Bruno Bouchard Ceremade

Romuald Elie
email: elie@ceremade.dauphine.fr

Ludovic Moreau
email: moreau@ceremade.dauphine.fr

A note on utility based pricing and asymptotic risk diversification

Keywords: Utility indifference pricing, diversification, risk aversion, entropy Mathematics subject classification (2010): 60H30, 91B16, 91B24, 91B30

In principle, liabilities combining both insurancial risks (e.g. mortality/longevity, crop yield,...) and pure financial risks cannot be priced neither by applying the usual actuarial principles of diversification, nor by arbitrage-free replication arguments. Still, it has been often proposed in the literature to combine these two approaches by suggesting to hedge a pure financial payoff computed by taking the mean under the historical/objective probability measure on the part of the risk that can be diversified. Not surprisingly, simple examples show that this approach is typically inconsistent for risk adverse agents. We show that it can nevertheless be recovered asymptotically when the number of sold claims goes to infinity and the absolute risk aversion of the agent goes to zero simultaneously. This follows from a general convergence result on utility indifference prices which is valid for both complete and incomplete financial markets. In particular, if the underlying financial market is complete, the limit price corresponds to the hedging cost of the mean payoff. If the financial market is incomplete but the agent behaves asymptotically as an exponential utility maximizer with vanishing risk aversion, we show that the utility indifference price converges to the expectation of the discounted payoff under the minimal entropy martingale measure.

Introduction

These last years have seen the explosion of the number of liabilities combining pure financial and pure insurancial risks. They typically have the following form: an insurance company sells to the client i a claim with discounted payoff g i paid at maturity T whose value depends on the evolution of some tradable financial assets S = (S t ) t≥0 and some additional idiosyncratic risk. The g i 's are usually not unconditionally independent, but still independent conditionally to S.

It is (essentially) the case of many variable annuities schemes in which death times or withdrawals policies can be assumed to be independent conditionally to the financial market's behavior, see e.g. [START_REF] Bauer | A universal pricing framework for guaranteed minimum benefits in variable annuities[END_REF]. This is also the case for crop revenue insurance schemes that depend on the production yield of the farmer and the market price of the crop, see e.g. [START_REF] Goodwin | Modeling price and yield risk. A comprehensive assessment of the role of risk in US agriculture[END_REF]. More examples can be found in [START_REF] Embrechts | Actuarial versus financial pricing of insurance[END_REF] or [START_REF] Becherer | Rational hedging and valuation of integrated risks under constant absolute risk aversion[END_REF]. In such a situation, and if the financial market formed by the financials assets S is complete, it is tempting to play on the ability to diversify the conditionally idiosyncratic risks and cover the systemic pure financial risk by dynamically trading on the market. If the g i 's are independent and identically distributed given S, then the price of each of these contingent claims could be defined as p := E Q [ḡ(S)] where ḡ(S) := E g i |S does not depend on i, and Q denotes the unique martingale measure on the pure financial market (i.e. restricted to S). The rationality behind this is the following: by an unformal application of the law of large numbers conditionally to S, we obtain the convergence G n /n := n i=1 g i /n → ḡ(S) a.s. for a large number n of sold contracts. In the above, the payoff ḡ(S) only depends on S and can thus be hedged dynamically by trading on the (complete) pure financial market. Hence, by replicating the mean payoff ḡ(S), we end up with a zero net position in mean (under the initial probability measure P). This solution has been originally proposed by Brennan and Schwartz [START_REF] Brennan | Alternative investment strategies for the issuers of equity-linked insurance with an asset value guarantee[END_REF][START_REF] Brennan | Pricing and investment strategies for guaranteed equitylinked life insurance[END_REF], and then applied several times, in particular in the literature on variables annuities, see e.g. [START_REF] Bauer | A universal pricing framework for guaranteed minimum benefits in variable annuities[END_REF], [START_REF] Boyle | Guaranteed annuity options[END_REF], [START_REF] Milevsky | Mortality derivatives and the option to annuitize[END_REF] or [START_REF] Milevsky | Killing the law of large numbers: Mortality risk premium and the sharpe ratio[END_REF]. However, it seems to ignore the fact that playing with the law of large numbers on the diversifiable part of the risk requires selling a large number of contracts, and therefore may lead to huge positions on the financial market. If the law of large numbers does not operate well enough, then the losses may be leveraged by an unfavorable evolution of the financial market. More generally, the classical central limit theorem that allows to control the asymptotic distribution of the risk in terms of the Gaussian law will in general not apply in this context. One classical solution for pricing such claims is to use the indifference pricing rule of Hodge-Neuberger [START_REF] Hodges | Optimal replication of contingent claims under transaction costs[END_REF], see e.g. [START_REF] Becherer | Rational hedging and valuation of integrated risks under constant absolute risk aversion[END_REF] for the exponential utility case. Not surprisingly, it typically does not lead to the price p defined as above, see Section 2.2 for trivial counter-examples. However, one should intuitively recover it when the number of sold contracts is large, so that the conditional law of large numbers can operate, and the risk aversion is small. In this note, we provide sufficient conditions under which the above holds true. Namely, we consider a family of utility functions (U n ) n , defined on R, with corresponding absolute risk aversion (r n ) n and indifference prices (np n ) n for the aggregate claims (G n ) n . Under mild assumptions detailed in Section 3.2, we show that n → ∞ and n r n ∞ → 0 implies p n → p, whenever the underlying financial market formed by the liquid financial assets S is complete. This follows from a more general asymptotic result derived in Section 3.1, which provides a formulation for the asymptotic unit price lim n p n in terms of the sequence of martingales measures minimizing the corresponding dual problems. The latter applies to incomplete financial markets without providing a clear identification of the asymptotic pricing measure, except when (U n ) n behaves asymptotically like a sequence of exponential utility functions. In this case, we show that p n → p e := E Q e [ḡ(S)] where Q e is the martingale measure with minimum relative entropy, see Section 3.3. This generalizes to our setting the well-known property that the exponential utility indifference price of a claim converges to the risk neutral price under Q e for vanishing risk aversion, see [START_REF] Rouge | Pricing via utility maximization and entropy[END_REF] and [START_REF] Becherer | Rational hedging and valuation of integrated risks under constant absolute risk aversion[END_REF]. Note that a similar result is obtained in [START_REF] Becherer | Rational hedging and valuation of integrated risks under constant absolute risk aversion[END_REF] for the indifference price of the mean payoff Ḡn := G n /n = n i=1 g i /n as n goes to infinity and the risk aversion is fixed, which is a completely different situation.

In the following, any assertion involving random variables has to be understood in the a.s. sense. Given a probability measure Q and a sigma-algebra G, we denote by

L 1 (Q, G) (resp. L ∞ (Q, G))
the space of Q-integrable (resp. Q-essentially bounded) random variables that are G-measurable. We omit the argument Q or G if it is clearly given by the context.

Diversification based pricing rules and risk aversion

In this section, we describe the financial market and elaborate on the relation between diversification and utility indifference pricing.

The market model

From now on, we fix a time horizon T > 0 to avoid unnecessary technical issues, although for some applications (e.g. mortality/longetivity linked contracts) it should in principle be infinite. We consider a model of a security market which consists of d stocks with price process described by a locally bounded càdlàg semi-martingale (S i ) 1≤i≤d defined on some complete filtered probability space (Ω, F, F := (F t ) 0≤t≤T , P), with F satisfying the usual assumptions and F = F T . As usual, we normalize the risk free rate to 0 for simplicity, which can always been done by considering discounted values.

A (self financing) strategy is defined as an element ϑ = (ϑ i ) 1≤i≤d of the set Θ of F-predictable S-integrable processes. Given an initial endowment x ∈ R and a strategy ϑ ∈ Θ, the induced wealth process X x,ϑ = (X x,ϑ t ) 0≤t≤T is given by

X x,ϑ t = x + t 0 ϑ u • dS u , 0 ≤ t ≤ T .
In order to avoid doubling strategies, we restrict as usual to strategies leading to bounded from below wealth processes. We denote by X b (x) the family of terminal values of wealth processes starting from x such that the above holds:

X b (x) := X x,ϑ T : ϑ ∈ Θ, X x,ϑ ≥ -κ on [0, T ] for some κ ∈ R . (2.1) Note that X b (x) = x + X b (0).
As usual, a probability measure Q is called an equivalent local martingale measure if it is equivalent to P and if S is a (F, Q)-local martingale. The family of equivalent local martingales will be denoted by M. We assume throughout this paper that

M = ∅, (2.2)
which ensures the absence of arbitrage opportunities (in the no-free lunch with vanishing risk sense), see [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF] for details. In the following, we will often use the notation Q * to denote a fixed element of M. Note that we do not impose that F T equals F S T , where F S = (F S t ) t≤T is the completion of the right-continuous filtration generated by S, in order to allow for additional randomness. However, we shall often consider that the pure financial market is complete in the following sense.

Definition 2.1. We say that the pure financial market is complete, in short (HCM) holds, if

E Q * [ξ] = E Q [ξ] for all Q ∈ M and ξ ∈ L ∞ (F S T ),
where L ∞ (F S T ) denotes the set of essentially bounded F S T -measurable random variables.

Remark 2.1. Under (HCM), we must have ξ ∈ X b (E Q [ξ]) for all ξ ∈ L ∞ (F S T ). See [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF].

Remark 2.2. Note that, if F S T F T , then (HCM) only implies that the pure financial market is complete in the sense of Remark 2.1, and not that M is reduced to a singleton. As an example, assume that we can find A ∈ F T such that P [A] > 0 and A is independent of F S T given F s for all s ∈ [0, T ] under P.

E H ε A S (k) t |F s = εE [H * |F s ] S (k) s + (1 -ε)E E H * S (k) t |F S T 1 A |F s /P [A] = εE [H * |F s ] S (k) s + (1 -ε)E E H * S (k) t |F S T |F s E [1 A |F s ] /P [A] = (εE [H * |F s ] + (1 -ε)E [H * (S)|F s ] P [A|F s ] /P [A]) S (k) s = E [H ε A |F s ] S (k) s for 0 ≤ s ≤ t ≤ T , which shows that the measure Q ε A defined by dQ ε A /dP = H ε A belongs to M. In general Q ε A = Q * .
Remark 2.3. The same arguments as in Remark 2.2 imply that Q * (S) defined by dQ * (S) = E dQ * /dP|F S T dP belongs to M. Note for later use that dQ * (S) = E dQ/dP|F S T dP for any Q ∈ M when (HCM) holds.

Remark 2.4. Assume that F can be written as (F S t ∨F ⊥ t ) t≤T for some filtration F ⊥ = (F ⊥ t ) t≤T independent of F S under P, and satisfying the usual conditions. Then, any A ∈ F ⊥ T is independent of F S T given F s for all s ∈ [0, T ] under P. Indeed, under the above assumption, F s is generated by elements of the form

B S s ∩ B ⊥ s with B S s ∈ F S s and B ⊥ s ∈ F ⊥ s . Given ξ ∈ L ∞ (F S T ), we then have E ξ1 B S s ∩B ⊥ s = E E ξ|F S s 1 B S s E 1 B ⊥ s = E E ξ|F S s 1 B S s ∩B ⊥ s , so that E ξ|F S s = E [ξ|F s ]. Similarly, E 1 A |F ⊥ s = E [1 A |F s ]. Moreover, E ξ1 A 1 B S s ∩B ⊥ s = E ξ1 B S s E 1 A 1 B ⊥ s = E E ξ|F S s 1 B S s E E 1 A |F ⊥ s 1 B ⊥ s
which, combined with the above assertions, leads to

E ξ1 A 1 B S s ∩B ⊥ s = E E ξ|F S s 1 B S s E 1 A |F ⊥ s 1 B ⊥ s = E E [ξ|F s ] E [1 A |F s ] 1 B S s ∩B ⊥ s . Hence, E [ξ1 A |F s ] = E [ξ|F s ] E [1 A |F s ].

Diversification and utility based pricing

We are interested in the pricing by utility indifference, see [START_REF] Hodges | Optimal replication of contingent claims under transaction costs[END_REF], of aggregated claims of the form

G n := n i=1 g i , n ≥ 1,
where (g i ) i≥1 is a given sequence of random variables.

Although the specific structure of the G n 's is not so important from the mathematical point of view, we have in mind that each g i corresponds to a contingent claim sold by an insurance company to a specific agent i, and that the g i 's have the same law and are independent conditionally to F S T under P. This means that the g i 's may depend of two sources or risks. One related to the pure financial market behavior, i.e. S, the other one coming from an external source of randomness which only depends on the agent i.

Example 2.1 (Revenue insurance). Let S 1 denote the spot price of one quintal of wheat on the financial market and let Y i denotes the number of quintals produced by the farmer i at time T . The payoff of a revenue insurance takes the form g i = [K -Y i S T ] + for some strike K > 0 fixed in advance. It compensates the losses incurred by the farmer i if his revenue Y i S T , induced by the sale of the production at the spot price S T at time T , is less than a targeted level K. If the wheat market contains enough futures and provides enough liquidity, we can consider that it is complete. Moreover, we can also consider that the global level of production (at the level of a sufficiently large area) is already reflected into the prices so that the Y i 's can be assumed to be independent given F S T .

Example 2.2 (Mortality derivatives). A simple example takes the form g i = f (S, ζ i ) where f is a real valued measurable map on ID × ([0, T ] ∪ {∞}), with ID denoting the set of càdlàg R d -valued functions on [0, T ] (endowed with the Skorohod topology), and ζ i is a [0, T ]-valued random variable denoting the time of death of i if it is before T and taking the value ∞ otherwise. Again, one source of randomness comes from the financial market, while the ζ i 's can generally be assumed to be independent and with the same law (at least among a given sub-population).

Under the above interpretation, the global liability of the insurance company is G n if the contracts have been subscribed by the clients 1 to n. If the insurance company does not differentiate her clients, then she has to fix the same price p n to each of them.

If the global market was complete, meaning that M = {Q * }, and the law of G n /n under Q * was independent on n, then p n should be equal to p :

= E Q * [G n /n].
Obviously the completeness of the global market typically fails for the examples we have in mind. Still, as explained in the introduction, this pricing rule has been proposed in the literature for the case where (HCM) holds and (g i ) i≥1 is a sequence of independent and identically distributed random variables given F S T . The later has essentially to be understood in the sense that one can appeal to the law of large numbers, at least conditionally to F S T , so that

G n /n → ḡ P -a.s. for some ḡ ∈ L ∞ (F S T ). (2.3)
Under (HCM), one can indeed find ϑ ∈ Θ such that

X p,ϑ T = ḡ for p = E Q * [ḡ] ,
recall Remark 2.1. Then, (2.3) implies that

X n p,nϑ T -G n /n → 0 P -a.s.
i.e. by replicating nḡ we end up with a hedging error that converges P -a.s. to 0. This is achieved by considering the strategy nϑ and starting from the n initial premiums, each equal to p. This is however inconsistent with the typical behavior of a risk adverse agent. In particular, it has no reason to be in accordance with a (unit) utility indifference price defined by

p n (G n , U ) := inf{p ∈ R : sup X∈X b (np) E [U (X -G n )] ≥ sup X∈X b (0) E [U (X)]}, (2.4)
where U is a concave non-decreasing function viewed as a utility function, and where we restrict to a 0 initial endowment (before selling the claims) without loss of generality since any fixed initial wealth can be incorporated in the utility function by a simple translation argument.

We conclude this section with simple counter-examples. The first one concerns utility functions with bounded from below domains.

Example 2.3 (Utility with bounded from below domain). Let U be concave non-decreasing with values in

R ∪ {-∞} such that |U (0)| + |U (∞)| < ∞ and U (-c) = -∞ for some c > 0.
Let p n := p n (G n , U ) be defined as in (2.4). Since U (0) > -∞ and U is bounded from above, for each ε > 0, there must exist ξ

n,ε ∈ X b (0) such that np n + ε + ξ n,ε ≥ G n -c. This implies that p n ≥ sup Q∈M E Q [G n /n] -c/n and therefore lim inf n→∞ p n ≥ lim inf n→∞ sup Q∈M E Q [G n /n] . (2.5)
Let us now concentrate on the case where (g i ) i≥1 is defined as in Example 2.2 with f bounded. Assume that the ζ i 's form an independent family with a common law under P, and that σ(ζ i , i ≥ 1) ⊂ F ⊥ T with F ⊥ satisfying the conditions of Remark 2.4. Note that

E H * (S)f (S, ζ i )|F ⊥ T = E H * (S)f (S, ζ i )|ζ i , i ≥ 1 ,
where H * (S) is defined as in Remark 2.2. Set ψ := esssup E H * (S)f (S, ζ 1 )|ζ 1 . For k, n ≥ 1 and ε ∈ (0, 1), we define

A n k := {E H * (S)f (S, ζ i )|ζ i ≥ ψ -k -1 , for all i ≤ n} and H ε n,k := εH * + (1 -ε)H * (S)1 A n k /P [A n k ]. Note that P [A n k ] > 0 since the ζ i '
E Q ε n,k [G n /n] = ψ.
Clearly, the above lower bound is typically strictly larger than

E Q * E f (S,
] ⊂ R under P, then ψ = E [H * (S)f (S, y max )] = E Q * [f (S, y max )] = E Q * [max{f (S, y), y min ≤ y ≤ y max }]. Under (HCM),
the later is the hedging price of max{f (S, y), y min ≤ y ≤ y max }, recall Remark 2.1, so that p n ≤ ψ for all n ≥ 1, and therefore p n → ψ.

In general, going through utility function with unbounded domain does not help, as show in our second example.

Example 2.4 (Exponential utility function). Let U be an exponential utility function of the form U η (y) = -e -ηy , η > 0. Assume that the g i 's have the form taken in Example 2.3, that

F ⊥ t = σ(F ⊥,i t , i ≥ 1), t ≤ T , for some filtrations (F ⊥,i ) i≥1 such that F S T , F ⊥,1 T , F ⊥,2
T , . . . are independent under P, and that ζ i is F ⊥,i T -measurable for each i ≥ 1. Then, it can be shown that p η n := p n (G n , U η ) = p 1 (g 1 , U η ) =: p η whenever there exists a unique element of M with finite relative entropy and whose density is F S T -measurable, see Theorem 4.10 in [START_REF] Becherer | Rational hedging and valuation of integrated risks under constant absolute risk aversion[END_REF]. It is in particular the case if (HCM) holds and Q * (S) defined in Remark 2.3 has a finite relative entropy. On the other hand, it is well-known that p η converges to the super-hedging price of g 1 as η → ∞, see [START_REF] Delbaen | Exponential hedging and entropic penalties[END_REF] and [START_REF] Becherer | Rational hedging and valuation of integrated risks under constant absolute risk aversion[END_REF]. This implies that, for all ε > 0, we can find η ε large enough so that

lim n p ηε n = p ηε ≥ sup Q∈M E Q g 1 -ε. For ε > 0 small enough, this is again typically greater than E Q * E f (S, ζ 1 )|F S T .

Asymptotic diversification rule

As pointed out in the previous examples, the convergence of the mean aggregated claim G n /n to a replicable claim ḡ ∈ L ∞ (F S T ) is not enough in order to ensure the convergence of its unit utility indifference price p n (G n , U ) to E Q * [ḡ]. Intuitively, this can be recovered only if the risk aversion vanishes as the number of sold claims goes to infinity. Hence, we suppose from now on that the preferences of the agent are characterized by a sequence of utility functions (U n ) n≥1 , which depends on the number n of sold claims (g i ) 0≤i≤n . The purpose of this section is to investigate the asymptotic behavior of the corresponding unit utility indifference price when n times the absolute risk aversion of U n vanishes to 0 as n goes to ∞. We first provide a general characterization of the asymptotic unit utility indifference price in terms of the sequence of associated dual pricing measures. Whenever the pure financial market is complete, i.e. (HCM) is satisfied, this limit identifies to the risk neutral price of ḡ. When the pure financial market is incomplete, it does not seem possible to obtain a precise characterization of the limit price, except when (U n ) n≥1 behaves asymptotically like a sequence of exponential utility functions with vanishing absolute risk aversion. In this case, we prove that the asymptotic price coincides with the price of ḡ under the minimal entropy martingale measure.

General convergence result

In this section, we consider a sequence of twice continuously differentiable, strictly concave and increasing utility functions (U n ) n≥1 defined on the whole real line and satisfying the Inada conditions:

U n (-∞) = lim x→-∞ U n (x) = ∞ and U n (∞) = lim x→∞ U n (x) = 0 , n ≥ 1 . (3.1)
Besides, we suppose that all the utility functions have a reasonable asymptotic elasticity as defined in [START_REF] Schachermayer | Optimal investment when wealth may become negative[END_REF], i.e. lim sup

x→∞ xU n (x) U n (x) < 1 , lim inf x→-∞ xU n (x) U n (x) > 1 , n ≥ 1 . (3.2)
We finally introduce the convex conjugates of the U n 's defined by

V n : y ∈ (0, ∞) → sup x∈R {U n (x) -xy},
and assume that the dual problems are finite:

{(y, Q) ∈ (0, ∞) × M : E [V n (ydQ/dP)] < ∞} = ∅ for all n ≥ 1. (3.3)
Under the additional uniform boundedness assumption

sup n≥1 G n /n L ∞ < ∞, (3.4)
the unit utility indifference prices p n (G n , U n ) given by (2.4) are well defined for any n ≥ 1 and existence for the optimal dual probability and multiplier, given by

(y 0 n , Q 0 n ) := arg min E V n y dQ dP , (y, Q) ∈ (0, ∞) × M , (3.5)
is guaranteed, see e.g. Theorem 3.1, Remark 3.3 and Proposition 3.1 in [START_REF] Bouchard | Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility[END_REF] (see also [START_REF] Schachermayer | Optimal investment when wealth may become negative[END_REF] for a fomulation in terms of absolutely continuous local martingale measures).

In the rest of this section, we work under the standing assumption: In order to derive the asymptotic behavior of the unit indifference price, we now also suppose that the agent tends to become risk neutral when selling a large number of claims. More precisely, we shall work under the following additional condition on the asymptotic absolute risk aversion:

n r n ∞ -→ n→∞ 0 , with r n : x → - U n (x) U n (x) , ( 3
lim inf n→∞ p n (G n , U n ) = lim inf n→∞ E Q 0 n [G n /n] and lim sup n→∞ p n (G n , U n ) = lim sup n→∞ E Q 0 n [G n /n] .
Proof. We set p n := p n (G n , U n ) for ease of notations. We only provide the proof for the lim inf, the other one being similar.

1. Given n ≥ 1, it follows from standard duality arguments, see [START_REF] Owen | Utility based optimal hedging in incomplete markets[END_REF] and [START_REF] Bouchard | Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility[END_REF], that inf y>0,Q∈M

E V n y dQ dP + ynp n -y dQ dP G n = inf y>0,Q∈M E V n y dQ dP = E V n y 0 n dQ 0 n dP , recall (3.5). Taking in particular (y, Q) = y 0 n , Q 0 n , this implies that E V n y 0 n dQ 0 n dP + y 0 n np n -y 0 n dQ 0 n dP G n ≥ E V n y 0 n dQ 0 n dP ,
and therefore

p n ≥ E Q 0 n [G n /n].
2. On the other hand, it follows from [START_REF] Schachermayer | Optimal investment when wealth may become negative[END_REF] that, for n ≥ 1, we can find Xn ∈ L 0 (F T ) such that sup

X∈X b (0) E [U n (X T )] = E U n Xn
and for which there exists a sequence of optimizers

X k,n k≥1 ⊂ X b (0) such that U n X n,k T L 1 (P) -→ k→∞ U n Xn . (3.7)
In order to upper-bound p n , we then introduce the following candidate

π n := inf p ∈ R : E U n np + Xn -G n ≥ E U n Xn .
a. We first check that π n ≥ p n , for all n ≥ 1. (3.8) To this purpose, it suffices to fix n ∈ N and show that

U n np + X n,k T -G n L 1 (P) -→ k→∞ U n np + Xn -G n ∈ L 1 (P), for all p ∈ R . (3.9)
To see that the above holds, first note that

U n np + X n,k T -G n -U n np + Xn -G n = X n,k T Xn U n (np + t -G n )dt . (3.10)
Consider now the relation

log U n (np + t -G n ) -log U n (t) ≤ n U n U n ∞ p - G n n , t, p ∈ R,
which, together with (3.4) and (3.6) leads to the existence of a constant C p (which only depends on p) such that

U n (np + t -G n ) ≤ C p U n (t), for all t ∈ R. (3.11)
Plugging (3.11) into (3.10) gives

U n np + X n,k T -G n -U n np + Xn -G n ≤ C p U n X n,k T -U n Xn .
Hence, (3.9) follows from (3.7), which proves (3.8). b. We now conclude the proof by providing a lower bound for lim inf n→∞ π n . By definition of π n , the continuity of the non-increasing function U n , (3.9) and the monotone convergence theorem, we have

E U n nπ n + Xn -G n = E U n Xn . (3.12)
This implies that

E U n Xn = E U n Xn + U n Xn (nπ n -G n ) + 1 2 U n (ξ n ) (nπ n -G n ) 2 ,
where ξ n is a random variable lying in the (random) interval I n formed by Xn and nπ n + Xn -G n .

We now use the fact that U n Xn = y 0 n dQ 0 n dP , recall (3.5) and see [START_REF] Schachermayer | Optimal investment when wealth may become negative[END_REF] and [START_REF] Bouchard | Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility[END_REF], to deduce that

E Q 0 n   π n - G n n - nr n (ξ n ) 2 U n (ξ n ) U n Xn π n - G n n 2   = 0. ( 3.13)
We shall prove in c. below that

sup n≥1 U n (ξ n ) U n Xn π n - G n n 2 < ∞. (3.14)
Combining this last estimate with (3.6) and (3.13) leads to

lim inf n→∞ π n ≤ lim inf n→∞ E Q 0 n [G n /n] ,
which together with (3.8) and step 1. concludes the proof. c. It remains to prove the claim (3.14). To see that it holds, we first appeal to (3.12) to deduce that E U n ( ξn ) (nπ n -G n ) = 0 for some random variable ξn such that ξn ∈ I n . Since U n is strictly increasing, we deduce from (3.4) that

|π n -G n /n| = E U n ( ξn )|G n |/n E U n ( ξn ) + |G n /n| ≤ C E U n ( ξn ) E U n ( ξn ) + C = 2C , n ≥ 1, (3.15)
for some constant C > 0. Similarly, since ξ n ∈ I n , we have

log (U n (ξ n )) -log U n Xn ≤ U n U n ∞ ξ n -Xn ≤ n U n U n ∞ π n - G n n , n ≥ 1 ,
which is bounded uniformly in n thanks to (3.6) and (3.15).

2 Remark 3.1. Let (p ϕ(n) ) n≥1 be a convergent subsequence of (p n (G n , U n )) n≥1 .
Then, the same arguments as above show that lim

n→∞ p ϕ(n) = lim n→∞ E Q 0 ϕ(n) [G ϕ(n) /ϕ(n)] whenever ϕ(n)r ϕ(n) → 0.
Remark 3.2. Note that a similar result can be obtained for the indifference price of the aggregated risk p 1 (G n , U n ) under the weaker condition r n ∞ → 0, whenever (G n ) n≥1 is assumed to be uniformly bounded in L ∞ . Indeed, a straightforward adaptation of the above proof shows that, under the above conditions,

lim inf n→∞ p 1 (G n , U n ) = lim inf n→∞ E Q 0 n [G n ] and lim sup n→∞ p 1 (G n , U n ) = lim sup n→∞ E Q 0 n [G n ] .
This provides a general convergence result for bounded sequences of contingent claims when the absolute risk aversion vanishes in the sup norm, which is of own interest.

Semi-complete markets

The representation of the asymptotic unit utility indifference price presented in Theorem 3.1 does not provide a-priori an exact formulation, except in particular cases. When the pure financial market is complete, i.e. (HCM) is satisfied, and (2.3) holds, we verify hereafter that it coincides with the price under the risk neutral measure Q * of the replicable claim ḡ.

Corollary 3.1. Let the conditions of Theorem 3.1 hold. Assume further that (2.3) and (HCM) are satisfied. Then the sequence of unit utility indifference prices satisfies

lim n→∞ p n (G n , U n ) = E Q * [ḡ].
Proof. In view of Theorem 3.1, it suffices to show that (HCM) implies that the minimal dual measure Q 0 n , see (3.5), coincides with Q * (S) as constructed in Remark 2.3, and to apply the dominated convergence theorem, recall (3.4) and (2.3):

lim n→∞ p n (G n , U n ) = lim n→∞ E Q 0 n [G n /n] = lim n→∞ E Q * (S) [G n /n] = E Q * (S) [ḡ] = E Q * [ḡ] ,
where the last equality follows from the fact that ḡ is F S T -measurable. To see this, we use the convexity of V n to obtain that E V n (y 0

n dQ 0 n dP ) ≥ E V n (y 0 n H 0 n (S)
(G n , U n ) = E Q * E g 1 |F S
T . We thus retrieve asymptotically the hybrid pricing rule which consists in taking the mean on the part of the risk that can be diversified and computing the hedging price of the resulting replicable claim.

Remark 3.4. In [START_REF] Becherer | Rational hedging and valuation of integrated risks under constant absolute risk aversion[END_REF], the author refers to semi-complete product models to designate situations where the filtration F and M have the structure specified in Example 2.4, in particular: (HCM e ): there exists only one element Q e (S) of M with finite relative entropy and whose density with respect to P is F S T -measurable. This later condition is weaker than (HCM), up to the restriction related to the finite relative entropy. However, if we are only interested in utility functions of exponential type, the same argument as in the proof of Corollary 3.1 above shows that Q 0 n = Q e (S) whenever (HCM e ) holds. Assuming further that the conditions of Theorem 3.1 and (2.3) hold, this leads to

lim n→∞ p n (G n , U n ) = E Q e (S) [ḡ].

Incomplete markets and asymptotically exponential utility behaviors

In a general incomplete framework, it seems to be hopeless to interpret the limit of E Q 0 n [G n /n] as the expectation under some martingale measure of ḡ, the limit of G n /n, except if the U n 's are all of exponential type, compare with Remark 3.4.

In this section, we show that the convergence result of Remark 3.4 remains true even if the utility functions do not have a constant absolute risk aversion but only asymptotically behave like a sequence of exponential utility functions in the following sense. converging toward 0 such that

0 < η 2 n ≤ r n (x) ≤ η 1 n for all x ∈ R and n ≥ 1, (3.16)
lim n→∞ η 2 n /η 1 n = 1. (3.17)
Remark 3.5. The existence of the sequence (η 1 n ) n≥1 converging to zero is exactly the content of the assumption (3.6). Assumption 3.2 implies that the function r n is asymptotically bounded in between two sequences converging to zero with the same first order convergence rate. In particular, this assumption includes the case of agents with utility functions of the form U n : x → -(λ n ) -1 e -λnx -(µ n ) -1 e -µn(x+x0) , for n ≥ 1 and x 0 > 0 as long as the positive sequences (λ n ) n≥1 and (µ n ) n≥1 are equivalent as n goes to ∞.

It follows from Lemma 3.1 below that (3.3) is equivalent to

Q ∈ M : E dQ dP log dQ dP < ∞ = ∅, (3.18
p n (G n , U n ) = E Q e [ḡ]
.

We can now use the fact that V 1 n and V 2 n interpret as Fenchel transforms of exponential utility functions, and that the dual probability associated to any exponential utility function is the minimal entropic one Q e , to deduce from (3.23) that the sequence of dual martingale measures (Q α n ) n≥1 associated to (U α n ) n≥1 achieves asymptotically the mininal relative entropy, under Assumption 3.2 and for a suitable choice of the sequence (α n ) n≥1 . We shall see later that this implies convergence to Q e is the total variation norm. Taking the infimum over y > 0 and recalling (3.27), this shows that

E V 2 n ŷn dQ e dP -V 1 n ŷn dQ e dP = (1 -e -ρ)α n 1 η 2 n - 1 η 1 n , so that lim n→∞ E V 2 n ŷn dQ e dP -V 1
E V 1 n ŷn dQ e dP ≤ E V 1
α n 1 η 2 n - 1 η 1 n = 1 {η 2 n =η 1 n } η 1 n η 2 n -1 -→ n→∞ 0 and α n η 1 n = η 2 n η 2 n -η 1 n 1 {η 2 n =η 1 n } + 1 {η 2 n =η 1 n } η 1 n -→ n→∞ ∞.
Coming back to (3.33), this implies that ρ n → ρ as n → ∞, i.e. (3.26) holds.

We are now in position to complete the proof of Theorem 3.2.

Proof of Theorem 3.2. In the following, we let (α n ) n≥1 be as in Lemma 3.2. 1.

We first deduce from Lemma 3.2 that (Q α n ) n≥1 converges to Q e in the norm of total variation. Since Q e minimizes the entropy with respect to P over the convex set M, it follows from Theorem 2.2 in [START_REF] Csiszár | I-divergence geometry of probability distributions and minimization problems[END_REF]

E Q e [ḡ]
. This shows that (3.20) holds. We conclude by using (3.22) and Theorem 3.1.

Assumption 3 . 1 .

 31 The conditions (3.1), (3.2), (3.3) and (3.4) hold.

. 6 )

 6 and r n ∞ := sup x∈R |r n (x)|. Theorem 3.1. Let Assumption 3.1 and (3.6) hold. Then, the sequence of utility indifference prices satisfies

Assumption 3 . 2 .

 32 There exist two sequences of strictly positive numbers η 1 n n≥1 and η 2 n n≥1

  any Q ∈ M. Picking in particular Q = Q α n

n η 1 ne 1 n η 2 n η 2 n -η 1 n 1 {η 2 n =η 1 n } + 1 {η 2 n =η 1 n

 11211211 y n > 0 associated to the middle term is given byy n := α n e -ρnwith ρ n := E dQ the definitions in (3.28) and (3.32) of ŷn and y n , is equivalent toα -ρ -e -ρn -→ the sequence (α n ) n≥1 as α n := η } , n ≥ 1 . (3.34) By Assumption 3.2, it satisfies

  S -stopping times (τ k ) k≥1 such that S (k) := S •∧τ k is bounded on [0, T ] for all k ≥ 1, we have

	T	and
	H ε A := εH

Set H * := dQ * /dP, H * (S) := E H * |F S * + (1 -ε)H * (S)1 A /P [A] for some ε ∈ (0, 1]. Then, for any increasing sequence of

F

  s are independent and have the same law under P. Then, according to Remark 2.4 and Remark 2.2, Q ε n,k := H ε n,k • P ∈ M.

	Recalling (2.5), this implies that
	lim inf n→∞	p n ≥ lim inf n→∞	lim k→0	lim ε→0

  ζ 1 )|F S T , while applying the law of large numbers conditionally to F S T implies that G n /n → ḡ = E f (S, ζ 1 )|F S T P -a.s. For instance, if f is lower-semicontinuous, non-decreasing with respect to its second parameter, and if each ζ i has a support equal to [y min , y max

  Remark 3.3. If the g i 's have the same law and form an independent family conditionally to F S T under P, then applying the law of large numbers conditionally to F S T implies ḡ = E g 1 |F S T , so that lim n→∞ p n

				) where
	H 0 n (S) := E	dQ 0 n dP |F S T	= dQ * (S)/dP by Remark 2.3, under (HCM). Since Q * (S) ∈ M by
	Remark 2.3 again, this proves the claim.	2

  In the following, we denote by Q e the element of M that minimizes the relative entropy, The map y > 0 → y ln y being strictly convex, it follows from Remark 2.3 that dQ e /dP is F S T -measurable, recall the argument used in the proof of Corollary 3.1. However, we do not impose in(3.18) the uniqueness condition of (HCM e ) in Remark 3.4. Theorem 3.2. Let Assumption 3.2, Assumption 3.3 and (2.3) be in force. Then the sequence of unit utility indifference prices satisfies

										)
	whenever Assumption 3.2 holds. Hence, Assumption 3.1 can now be formulated as
	Assumption 3.3. The conditions (3.1), (3.2), (3.18) and (3.4) hold.
	E	dQ e dP	log	dQ e dP	= inf Q∈M	E	dQ dP	log	dQ dP	,
	and whose existence is guaranteed by Theorem 2.2 in [8].
	Remark 3.6. lim n→∞

  Lemma 3.2. Let Assumption 3.2 and (3.18) hold. Then, there exists a sequence of positive numbers (α n ) n≥1 such that

			lim n→∞	E	dQ α n dP	log		dQ α n dP	= E	dQ e dP	log	dQ e dP	.	(3.26)
	Proof. For i = 1, 2, direct computations leads to
	inf y>0,Q∈M	E V i n y	dQ dP	=	inf y>0,Q∈M	y η i n	E	dQ dP	log	dQ dP		+	y η i n	log	y α n	+	α n -y η i n
				= inf y>0	y η i n	E	dQ e dP	log	dQ e dP	+	y η i n	log	y α n	+	α n -y n η i	(3.27)
				= E V i n ŷn	dQ e dP	,
	where the common minimizer ŷn > 0 is given by
			ŷn := α n e -ρ		with ρ := E	dQ e dP	log	dQ e dP	.	(3.28)
	Also note that

Remark 3.7. When the g i 's satisfy the conditions of Remark 3.3, the above result shows that the unit indifference prices converges to E Q e [E g 1 |F S T ]. Again, this consists in taking the mean over the part of the risk that can be diversified and computing the price under the minimal entropy martingale measure of the pure financial remaining claim.

In the rest of this section, we provide the proof of Theorem 3.2. In view of Theorem 3.1, it would suffice to show that

where we recall that Q 0 n is defined in (3.5). In the following, we actually prove that

where

in place of U n . In the above, (α n ) n≥1 is a sequence of positive numbers to be chosen later on, see (3.34) in the proof of Lemma 3.2 below. This trick is inspired from [START_REF] Carassus | Risk-averse asymptotics for reservation price[END_REF] and allows to reduce to the case where

Obviously, since U α n is an affine transformation of the utility function U n , we have We first provide upper and lower bounds for the Fenchel transform V α n of U α n in terms of Fenchel transforms of exponential utility functions with risk aversion η 1 n and η 2 n .

Lemma 3.1. Let Assumption 3.2 hold. Then, for each n ≥ 1,

where the functions V 1 n and V 2 n are defined by

Proof. It follows from the definition of (V i n ) i=1,2 and (3.21) that

, we deduce from (3.16) in Assumption 3.2 that

Together with the strict convexity of V α n , of each V i n , and the relation η

We now simply deduce from the right-hand side of (3.24) and (3.25) that

We conclude by using the left-hand side of (3.24).


Abhijit Sarkar

Laurent Blondé

Patrick Le Callet

Florent Autrusseau

Jürgen Stauder

Patrick Morvan

MODERN DISPLAYS: WHY WE SEE DIFFERENT COLORS, AND WHAT IT MEANS?

Keywords: Color Vision, Cone Fundamentals, Color Matching, Observer Variability, Displays

INTRODUCTION

The perception of color arises due to the response of the human visual system to light, and the interaction of light with objects. Thus, the perception of color requires three components, a light source, an object and an observer. Contributions of relative spectral power of each component, integrated over a range of wavelengths, constitute a mathematical representation of the visual response. This representation, referred to as tristimulus values, forms the basis of all colorimetric computations in basic and applied colorimetry. As shown in Eq. ( 1), these values are obtained by the integration of the spectral power distribution of a standard light source (S λ ), the reflectance factor of the object (R λ ), and the color matching functions of an average, standard observer ( λ x , λ y and λ z ) [START_REF] Berns | Billmeyer and Saltzman's principles of color technology[END_REF]. Here, k is a normalization factor that assigns the luminance of white an arbitrary value of 100.

λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ∆ = ∆ = ∆ = ∆ = ∑ ∑ ∑ ∑ y S k z R S k Z y R S k Y x R S k X 100 ; ;
(1) When two color stimuli produce the same visual response, i.e. identical tristimulus values, a visual match is obtained. (Note: small differences in tristimulus values cannot be perceived under normal circumstances, so the tristimulus values do not strictly need to be identical for a visual match. However, this perceptibility issue is not relevant for our current discussion.) By virtue of the wavelength-wise integration process, two stimuli with very different spectral power distribution can give rise to identical visual response, leading to a metameric match. However, such a match established by one observer can, and quite often does lead to a mismatch for a different observer, as the second observer has a different set of color matching functions than the former. This phenomenon is commonly termed as observer metamerism.

CIE standard observers and observer metamerism

Either of the two standard observers, as defined by the CIE (Commission Internationale de l'Eclairage), is currently used in applied colorimetry. The first is the 1931 CIE 2° standard observer, and the second is the 1964 CIE 10° standard observer (hereafter referred to as 10° SO). The latter is more suitable for large-field color stimuli, as encountered in most practical industrial applications.

The use of a standard observer in colorimetric computations is essentially based on the assumption that the whole population of color normal observers can be reasonably represented by a single observer model. In 1989, CIE recognized the variability among individual observers by introducing the concept of standard deviate observer [START_REF] Cie | Special Metamerism index: Change in observer[END_REF], but the model significantly under-predicted inter-observer variability [START_REF] Alfvin | Observer Variability in Metameric Color Matches Using Color Reproduction Media[END_REF], and was never adopted by the industry.

Thus, applied colorimetry in its current form does not have any provision of incorporating observer variability into the computations.

CIE 2006 Physiologically-Based Observer

In 2006, CIE's technical committee TC 1-36 published a report [START_REF] Cie | Fundamental Chromaticity Diagram with Physiological Axes -Part I[END_REF] (described hereafter as CIE06) on the choice of a set of Color Matching Functions (CMFs) and estimates of cone fundamentals for the color-normal observers. The cone fundamentals represent cone spectral sensitivities at the corneal plane, while color matching functions represent spectral tristimulus values of monochromatic stimuli of unit radiant power that embodies the color matching properties of the observer eye in a given system of primary stimuli R, G and B [START_REF] Wyszecki | Color Science, Concepts and Methods, Quantitative Data and Formulae[END_REF] (pp. 124). CMFs can be obtained through a linear transformation of the cone fundamentals. The CIE06 model is largely based on the work of Stockman and Sharpe [START_REF] Stockman | The spectral sensitivities of the middle-and long-wavelength-sensitive cones derived from measurements in observers of known genotype[END_REF]. Starting from 1959 Stiles-Burch (described hereafter as S&B) 10° CMFs [START_REF] Stiles | Colour-matching investigation: final report[END_REF], it defines 2° and 10° reference observers and provides a convenient framework for calculating average cone fundamentals for any field size between 1° and 10° and for an age between 20 and 80.

The introduction of CIE06 model is perhaps one of the most fundamental contributions in the field of color science since the establishment of 10° SO in 1964. This model was introduced to address the need of modeling the variability in observer color vision. Recent studies by current authors, as will be reviewed shortly, show that an age-dependent observer model may not be very effective for practical color imaging applications. However, this model provides an opportunity to further refine the 10° SO functions.

OBSERVER VARIABILITY IN THE CONTEXT OF DISPLAY COLORIMETRY

The relevance of the observer metamerism issue is quite dependent on the application context. In cross-media color reproduction, i.e. when a display color is compared with its printed version on paper, the significance of observer variability has not been found to be significant [START_REF] Oicherman | Effect of observer metamerism on colour matching of display and surface colours[END_REF]. This is presumably true for most of the industrial applications of the past several decades, be it printing, photography, painting or textile. Thus, taking into account individual observer variability in applied colorimetry did not warrant a serious consideration in the past. However, this limitation has become non-trivial with the advent and wide-spread adoption of modern wide-gamut consumer displays. Colors on two displays with very different spectral characteristics are highly metameric in nature, often resulting in colors that are a satisfactory match for one observer, and an unacceptable match for another. The observer metamerism is particularly significant when colors are compared on a display with broadband primaries (e.g. a CRT), and on another display with narrow-band primaries. Many modern Liquid Crystal Displays (LCDs) are fitted with Light Emitting Diode (LED) backlight (or sometimes, laser primaries) in order to achieve more vivid, more saturated and brighter colors. These displays are particularly susceptible to observer variability [START_REF] Fairchild | Mean Observer Metamerism and the Selection of Display Primaries[END_REF][10], since their peaky primaries can cause noticeable shift in the chromaticities of perceived colors with relatively minor change in the visual characteristics of the observer. Not surprisingly, similar observer metamerism issue has been observed when narrow band RGB-LEDs were matched with broadband lights [START_REF] Csuti | Colour matching experiments with RGB-LEDs[END_REF]. Note that none of the traditional industrial color applications mentioned before involved a color system with spectral characteristics similar to modern displays or the LEDs.

LIMITATIONS OF AN AGE-DEPENDENT OBSERVER IN AN APPLIED CONTEXT

CIE06 age parameter does not necessarily correspond to real observer ages. In other words, predicted model functions that best match the real observer data may not always be obtained using real observer ages. This may happen because of random observer variability, and/or because of the exclusion of one or more age-independent physiological factors from the CIE06 model [12][13]. CIE committee TC 1-36 also recognized this restriction by pointing out that CIE06 fundamental observer was a theoretical construct [START_REF] Cie | Fundamental Chromaticity Diagram with Physiological Axes -Part I[END_REF].

The CIE06 age parameters that resulted in the best predictions of individual S&B observer cone fundamental data were determined. This was done by computing the correlation coefficients between the normalized cone fundamentals for each S&B observer and those corresponding to all possible CIE06 age parameter values between 20 and 80 (a total of 61). For each S&B observer, the corresponding CIE age was the one yielding the highest correlation coefficient for a given cone fundamental. This process was repeated for all three cone fundamentals and for all 47 S&B observers. In Figure 1, the actual ages of 47 S&B observers have been plotted against the CIE06 predicted ages obtained using the correlation coefficient (CORR) method, applied separately to L, M and S. Evidently, no direct correspondence exists between the real and predicted ages that can be explained through a mathematical function. As will be seen in the next section, using real observer ages in the CIE model can lead to larger error in average observer prediction than what results from using the 10° SO. For this reason, there is limited usefulness of using an age-dependent observer in practical industrial applications.

PREDICTING INTRA-GROUP AVERAGE OBSERVERS WITH CIE06 MODEL AND 10° SO

The introduction of the physiologically based fundamental observer model by CIE warrants a comparative evaluation of the model and the 10° SO, to determine which is a better representation of average real observer data: CMFs derived from the CIE06 model, or the 10° SO. This issue was explored through a theoretical analysis performed in the context of display colorimetry. Experimental data from the 1959 S&B 10° CMFs involving 47 observers were used. Out of the 47 S&B observers, three age-groups were identified. Group-1 was formed by six observers with ages between 22 and 23, Group-2 consisted of ten observers with ages between 27 and 29 and another six observers with ages between 49 and 50 were placed in Group-3.

Because of space constraint, only x-CMF plots for Group-1 and -3 are shown (Figure 2), for which variations of model predictions and their potential impact on color perception were found to be significant. Intra-group minimum, maximum and average values are shown along with the 10° SO and CIE06 model predictions, with and without the age correspondence.

Table 1 lists the results of a statistical comparison of the S&B observer CMF data, 10° SO and CIE06 model predictions with and without age correspondence. Values corresponding to x-, y-and z-functions (corresponding to long-, medium-and short-wavelengths) for each group are shown. All values are computed by weighting wavelengthwise values by a scale-factor, followed by averaging over all wavelengths. The scale factor used was the respective intragroup average S&B observer data divided by their sum over all wavelengths, so that more weights were assigned to the values around the peak than those in the lower end of the ordinate. Note that since the x-, y-and z-CMFs do not have the same ordinate scale, the rows should not be compared as such. The 3 rd column in this table shows the intra-group standard deviation of the S&B data (note that standard deviation has the same units as the data), signifying intragroup observer variability. Following three columns list absolute difference of various functions from the intra-group mean, averaged over all wavelengths. The three functions considered here are i) 10° SO, ii) CIE06 with real observer ages as input (no age correspondence), and iii) CIE06 with corresponding ages as input (with age correspondence, as described in the previous section). As shown in Figure 2, in case of x-CMFs for Group-1 and -3, both CIE06 model predictions and 10° SO deviate from the intra-group average. Based on the results shown in Table 1, CIE06 model with real observer ages generally performs similar to or worse than the 10° SO x-and y-CMFs. For Group-1 and -3, the age correspondence method improves CIE06 predictions, and is mostly better than the 10° functions. For Group-2, the prediction error is relatively low even without age correspondence, indicating CIE06 model's age parameter works well for the age group of 27-29. This is not surprising since the average observer age in the S&B study, on which CIE06 is based, was 32.

Fig. 3. Spectral Power Distribution of the CRT and the LCD used in the experiments

As far as the z-CMF is concerned, the CIE06 model produces better results than the 10° SO, even without age correspondence. On an average, the reduction in mean absolute difference is more than 50%. Within the constraints of current analysis, CIE06 seems to offer an improvement over the 10° SO z-function.

Any statistical method used to compare the model predictions with real observer data is incomplete without an analysis of the perceptual effect of the prediction errors. Thus, an additional analysis was performed to simulate the effect of the deviations of CIE06 and 10° SO from the average intra-group observer data on display color perception. In this analysis, two displays were used. The first was a Sony BVM32 Cathode Ray Tube (CRT) reference studio display (hereafter referred to as Ref-CRT). The second was a Hewlett-Packard DreamColor Wide-Gamut Liquid Crystal Display with LED backlight (hereafter referred to as WG-LCD). The spectral characteristics of the CRT and the LCD are significantly different (see Figure 3).

Figure 4 shows the perceptual color difference in terms of CIELAB delta-E 2000 values (an advanced color difference formula denoted as ∆E* 00 ) for 24 Colorchecker patches (an industry standard color dataset) simulated for WG-LCD, and computed with CIE06 CMFs and 10° SO. The magnitude of the ∆E* 00 values are represented along the abscissa, and the color differences for all 24 colors are plotted as bars along the ordinate. For each group, the colors were divided into three categories, long (magenta through orange yellow -see Figure 4), mid (cyan through yellow) and short (purplish blue through purple), and the 90 th percentile values were computed separately for each category (indicated in Figure 4). Neutral and skin colors are shown separately. Note that all these ∆E* 00 values are computational color differences with respect to the intragroup S&B average observer data, and simply help us compare model prediction errors in a space. These do not represent color differences perceived by a real observer. The reader should be warned that the Euclidean nature of the perceptual color space (and that of the color difference equation) is potentially distorted when the 10° SO is replaced by a different set of observer CMFs. So, there is a practical limitation in evaluating performances of different observer models in a perceptual color space. Further, because of the nonlinear transformations and interactions of x-, y-and z-CMFs in computing perceptual color difference, it is difficult to directly correlate the prediction errors in individual CMFs (Table 1) with perceptual color differences in Figure 4. Based on Figure 4, 10° SO generally performs better than the CIE06 model for Group-1 and -3, except in the blue/blue-purple region for Group-3. For Group-2, all predictions errors are relatively low. Age correspondence improves the results for Group-1 and -2, which are often better than those of the 10° SO. However, this is not so evident in case of Group-3. For Group-3, the prediction error is markedly higher for 10° SO in the blue/blue-purple region.

For Ref CRT, the difference between the CIE06 CMFs and the 10° SO in predicting intra-group average observer data was less apparent.

Overall, this analysis suggests that the CIE06 model, in its original form, does not offer an improvement over the 10° SO in predicting average observer x-and y-functions for various age-groups. However, for the z-function, CIE06 in all cases appears to perform better than the 10° SO in the tristimulus space.

A COLOR MATCHING EXPERIMENT USING TWO DISPLAYS

While metameric color matching data are available from several past studies on cross-media color reproduction, no experimental data have so far been available on display color matching, in particular, for displays with vastly different spectral characteristics. To address this need, a preliminary set of color matching experiments [START_REF] Sarkar | A color matching experiment using two displays: design considerations and pilot test results[END_REF] were performed on the same two displays used in the theoretical analysis. Because of the significant difference in the spectral characteristics of the CRT and the LCD (Figure 3), color matches were expected to vary significantly from one observer to the other.

Fig. 5. Experimental setup

The displays were placed perpendicular to each other, as shown in Figure 5. A front-surface reflection mirror was placed in front of the CRT, and a mask was placed between the observer and the displays such that the observer's visual field consisted simply of a 10° circular area divided vertically in two equal parts (called bipartite field). The right half of the bipartite field was the LCD screen, and the left half was the CRT screen, seen through the mirror. The experiment was conducted both in dark and white surround conditions. For the latter, a white mask uniformly lit by an overhead projector was used. The observers were asked to adjust the color on the left half of the bipartite field (CRT) to match the color on the right half (LCD). After each match, the spectra of the colors on the two displays were measured. Each observer matched nine colors (these were pre-selected along physiologically significant axes in the Macleod-Boynton chromaticity diagram [START_REF] Sarkar | A color matching experiment using two displays: design considerations and pilot test results[END_REF]). A total of ten observers participated in the experiment.

For a more detailed discussion on the experiments, the readers are directed to reference [START_REF] Sarkar | A color matching experiment using two displays: design considerations and pilot test results[END_REF].

In order to analyze the experimental data, XYZ values were computed from the spectral data for both the LCD test colors and the CRT matching colors, using CIE 10° SO. For each observer, the XYZ values over all repetitions were averaged, and then were converted to CIELAB values. Finally, ∆E* 00 color difference between these two sets of CIELAB values (corresponding to CRT and LCD) were computed. These ∆E* 00 values represent the differences perceived by an observer identical to the CIE 10° SO between the LCD and CRT colors, while in reality they were satisfactory matches for individual observers. Table 2 lists these ∆E* 00 values. Figure 6 plots the ∆E* 00 color difference values corresponding to individual observer matches on LCD and CRT, for both pilot tests (without and with surround). There are nine colors shown along the xaxis. The color differences along the y-axis are represented in the form of bars, grouped by the number of observers. From the data in Table 2 and the plots in Figure 6, it is apparent that for several stimuli, the colors on the two displays that matched for individual observers were predicted by the CIE 10° SO as having a perceptible color difference. In a similar analysis it was found that when the colors, which were predicted by the standard observer to be a match, were shown on the two displays, they were significantly different from individual observer matches. The errors are the highest for the test color #4, a saturated blue (Table 2). In case of dark surround test, the mean, maximum and the 75 th percentile ∆E* 00 values between individual observer matches on LCD and CRT, across all stimuli and all observers, were 1.39, 3.41 and 1.86 respectively. In case of white surround, the values were 1.4, 3.47 and 1.84 respectively. Overall, no significant effect of surround on color matching is evident from the results.

The significance of the ∆E* 00 values depends on the context, viewing condition and the observer. While the values reported here are possibly low in case of complex images and surrounds, for a carefully controlled experimental setup such as ours, where uniform color stimuli are matched by experienced observers, a ∆E* 00 value much larger than 1.0 is likely to be perceptible. In color critical applications involving expert observers, such a difference will even be unacceptable. This was further confirmed during the course of the study, when color matches made by one observer, and confirmed to be a satisfactory match, was presented to others. In several instances, color match of any one observer was rejected by some others, and vice versa.

CONCLUSIONS

In the context of modern display colorimetry, when colors are compared on two displays with very different spectral power distributions, color perception can vary significantly among color normal observers. Thus, using the CIE 10° standard observer in colorimetric computation in such applied context can lead to an unacceptable color match for many color normal observers. In this paper, we explored this problem by summarizing the results from our ongoing study on observer variability, involving both theoretical analysis and visual experiments. Following are the two key inferences that emerge: 1) an age-dependent observer model such as CIE 2006 may not be very suitable for practical applications, since variations in color perception within a given age-group is significant, and cannot be predicted by such models; 2) in the tristimulus space, the short-wave sensitive (z-) average observer CMF is better predicted by the CIE 2006 model than by the CIE 10° standard observer. This needs to be further investigated with a larger set of observers belonging to various age-groups.

In the concluding remark, next generation colorimetry must offer a practical framework to take into account observer variability in highly color-critical professional applications, specifically those involving modern displays with narrow-band primaries. This is the ultimate goal of the present work.

Fig. 1 .

 1 Fig. 1. Age correspondence between CIE06 model's best prediction and 47 Stiles-Burch observers' visual data, actual observer age vs. CIE06 predicted age

Fig. 2 .

 2 Fig. 2. x-CMFs for the average S&B observer, CIE06 model predictions and 10° SO for Group-1 [top] and -3 [bottom]

Fig. 4 .

 4 Fig. 4. Simulated ∆E* 00 plots (90 th percentile) for ColorChecker patches shown on WG-LCD and computed for various CMFs corresponding to Group-1, -2 and -3 [top to bottom, respectively]

Fig. 6 .

 6 Fig. 6. ∆E* 00 Color difference between CRT and LCD observer matches as predicted by 10° SO, for dark surround (top) and white surround (bottom)

Table 1 .

 1 Comparison of CMF data from intra-group average S&B observer, 10° SO and CIE06 model predictions with and without age correspondence

	CMF	Grp. No.	Mean Intra-group S&B Std. Dev.	Mean Abs. Diff. From Mean Intra-group S&B Data CIE06 10° Std. No Age With Age Obs. Corres. Corres.
		1	10.11	5.68	6.53	2.51
	x	2	11.28	2.54	1.74	1.99
		3	9.12	9.93	10.58	6.06
		1	6.02	2.81	4.73	1.13
	y	2	6.68	2.28	2.42	2.43
		3		2.12	4.21	2.5
		1	22.7	19.25	8.22	7.55
	z	2	25.54	10.88	6.2	6.17
		3	21.43	11.71	5.21	3.99

Table 2 .

 2 Color difference (∆E* 00 ) values computed between the test colors on LCD and observer matches on CRT

	Stimulus ID	∆E* 00 for Av. Observer Matches (LCD and CRT) Pilot Test 2 Pilot Test 3
	1	1.11	1.34
	2	1.89	1.91
	3	1.45	1.37
	4	2.07	2.13
	5	1.07	1.26
	6	1.58	1.50
	7	1.05	0.98
	8	0.86	0.70
	9	1.40	1.36