which is satisfied by, say, the exponential utility. 8 More important, with Theorem 4.1, Theorem 3.1 shows that the investment behavior of the sophisticated RDU agent is indistinguishable from an EUT maximizer in a market where the market price of risk is revised from θ(•) to λ(•)θ(•). This finding may have important economic implications especially in the study of market equilibria.

Assumption 4 . 2 -

 42 (iii) and -(iv) are mild. Assumption 4.2-(i) can be relaxed with a more subtle analysis than the one to be given below; although we will not pursue in that direction.The first part of Assumption 4.2-(ii) is the most important of all, which regulates how the probability weighting function w(t, •) should behave, given the market, when t is sufficiently close to the terminal time T .9 Mathematically, this terminal behavior of weighting functions is translated into the singularity of the ODE (3.3) at T , which is why Assumption 4.2 is needed for a proof of the existence of (3.3). Luckily, we will show in Appendix B that all the parts of Assumption 4.2 are satisfied by a family of time-varying Tversky-Kahnamen's weighting functions.We first strengthen Lemma 2.7 under Assumption 4.2-(i).

Lemma 4 . 3

 43 Under Assumption 4.2-(i) in addition to the same assumption of Lemma 2.7, for any t ∈ [0, T ], h(t, •) has the following properties:

. 4 )

 4 Proof. By Lemma 4.3-(i), h(T -t, x) is increasing when x ≥ 0; hence lim sup t↓0 h(T -t, √ kt) ≤ lim sup t↓0 h(T -t, 1) < +∞, (4.5)

Proposition 4 . 5

 45 Under Assumption 4.2, equation (4.2) admits a solution y(•) ∈ C[0, δ] ∩ C 1 (0, δ] on a time interval [0, δ] for some δ > 0, with y(t) > 0 ∀t ∈ (0, δ].

Assumption 4 . 8

 48 assume that the ODE (3.3) admits a positive solution Λ(•) ∈ C[0, T ] ∩ C 1 [0, T ), and introduce the following additional assumption. For a.e. t ∈ [0, T ), w(t, •) is either convex or inverse S-shaped.

Lemma 4 . 9

 49 Under Assumptions 4.2 and 4.8, we have lim |y|→+∞ w p (t, N (y))N (y)

Theorem 4 .f

 4 10 Under Assumptions 4.2 and 4.8, the inequality (3.5) holds.Proof. We have shown above that (3.5) holds at t when w(t, •) is convex. Let us now focus on the case when w(t, •) is inverse S-shaped.First of all, it follows from Theorem 4.7 that the ODE (3.3) admits a positive solutionΛ(•) ∈ C[0, T ] ∩ C 1 [0, T ). Fix t ∈ [0, T ). Noting f (x) = -κI (κe -x )e -x where κ = κe -1 2 Λ(0) > 0, we can rewrite condition (c -Λ(t)y) d w p (t, N (y)) u (f (c -Λ(t)y)) )y-c ) w p (t, N (y)) Λ(t) + ∂w p (t, N (y)) ∂y dy = Λ(t)M 4 + M 5 , where M 4 = +∞ -∞ w p (t,N (y))N (y)

0 Nw

 0 (y)d[w p (t, N (y))] = (w p (t, N (y))N (y))| y=b y=0 -b a w p (t, N (y))dN (y) = w p (t, N (b))N (b) -t, N (y))ydN (y). Since w p (t, p) ≥ 0 for sufficiently large p, b 0 ∂w p (t,N (y)) ∂y N (y)dy is increasing in b for sufficiently large b. Hence lim b↑+∞ b 0 ∂w p (t,N (y)) ∂y N (y)dy exists. However, +∞ > h x (t, 0) ≡ E[w p (t, N (ξ))ξ], we conclude that lim b↑+∞ b 0 w t (N (y))ydN (y) exists and is finite. This implies that a := lim b↑+∞ w p (t, N (b))N (b) exists. It then follows from the fact that +∞ > h(t, 0) ≡ E[w p (t, N (ξ))] that a = 0, i.e., +∞ 0 ∂w p (t, N (y)) ∂y N (y)dy =p (t, N (y))ydN (y).Similarly, we can show that 0 -∞ ∂w p (t, N (y)) ∂y N (y)dy = 1 p (t, N (y))ydN (y). p (t, N (y))ydN (y) = h x (t, 0) ≥ 0.

DefineF

  (x) := N (x/ √2), and w(t, p) := F (N -1 (p)),

w ( 1 -w ( 1 -

 11 Before we move to the next assumption, for any probability weighting function w : [0, 1] → [0, 1], define (with a slight abuse of notation) h(x; w) := E[w (N (ξ))e xξ ], x ∈ R. It then follows from Lemma 2.7-(ii) that, for any odd number n ≥ 1, h (n) (0; w) = E[w (N (ξ); δ)ξ n ] (y))y n N (y)dy + +∞ 0 w (N (y))y n N (y)dy = 0 +∞ w (N (-y))(-y) n N (-y)d(-y) + +∞ 0 w (N (y))y n N (y)dy = -+∞ 0 N (y))y n N (y)dy + +∞ 0 w (N (y))y n N (y)dy = +∞ 0 [w (N (y)) -w (1 -N (y))]y n N (y)dy, N (y))y 2 N (y)dy + +∞ 0 w (N (y))y 2 N (y)dy = +∞ 0 [w (N (y)) + w (1 -N (y))]y 2 N (y)dy. (B.7) Proposition B.2 w satisfies Assumption 4.2-(i). Proof. With a slight abuse of notation, define h(x; δ) := E[w KT (N (ξ); δ)e xξ ], x ∈ R, where δ ∈ (0, 1]. Again, as t ∈ [0, T ] is fixed in Assumption 4.2-(i), we can drop t and need only to show that h (0; δ) ≥ 0 and h (0; δ) ≥ 0.Noting thatw T K (1 -p; δ) = δ(1 -p) δ-1 α(p; δ) -(1 -p) δ α (p; δ), we have w T K (p; δ) -w T K (1 -p; δ) = δp δ-1 α(p; δ) + p δ α (p; δ) -δ(1 -p) δ-1 α(p; δ) -(1 -p) δ α (p; δ) = δα(p; δ)[p δ-1 -(1 -p) δ-1 ] + α (p; δ)[p δ + (1 -p) δ ] = δα(p; δ)[p δ-1 -(1 -p) δ-1 ] + α (p; δ)α(p; δ) -δ = δα(p; δ)[p δ-1 -(1 -p) δ-1 ] + α(p; δ)[(1 -p) δ-1 -p δ-1 ] = (δ -1)α(p; δ)[p δ-1 -(1 -p) δ-1 ] > 0 ∀p ∈ (1/2, 1]. (B.8)Now Assumption 4.2-(i) holds by virtue of (B.6) where we take w = w P K (•; δ).Proposition B.3 If inf t∈[0,T ] δ(t) > 0, then w satisfies Assumption 4.2-(iv).Proof. We havew T K (p; δ) + w T K (1 -p; δ) = δp δ-1 α(p; δ) + p δ α (p; δ) + δ(1 -p) δ-1 α(p; δ) -(1 -p) δ α (p; δ) = δα(p; δ)[p δ-1 + (1 -p) δ-1 ] + α (p; δ)[p δ -(1 -p) δ ] > δα(p; δ)[p δ-1 + (1 -p) δ-1 ] > 2 -1 δ δ ∀p ∈ (1/2, 1]. It hence follows from (B.7) that h x (t, 0) > +∞ 0 2 -1 δ(t) δ(t)y 2 dN (y) = 2 -1 δ(t) δ(t)/2 ≥ inf t∈[0,T ] [2 -1 δ(t) δ(t)]/2 > 0,owing to the condition that inf t∈[0,T ] δ(t) > 0.For Assumption 4.2-(iii), we need the following lemma.Lemma B.4 For any ∈ (0, 1), we have E[N (ξ) -e xξ ] < +∞ ∀x ∈ R.

  Since E[N (ξ) -e xξ 1 ξ≥-1 ] ≤ e x E[N (ξ) -] = e x 1-, we only need to focus on E[N (ξ) -e xξ 1 ξ<-1 ]. By the fact that for any γ > 0, lim y→-∞ N (y)e γy = lim y→-∞ N (y) -γe -γy = 0, we deduce that there exists a constant K > 0 such that N (y) (1-)/2 e xy ≤ K ∀y < -1. Thus E[N (ξ) -e xξ 1 ξ<-1 ] ≤ KE[N (ξ) -(1+ )/2 ] < +∞.The proof is complete.Proposition B.5 If inf t∈[0,T ] δ(t) > 0, then w satisfies Assumption 4.2-(iii).Proof. Take := 1∧inf t∈[0,T ] δ(t) > 0. For any t ∈ [0, T ], it follows from the bound of w T K (p; δ)(see the proof of Proposition B.1) that h(t, 2) = E[w T K (N (ξ); δ(t))e 2ξ ] ≤ E[N (ξ) δ(t)-1 e 2ξ ] + E[N (-ξ) δ(t)-1 e 2ξ ] = E[N (ξ) δ(t)-1 e 2ξ ] + E[N (ξ) δ(t)-1 e -2ξ ] ≤ E[N (ξ) -1 e 2ξ ] + E[N (ξ) -1 e -2ξ ].

Lemma B. 6 2 N 2 N 2 = lim y→+∞ δ 2 N 1 0 1 y

 6222211 There exists δ ∈ (0, 1) such that sup δ∈[ δ,1] +∞ 0 ln[N (y) δ + N (-y) δ ]dy < +∞. Proof. Fix δ ∈ (0, 1]. Observe that lim y→+∞ ln[N (y) δ + N (-y) δ ] (y) δ-1 -N (-y) δ-1 N (y) δ + N (-y) δ (-y) δ-1 N (y) δ + N (-y) δ ≤ lim y→+∞ N (y) N (-y) 1-δ y -3 δ (y) 1-δ/2 N (-y) 1-δ N (y) δ/2 y -3 = 0.Hence, there exists a constant K > 0 (which may depend on δ), such that ln[N (y) δ + N (-y) δ ] < Ky -2 ∀ y > 1.As a result,+∞ 0 ln[N (y) δ + N (-y) δ ]dy ≤ ln(N (y) δ + N (-y) δ )dy + K T -2 dy < +∞.Note that the integrand on the left hand side is decreasing in δ; hence the above finiteness is uniform when δ is sufficiently close to 1.Recall we have definedh(x; δ) = E[w T K (N (ξ); δ)e xξ ], x ∈ R, δ ∈ (0, 1]. Lemma B.7 We have h (0; δ) = 1 -δ δ +∞ 0 ln[N (y) δ + N (-y) δ ]dy + o 1 -δ δwhen δ is sufficiently close to 1.Proof. By (B.6) and (B.8), we haveh (0; δ) = +∞ 0 (δ -1)α(N (y); δ)[N (y) δ-1 -N (-y) δ-1 ]ydN ((y); δ)yd[N (y) δ + N (-y) (y) δ + N (-y) δ ] 1-1/δ = +∞ 0 {1 -[N (y) δ + N (-y) δ ] 1-1/δ }dy,where we have used the fact that lim y→+∞ {1 -[N (y) δ + N (-y) δ ] 1-1/δ }y = 0.Applying the general Taylor expansion x = 1+ ln x+o(| |)(ln x) 2 for x > 0 and sufficiently small | |, we deduce+∞ 0 {1 -[N (y) δ + N (-y) δ ] 1-1/δ }dy = 1 -δ δ +∞ 0 ln[N (y) δ + N (-y) δ ]dy + o 1 -δ δ ,where we have used the finiteness +∞ 0 {ln[N (y) δ + N (-y) δ ]} 2 dy < +∞, which follows from the inequalities 0 < ln[N (y) δ + N (-y) δ ] < ln 2 and Lemma B.6. Proposition B.8 If lim sup t↑T 1 -δ(t) δ(t) √ T -t = 0 and lim inf t↑T |θ(t)| 2 > 0, then w satisfies Assumption 4.2-(ii).

  t 1 ). Grownwall's inequality then yields y(t 1 ) ≤ x 1 e c 2 t 1 < c 3 , which is a contradiction. Hence t 1 = T and y(t) ≤ c 3 on [δ, T ]. In this case equations (4.8) and (4.7) coincide.

	Combing Propositions 4.5 and 4.6, we arrive at
	Theorem 4.7 Under Assumption 4.2, the equation (3.3) admits a solution

The same approach is used to derive the stochastic maximum principle for optimal stochastic controls; see, e.g.,[START_REF] Yong | Stochastic Controls: Hamiltonian Systems and HJB Equations[END_REF].

Through an experiment with a large number of subjects, Noussair et al (2014) observe that the majority of individuals' decisions are consistent with prudence.

To save space, we will omit to write out the dependence of m and Y in y from this point of the proof, except when it is important to spell out this dependence. However, the reader is urged to bear in mind this dependence while reading the proof.

Recall that t is fixed; so any F t -measurable random vector k is almost surely deterministic conditional onF t .

This is based on the fact (which can be easily shown) that given α > 0, for any a > 0 there is a constant c(a) > 0 such that e -α ln 2 x ≤ c(a)x -a ∀x > 1.

Indeed, if this condition fails, than it is usually an indication that the original problem is not well-posed and/or an optimal solution is not attainable; see[START_REF] Jin | A convex stochastic optimization problem arising from portfolio selection[END_REF] for a detailed analysis on this constraint.

We believe that this is a distinctive feature of the continuous-time setting. In the discrete-time case, there is no infinitesimal issue of the weighting functions close to the terminal time.

In[START_REF] Agarwal | A survey of recent results for initial and boundary value problems singular in the dependent variable[END_REF], f * is assumed to be (jointly) continuous; but from the proofs therein, we can easily weaken it to this current version.

Partially supported by Lebesgue Center of Mathematics "Investissementsd'avenir"program-ANR-11-LABX-0020-01, by ANR CAESARS (Grant No. 15-CE05-0024) and by ANR MFG (GrantNo. 16-CE40-0015-01). The research of this author was partially supported by research grants from Oxford-Nie Financial Big Data Lab and Oxford-Man Institute of Quantitative Finance. The research of this author was supported through start-up grants at both University of Oxford and Columbia University, and research funds from Oxford-Nie Financial Big Data Lab, Oxford-Man Institute of Quantitative Finance, and Nie Center for Intelligent Asset Management.

Appendices A Existence of Solutions to a Class of ODEs

In this appendix we present some general existing results on a class of ODEs, taken from [START_REF] Agarwal | A survey of recent results for initial and boundary value problems singular in the dependent variable[END_REF]. Consider the following ODE    y (t) = f (t, y(t)), t ∈ (0, T ],

where f may not be defined at t = 0.

Denote by AC[0, s] the set of absolutely continuous functions on [0, s] where s > 0. The following assumption is introduced in Agarwal and O'Regan (2004).

2. There is α ∈ AC[0, t 0 ] ∩ C 1 (0, t 0 ] with α(t) ≤ β(t) ∀t ∈ [0, t 0 ] and α(0) ≤ 0 such that α (t) ≤ f (t, α(t)), t ∈ (0, t 0 ].

The function

is, in the region (0, t 0 ] × R, continuous in y for any t and measurable in t for any y,

The following two propositions, both drawn from

admits a solution y ∈ AC[t 0 , T ].

B Tversky-Kahnamen's Probability Weighting Functions

In this appendix we verify that a class of time-varying Tversky-Kahnamen (TK) probability weighting functions satisfy all the technical assumptions required in the paper.

First of all, the original TK weighting function, introduced in Tversky and Kahnamen (1992), is

where δ ∈ (0, 1] is a parameter. This is an inverse S-shaped function, with w T K (p; δ) > 1 when p is close to both 0 and 1. Moreover, a smaller δ implies a stronger degree of probability weighting. When δ = 1, there is no probability weighting.

We now vary the parameter δ over time to generate a family of time-dependent TK functions. Given a measurable function δ : [0, T ] → (0, 1] with 0 < δ(t) < 1 for t ∈ [0, T ) and

The purpose of this appendix is to show that w satisfies all the assumptions in the paper, under proper conditions on δ(•).

Clearly, w satisfies Assumption 2.3-(i).

Proposition B.1 w satisfies Assumption 2.4-(iv).

Proof. As t ∈ [0, T ) is fixed in Assumption 2.4-(iv), from the construction of w it suffices to prove the conclusion for w T K (•; δ) with δ ∈ (0, 1) fixed.

Proof. This follows immediately from Lemmas B.6 and B.7.

The second condition in the above is satisfied if we assume that θ(•) is left continuous at T and θ(T ) = 0, which means that we do not have a trivial market at T . The first condition means that as t approaches the terminal time T , δ(t) should approach δ(T ) ≡ 1 faster than √ T -t.

To summarize, the family of time-varying TK weighting functions satisfy all the assumptions of the paper if the measurable function δ(•) satisfies inf t∈[0,T ] δ(t) > 0, lim sup


Caroline Urmès

Joanne Obeid

Jean-Marc Schweitzer

Amandine Cabiac

Carine Julcour

Yves Schuurman

Periodic reactor operation for parameter estimation in catalytic heterogeneous kinetics. Case study for ethylene adsorption on Ni/AbO 3

Introduction

Periodic operation of chemical reactors has been of interest to chemical engineers in the last few decades, as it allows in certain cases to increase the reactor performance (Silveston et al., 1995). More recently, the importance of reactor/reaction dynamics has been stressed in the area of renewable resources where large flue tuations in flow and gas composition can occur (Katz et al., 2017). At the laboratory level, periodic operation of a catalytic reactor allows to get more detailed insight into the reaction mechanism and better parameter estimates of the underlying reactions

with T R ðe þ 1 e ð Þe p Þ L R vsg and T R 2 ð1 eÞ L R vsg
The transfer function of the whole reactor system in the fre quency domain becomes (with s jw):

H jw ð Þ e T R T R 2 C NM k 1 k 1 T 2 1 1þw 2 T 2 1 jw e T R 2 C NM k 1 k 1 T 3 1 w 2 1þw 2 T 2 1 ð14Þ
The following expressions are then used to obtain the gain and the phase shift:

H w ð Þ X 2 R ðwÞ þ X 2 I ðwÞ
q with X R the real part of H jw ð Þ and X I the imaginary part of H jw ð Þ. / arctan X I ðwÞ X R ðwÞ
( )

The corresponding expressions of phase shift and the gain are given in Table 2. Both the expressions for the gain and phase shift contain the individual adsorption and desorption rate constants, as well as the number of active sites. They also depend on the overall residence time in the catalytic reactor. Note that the gain and the phase shift are defined between the flow at the reactor inlet and outlet.

The expressions of the gain and the phase shift for several cases are reported below (Table 2).

The real benefit of this method is that an analytical expression allows to directly extract the parameters from the transient exper iments. The method can also be applied for more complicated reac tions. However, these expressions are not always easy to obtain, especially for more complicated reaction networks. Therefore another more versatile methodology will be preferred.

Frequency analysis

The numerical solution of the differential equations in the time domain is a very versatile method, but it implies to run the code for each frequency tested. In order to evaluate the response of the sys tem as a function of the frequency, multiple runs are thus neces sary, which will become very time consuming. An alternative method is to perform the estimation of the gain and the phase shift using a frequential resolution. This methodology consists of obtaining a numerical steady state solution and then of estimating the system response to an excitation in the form of an oscillation for a predefined frequency range. The mathematical development will then lead to a complex valued linear system.

Under dynamic operation, the model can be written as:

d y ! dt f ! ð y ! ; y E ! Þ with : y ! y ss ! þ x ! and y E ! y ss E ! þ x E !
where y ! is the vector of parameters to be calculated (in our case, the concentrations)

y E
! is the vector of the inlet parameters

x ! is the resulting perturbation vector on the parameters The following linearization of the system can be performed:

x E ! is
dðy ss 1 þx 1 Þ dt
f 1 y ss 1 ; y ss 2 ; . . . ; y ss n ; y ss E 1 ; y ss E 2 ; . . . ; y ss

E k þ @f 1 @y 1 x 1 þ @f 1 @y 2 x 2 þ . . . þ @f 1 @y n x n þ @f 1 @y E 1 x E 1 þ @f 1 @y E 2 x E 2 þ . . . þ @f 1 @y E k x E k : : : dðy ss
n þxnÞ dt f n y ss 1 ; y ss 2 ; . . . ; y ss n ; y ss E 1 ; y ss E 2 ; . . . ; y ss

E k þ @f n @y 1 x 1 þ @f n @y 2 x 2 þ . . . þ @f n @y n x n þ @f n @y E 1 x E 1 þ @f n @y E 2 x E 2 þ . . . þ @f n @y E k x E k
It can be thus written using a matrix formalism:

_ x 1 . . . _ x n 0 B B @ 1 C C A J x 1 . . . x n 0 B B @ 1 C C A þ B x E 1 . . . x E k 0 B B @ 1 C C A or _ X J:X þ B:X E
where J and B are the following Jacobian matrixes:

Table 2 Analytical expressions for the gain and phase shift for several cases.

Molecular adsorption without mass transfer limitations

/ w ÀT R À T R2 CNM k1 k 1 T 2 1 1þw 2 T 2 1 And HðwÞ j j e TR 2 C NM k 1 k 1 T 3 1 w 2 1þw2 T 2 1 With T R ðe þ 1 À e ð Þ epÞ

  chemical reactors has usually been analyzed for a potential gain in performance. However, at the laboratory scale, periodic operation of a catalytic reactor can be used for kinetic studies with better parameter estimation, which is inherent to transient kinetic studies. Different mathematical methods are developed and compared in this study to simulate and model con centration cycling in a fixed bed catalytic reactor. Different cases including dissociative and rnolecular adsorption are considered with or without external and internai mass transfer limitations. The analysis of these cases gives information on the contribution of the different parameters on the phase lag and gai n loss. This information can be used to better plan experimental kinetic studies. Ethylene adsorption over a Ni{AhO3 catalyst has been studied experimentally in a dedicated set up with a high speed Infrared analyzer capable of analyzing the outlet gas mixture with a data acquisition frequency up to 30 Hz. The data are modeled with one of the analytical expressions developed in this work to estimate the equilibrium adsorption constant

  the vector of the perturbations performed on the inlet parameters of steady state values of the parameters After steady state values have been calculated, the only param eters to be solved are the components of x ! :


François Benezit

Paul Loubet

Florence Galtier

Charlotte Pronier

Nezha Lenzi

Zineb Lesieur

Stéphane Jouneau

Gisèle Lagathu

Anne-Sophie L'honneur

Vincent Foulongne

François Bénézit

Christine Vallejo

Sophie Alain

Xavier Duval

Nawal Houhou

Yolande Costa

Philippe Vanhems

Sélilah Amour

Fabrice Carrat

Bruno Lina

Odile Launay

MD Pierre Tattevin
email: pierre.tattevin@chu-rennes.fr

Fluvac Study Group

Non-influenza respiratory viruses in adult patients admitted with influenza-like illness a 3-year prospective multicenter study

Keywords: Influenza-like illness, Influenza, picornavirus, respiratory syncytial virus, coronavirus, human metapneumovirus

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Non-influenza respiratory viruses (NIRV) are responsible for a substantial proportion of influenza-like illness (ILI), and pneumonia [START_REF] Ruuskanen | Viral pneumonia[END_REF][START_REF] Luchsinger | Community-acquired pneumonia in Chile: the clinical relevance in the detection of viruses and atypical bacteria[END_REF], although epidemiological data on their burden are scarce. The pathogenicity of viruses recently discovered, such as human metapneumovirus in 2001, or bocavirus in 2005 [START_REF] Van Den Hoogen | A newly discovered human pneumovirus isolated from young children with respiratory tract disease[END_REF][START_REF] Allander | Cloning of a human parvovirus by molecular screening of respiratory tract samples[END_REF], remains poorly characterized. Even NIRV discovered earlier (e.g. parainfluenza virus, respiratory syncytial virus (RSV), or adenovirus), have attracted limited clinical interest thus far, as a result of the absence of rapid tests routinely available, and the lack of vaccine or antiviral treatment with proven clinical efficacy [START_REF] Ruuskanen | Viral pneumonia[END_REF][START_REF] Luchsinger | Community-acquired pneumonia in Chile: the clinical relevance in the detection of viruses and atypical bacteria[END_REF][START_REF] Pavia | What is the role of respiratory viruses in community-acquired pneumonia?: What is the best therapy for influenza and other viral causes of community-acquired pneumonia?[END_REF][START_REF] Galván | Review of Non-bacterial Infections in Respiratory Medicine: Viral Pneumonia[END_REF]. The advent of highly sensitive and specific point-of-care tests for simultaneous detection of the main respiratory viruses is an opportunity to investigate the role of NIRV in common acute respiratory febrile illness, including the risk factors for, and the characteristics of, NIRV-associated ILI. These data may have important implications, including i) screening and early implementation of respiratory isolation for patients with NIRV to prevent nosocomial transmission; ii) identification of the major NIRV that should be targeted for development of antivirals or vaccines. We aimed to determine the characteristics of, and the risk factors for NIRV among adults admitted with ILI during influenza seasons in France.

Methods

Study design and population

We analyzed cases of laboratory-confirmed NIRV infection during three consecutive Hospital, Lyon). Patients were invited to participate during periods of influenza circulation (from December to March). We collected data on non-institutionalized adults (>18 years), who were hospitalized for at least 24 h with ILI, with symptoms onset less than 7 days prior to sampling, through an active surveillance system staffed by trained healthcare professionals.

ILI was defined as a combination of two criteria: (i) at least one of the following symptoms: fever (≥38°C), headache, myalgia or malaise, and (ii) at least one of the following respiratory symptoms: cough, sore throat, or shortness of breath (dyspnea) [START_REF]Influenza case definitions n[END_REF]. The characteristics and outcome of patients with influenza [START_REF] Rondy | -12 seasonal influenza vaccines effectiveness against confirmed A(H3N2) influenza hospitalisation: pooled analysis from a European network of hospitals. A pilot study[END_REF][START_REF] Loubet | Factors associated with poor outcomes among adults hospitalized for influenza in France: A three-year prospective multicenter study[END_REF], and RSV [START_REF] Loubet | Clinical characteristics and outcome of respiratory syncytial virus infection among adults hospitalized with influenza-like illness in France[END_REF] in this cohort have been previously reported.

Clinical data

We collected data on demographic, chronic underlying diseases, hospital admission during the previous 12 months, smoking status, hospitalization ward, and main characteristics of current ILI, including date of onset, date of admission, length of hospital stay, and outcome (i.e. complications, and in-hospital mortality). Data were collected on a standardized questionnaire from medical records, and face-to-face interviews with patients.

Virology

Tests for respiratory viruses were performed in nasopharyngeal swabs from all patients by means of multiplex reverse transcription-polymerase chain reaction (mRT-PCR). DNA were extracted with the automated Easymag system from BioMérieux (Marcy l'Etoile, France), and influenza viruses were detected with an in-house real-time RT-PCR protocol [START_REF] Duchamp | Pandemic A(H1N1)2009 influenza virus detection by real time RT-PCR : is viral quantification useful?[END_REF]. Samples were also screened for a panel of NIRV: adenovirus (52 serotypes), human bocaviruses 1-4, human coronaviruses (229E, NL63, OC43, HKU1), human metapneumoviruses 1-4, parainfluenza viruses 1-4, picornavirus, and RSV, by real-time PCR, using the Respiratory Multiwell System (MWS) r-gene® on an ABI 7300 analyzer.

Ethics

The FLUVAC study (clinicaltrials.gov NCT02027233) respected Good Epidemiological and Clinical Practices in clinical research and the Declaration of Helsinki, and was approved by institutional review board. All study participants provided informed consent for respiratory viruses testing and data collection.

Statistical analysis

We first described the characteristics of all patients hospitalized with ILI who tested positive for at least one NIRV. Results were expressed as mean and standard deviation (SD), or median and interquartile range (IQR) for quantitative variables, and n (%) for qualitative variables. Missing data for each variable were excluded from the denominator. Factors associated with NIRV infection were analyzed by using two different comparison groups: i) patients with negative tests for both NIRV and influenza; ii) patients with documented influenza. The Wilcoxon rank sum test or Fisher's exact test was used, as appropriate, for univariable comparisons. For multivariable analysis, we used a backward stepwise logistic regression model, using NIRV test results (positive/negative) as the dependent variable.

Covariables tested in the multivariable model were all variables with a P-value <0.2 in the univariable analysis. Results were expressed as odds ratios (OR) and adjusted OR (aOR) with 95% confidence intervals (CI). A P-value of 0.05 or less was considered statistically A c c e p t e d m a n u s c r i p t significant. All analyses were performed using R software (v3, R Foundation for Statistical Computing, Vienna, Austria) [START_REF]The R Project for Statistical Computing n[END_REF].

Results

Characteristics of patients with influenza-like illness, and virus distribution

Overall, 1452 patients hospitalized with ILI were enrolled. Median age was 70 years [IQR, 54-82], 780 patients were male (54%), 1155 (80%) had at least one chronic underlying disease (mostly respiratory or heart diseases), 661 (46%) had been hospitalized in the previous 12 months, and 644 (44%) had been vaccinated against influenza. Among the 1452 patients tested, 781 (54%) were positive for at least one respiratory virus. We excluded the 31 patients (2% of total), who had simultaneous detection of influenza virus, and NIRV. Among the 1421 remaining patients, influenza virus was detected in 535 (38%), and NIRV in 215 patients (15%), including 7 with two NIRV. The NIRV detected were picornavirus, n=61 (27%), RSV, n=53 (24%), coronavirus 229E, n=48 (22%), human metapneumovirus, n=40 (18%), adenovirus, n=12 (5%), and bocavirus, n=8 (4%). No parainfluenza virus was detected. Flow chart is presented in Figure 1. The proportion of positive tests for influenza and for NIRV remained similar during the two first influenza seasons; a significant increase of influenza cases was noted in 2014-2015 winter (Table 1).

Characteristics of patients with non-influenza respiratory viruses (Table 2)

The 215 patients with at least one NIRV detected, had a median age of 73 years [60-83], 116 (54%) were male, and 174 (81%) had at least one chronic underlying disease, mostly respiratory (n=114, 53%), or heart diseases (n=90, 42%). Fifty-two patients (24%) were current smokers, 44 (21%) were taking immunosuppressive drugs, and 4 were pregnant (27% A c c e p t e d m a n u s c r i p t of women less than 50-year-old). Mean duration of ILI at the time of admission was 3.0 days [START_REF] Luchsinger | Community-acquired pneumonia in Chile: the clinical relevance in the detection of viruses and atypical bacteria[END_REF][START_REF] Van Den Hoogen | A newly discovered human pneumovirus isolated from young children with respiratory tract disease[END_REF][START_REF] Allander | Cloning of a human parvovirus by molecular screening of respiratory tract samples[END_REF]. Main symptoms were fever (78%), cough (78%), dyspnea (74%), weakness or malaise (26%), headache (22%), and myalgia (21%). Ninety-four patients developed at least one complication during hospital stay, including pneumonia, n=68 (32%), and respiratory failure, n=47 (22%). Median length of stay was 8 days [START_REF] Pavia | What is the role of respiratory viruses in community-acquired pneumonia?: What is the best therapy for influenza and other viral causes of community-acquired pneumonia?[END_REF][START_REF] Galván | Review of Non-bacterial Infections in Respiratory Medicine: Viral Pneumonia[END_REF][START_REF]Influenza case definitions n[END_REF][START_REF] Rondy | -12 seasonal influenza vaccines effectiveness against confirmed A(H3N2) influenza hospitalisation: pooled analysis from a European network of hospitals. A pilot study[END_REF][START_REF] Loubet | Factors associated with poor outcomes among adults hospitalized for influenza in France: A three-year prospective multicenter study[END_REF][START_REF] Loubet | Clinical characteristics and outcome of respiratory syncytial virus infection among adults hospitalized with influenza-like illness in France[END_REF][START_REF] Duchamp | Pandemic A(H1N1)2009 influenza virus detection by real time RT-PCR : is viral quantification useful?[END_REF][START_REF]The R Project for Statistical Computing n[END_REF][START_REF] Tanner | Respiratory viral infections during the 2009-2010 winter season in Central England, UK: incidence and patterns of multiple virus co-infections[END_REF][START_REF] Ambrosioni | Epidemiology of viral respiratory infections in a tertiary care centre in the era of molecular diagnosis, Geneva, Switzerland, 2011-2012[END_REF][START_REF] Falsey | Respiratory Syncytial Virus Infection in Elderly and High-Risk Adults[END_REF][START_REF] Zwaans | The relevance of respiratory viral infections in the exacerbations of chronic obstructive pulmonary disease-a systematic review[END_REF][START_REF] Gunawardana | Experimental rhinovirus infection in COPD: Implications for antiviral therapies[END_REF]. ICU admission was required for 16 patients (11%). Eleven patients died during hospitalization (5%).

Comparison of patients who tested positive for non-influenza respiratory viruses, and i) patients with influenza; ii) patients with no respiratory virus detected (Table 2)

Patients who tested positive for NIRV were older than patients with influenza (median, 73 years versus 68, P=0.026), more likely to have chronic respiratory diseases (53% versus 45%, P=0.034), solid cancer (14% versus 9%, P=0.029), and to be on immunosuppressive drugs (21% versus 14%, P=0.028), but less likely to have diabetes (18% versus 25%, P=0.038).

Fever was less common with NIRV than influenza (79% vs 90%, P<0.001). Outcomes were similar, including complications, median length of stay, and in-hospital mortality. On multivariable analysis, only chronic respiratory disease (OR 1.5 [1.1-2.0], P=0.008), and diabetes (OR 0.5 [0.4-0.8], P=0.01), were significantly associated with NIRV.

When compared to patients with no detection of respiratory virus, patients who tested positive for NIRV were more likely to have chronic respiratory diseases (53% versus 44%, P=0.024), to be pregnant (27% versus 5%, P=0.021), and on immunosuppressive drugs (21% versus 15%, P=0.041), but less likely to have diabetes (18% versus 27%, P=0.011). Weakness was less common in patients with NIRV than in patients who tested negative for all respiratory viruses (26% vs 33%, P=0.047). Regarding outcome, pneumonia was more common in patients who tested positive for NIRV (32% vs 24%, P=0.019).

A c c e p t e d m a n u s c r i p t

Discussion

In this prospective multicenter study performed during three influenza seasons in France, at least one NIRV was found in 15% of patients admitted with ILI, the major players being picornavirus (27%), RSV (24%), coronavirus 229E (22%), and human metapneumovirus (18%). NIRV were more common in patients with chronic respiratory diseases, and less common in patients with diabetes, whatever the comparison group (patients with influenza, or patients with no respiratory virus), and these associations remained significant in multivariable analysis. Although patients with NIRV were more likely to develop pneumonia than patients with no respiratory virus, mortality was similar in patients with NIRV, in patients with influenza, and in patients with no respiratory virus, at 4-5%.

Few studies have evaluated the epidemiology of NIRV in patients with symptoms suggestive of acute respiratory tract infection. Tanner et al. performed a prospective study during the 2009-2010 winter season in Central England, both in hospitals, and in general practitioner offices [START_REF] Tanner | Respiratory viral infections during the 2009-2010 winter season in Central England, UK: incidence and patterns of multiple virus co-infections[END_REF]. The two main NIRV were RSV (31%), and picornavirus (24%), as in our study, but the authors did not test for coronavirus, and bocavirus. Ambrosioni et al.

conducted a two year-prospective study in one referral centre in Switzerland [START_REF] Ambrosioni | Epidemiology of viral respiratory infections in a tertiary care centre in the era of molecular diagnosis, Geneva, Switzerland, 2011-2012[END_REF], and their findings were in line with ours, with two main differences: i) coronaviruses were more heterogeneous, and included 229E, NL63, OC43, and HKU1 strains; ii) human metapneumovirus was rare, at around 5%, with no significant difference between upper and lower respiratory tract samples, and between adults younger than 65 years, and elderlies.

Previous studies have identified age >65 years as a risk factor for NIRV, as well as chronic respiratory diseases [START_REF] Ruuskanen | Viral pneumonia[END_REF][START_REF] Galván | Review of Non-bacterial Infections in Respiratory Medicine: Viral Pneumonia[END_REF][START_REF] Ambrosioni | Epidemiology of viral respiratory infections in a tertiary care centre in the era of molecular diagnosis, Geneva, Switzerland, 2011-2012[END_REF][START_REF] Falsey | Respiratory Syncytial Virus Infection in Elderly and High-Risk Adults[END_REF]. For the latter, experimental studies suggested that NIRV may trigger acute exacerbation of chronic obstructive pulmonary diseases (COPD)

A c c e p t e d m a n u s c r i p t [START_REF] Zwaans | The relevance of respiratory viral infections in the exacerbations of chronic obstructive pulmonary disease-a systematic review[END_REF][START_REF] Gunawardana | Experimental rhinovirus infection in COPD: Implications for antiviral therapies[END_REF][START_REF] Mallia | Experimental rhinovirus infection as a human model of chronic obstructive pulmonary disease exacerbation[END_REF]. Interestingly, chronic respiratory diseases remained associated with NIRV in our study, even when the comparison group was patients with influenza, and even in multivariable analysis. This suggests a specific pathogenicity of NIRV in this population, that may be more prone to decompensate during NIRV infection than during influenza. Treatment with immunosuppressive drugs was also associated with NIRV in our study, whatever the comparison group, but the association was no longer significant on multivariable analysis, which suggests that confounding factors are involved. The lower prevalence of diabetes mellitus in patients with NIRV, as compared to patients with influenza, and to patients with no respiratory virus, in our study, was not expected. Diabetes has been associated with a broad range of infectious diseases [START_REF] Shah | Quantifying the Risk of Infectious Diseases for People With Diabetes[END_REF][START_REF] Pearson-Stuttard | Diabetes and infection: assessing the association with glycaemic control in population-based studies[END_REF], including upper respiratory tract infection in some studies [START_REF] Shah | Quantifying the Risk of Infectious Diseases for People With Diabetes[END_REF], but not all [START_REF] Muller | Increased risk of common infections in patients with type 1 and type 2 diabetes mellitus[END_REF]. Although the lower prevalence of diabetes in patients with NIRV as compared to patients with influenza could be related to stronger association between diabetes, and influenza, than between diabetes, and NIRV, we have no explanation for the lower prevalence of diabetes in patients with NIRV as compared to patients with no respiratory virus.

Our study has limitations. Firstly, we have no data to support causality between the presence of NIRV in upper respiratory samples, and the ILI that required admission. Other pathogens may be involved (e.g. bacteria), and NIRV may merely be a bystander infectious agent with no role in the symptoms reported. The lack of a control population with no respiratory symptoms is another limitation that precludes any conclusion on the pathogenicity of NIRV. Secondly, we only performed systematic tests for influenza, and a selection of NIRVs, although other pathogens have been associated with acute respiratory infections [START_REF] Jain | Community-Acquired Pneumonia Requiring Hospitalization among U.S. Adults[END_REF].

Thirdly, our study was limited to influenza seasons, in one country, and its findings may not apply to other settings. However, our study has strengths, including its prospective, multicentre design, and the standardization of viral tests for all adult patients admitted with A c c e p t e d m a n u s c r i p t predefined ILI, during three consecutive influenza seasons. These strengths, and the limited number of missing data, reduce the risk of potential biases. Our study adds another brick in the wall by contributing to better characterization of the burden of NIRV in adult patients with suspicion of acute respiratory tract infections. The identification of transmissible, NIRV, in 15% of adult patients admitted with ILI would advocate for more systematic testing of these patients, especially those with chronic respiratory diseases. This would allow early respiratory isolation, as to prevent nosocomial transmission of NIRV, which may have severe consequences on patients with chronic respiratory diseases, and patients on immunosuppressive drugs.

  influenza seasons(2012/2013, 2013/2014, and 2014/2015), in a post-hoc analysis of patients hospitalized with community-acquired ILI, and enrolled in the FLUVAC study. FLUVAC is a prospective observational study of influenza vaccine efficacy conducted in six university

A

  c c e p t e d m a n u s c r i p t hospitals in France (Cochin Hospital, Paris; Bichat Hospital, Paris; Pontchaillou Hospital, Rennes; Dupuytren Hospital, Limoges; Montpellier University Hospital; Edouard Herriot

  Bronchoalveolar lavage fluid samples and tracheal aspirates ordered by the physician in charge were also tested. Samples were initially tested in the virology laboratory of each participating hospital by means of real-time influenza A & B PCR. All samples were then sent to the French National Influenza Reference Center (CNR-Lyon) for confirmation. RNA and

Fig. 1

 1 Fig. 1 Flow chart of patients hospitalized with influenza-like illness, and viruses detected in

A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p t

Table 1

 1 Number and percentage of patients hospitalized with influenza-like illness who tested positive for non-influenza respiratory viruses, and for influenza, by year.Chi square test was done to compare the evolution, with 2012/2013 as the reference season. NS , non significant; *, P<0.05; **, P<0.001;

	NIRV, non-influenza respiratory virus

Ref

, reference;

Acknowledgements

The current work received no funding. However, the study sites received funding from Sanofi Pasteur and Sanofi Pasteur MSD for the FLUVAC study. Vaccine producers had no role in the study design, data analysis, decision to publish or preparation of the manuscript

Compliance with ethical standards

Conflict of interest

The authors declare no competing interest related to the study. O Launay is an investigator for clinical trials sponsored by Janssen and other companies and received travel support to attend scientific meetings from pharmaceutical companies.