Envelope theorems in Banach lattices

Anna BattauzDepartment of Finance, Bocconi University

Marzia De DonnoDepartment of Decision Sciences, Bocconi University

Fulvio OrtuDepartment of Finance and IGIER, Bocconi University

March 11, 2011

Abstract

We derive envelope theorems for optimization problems in which the value functiontakes values in a general Banach lattice, and not necessarily in the real line. We imposeno restriction whatsoever on the choice set. Our result extend therefore the ones of

Milgrom and Segal (2002). We apply our results to discuss the existence of a well-defined notion of marginal utility of wealth in optimal consumption-portfolio problemsin which the utility from consumption is additive but possibly state-dependent and,most importantly, the information structure is not require to be Markovian. In thisgeneral setting, the value function is itself a random variable and, if integrable, takesvalues in a Banach lattice so that our general results can be applied.

1 Introduction

Envelope theorems constitute one of the genuine workhorses of economics, and their appli-

cations are truly ubiquitous. Over the years, in fact, several extensions of the traditional

envelope theorems have emerged, as a response to the necessity of analyzing the behavior

of the value function of optimization problems lacking the assumptions for the applicability

of the standard envelope results of any graduate textbook. To our knowledge, the most

general set of envelope theorems currently available in the literature is due to Milgrom

and Segal (2002), who develop envelope results that do not require any assumption on the

choice set of the optimization problem. In particular, they first show that the traditional

envelope formula holds at any differentiability point of the value function, and then they

establish conditions for the (left, right or full) differentiability of the value function.

1

The contribution of this paper is to extend Milgrom and Segal’s results by allowing the

objective function of the optimization problem under scrutiny to take values in a general

Banach lattice, instead of simply in the set of real numbers. In doing so, we still allow

the choice set to be arbitrary, while we let our parameters to belong to a general Banach

space. In this setting, we show that all the results in Milgrom and Segal (2002) can

be nicely extended, provided that the standard notion of differentiability for real-valued

functions is suitably replaced by the more general notion of Fréchet-differentiability. Given

a function from a Banach spaces into another Banach space, in fact, the Fréchet differential

constitutes the right tool to analyze the incremental effect on the value of the function due

to an increment in the independent variable.

The obvious question at this point is: why Banach lattices? This question can be

answered from two angles. A first, more general angle involves the long standing tradition

of using the lattice structure in economic theory, in particular in general equilibrium theory

and in its applications to finance theory (see e.g. the references in Aliprantis, Monteiro

and Tourky, 2004). In his fundamental contribution to general equilibrium theory, Mas-

Colell (1986) analyzes the existence problem in economies where the commodity space is a

general topological vector lattice (of which Banach Lattices are clearly special cases). The

fundamental motivation for his approach lies in the mounting interest for economies with

infinitely many commodities, such as Arrow-Debreu economics with infinitely-lived agents,

or in the analysis of commodity differentiation, or in the arbitrage literature dating back

to Black and Scholes (1973) and Harrison and Kreps (1979). The utility of employing the

techniques associated with the lattice structure comes from the fact that, in many of these

applications, the positive orthant would have empty interior, a fact that could produce

insurmountable difficulties unless the lattice structure is taken to full bearing.

A second angle comes from a specific, interesting problem in asset pricing that we en-

countered when working on Battauz et al. (2011), and that actually motivated the present

paper. Consider a standard, multi-period optimal consumption-portfolio problem for an

agent with finite life. Let the agent’s utility function be time-additive but possibly state-

dependent. Crucially, depart from the standard treatments available in the literature by

not imposing the information structure to be Markovian. More precisely, take for given the

information structure modelled as a filtration of a standard probability space, and assume

that prices and dividends are any stochastic processes adapted to the given, general filtra-

2

tion. To any time t associate a value function that, given the level of wealth accumulated

up to t, gives the maximum remaining utility (continuation utility) conditional on following

an optimal consumption-portfolio strategy from t on. The question is then: is the marginal

utility of wealth well defined and, if so, how does it compare to the marginal utility from

consumption at time t? In a Markovian framework, the usual envelope result could be

invoked to obtain that, under standard conditions, the marginal utility of wealth is indeed

well defined and it coincides with the marginal utility from consumption. This conclusion

would easily derive from the fact that the, in a Markovian setting, the value function is

a real-valued function of the wealth level (and, possibly, of other state variables). In a

our general setting, however, this is not the case, since the value function itself would be

a random variable, obtained by taking the ’right type’ of sup over all possible controls,

i.e. consumption paths and investment strategies. Upon requiring the value function to

be at least integrable, the ’right type’ of sup manifests itself upon recognizing that the

set of integrable random variables is a nice Banach lattice, with the sup among random

variables defined via the standard pointwise max operator. At this point, our general

envelope results for Banach lattices can be taken to bear on this problem, to show that

under a suitable set of conditions the value function is in fact (Fréchet)-differentiable, and

its (Fréchet) differential equals the (Fréchet) differential of the utility function, when the

latter exists.

The remainder of the paper is as follows. In the next section, we introduce the notation

and definitions to set up the Banach lattice-valued optimization problem, we discuss the

assumptions underlying our results and we then prove our extension of the envelope theorem

to Banach lattices. In Section 3, we describe the asset pricing application of our result,

that is we introduce a general optimal consumption-portfolio problem and we apply our

general results from Section 2 to discuss the conditions under which the marginal utility of

wealth is well defined and coincides with the marginal utility from consumption. Section

4 concludes. The appendix contains some basic material on Banach lattices useful for the

problem at hand.

3

2 The general results

To prove their envelope theorem for arbitrary choice sets, Milgrom and Segal (2002) assume

their parameter to lie in [0, 1] but make clear that their approach applies to parameters

lying in a more general normed vector space. Here we extend their results by allowing the

objective function, and, as a consequence the value function, to take values in a Banach

lattice and not just in R, while maintaining the arbitrariness of the choice set.

In our extended framework, the usual notion of differentiability must be replaced with

the notion of differentiability in Banach spaces, namely Fréchet differentiability. We recall

this definition hereafter.

Definition 1 Let X,Y be Banach spaces and U ⊂ X an open set. A function G : U → Y

is Fréchet-differentiable at a given point x∗ ∈ U if there exists a continuous (bounded)

linear operator DxG : X → Y such that

limkykX→0

kG(x+ y)−G(x)−DxG(y)kY

kykX= 0 (1)

The operator DxG is called the Fréchet differential of G in x.

It is also useful to recall the following definition:

Definition 2 We say that G admits directional derivative at x along a direction y ∈ X

if the following limit exists:

DxG(y) = lim

h→0

G(x+ hy)−G(x)

h(2)

where the limit is meant in Y -norm (as h tends to 0 in R).

G is said to be Gateaux differentiable at x if it admits a directional derivative along

all directions at x and in this case the function DxG is called the Gateaux differential of

G in x.

If G is Fréchet differentiable, then it is also Gateaux differentiable and DxG(y) = Dx

G(y) for

all y ∈ X.

Let X be a Banach space and take an open set U in X as the set of parameters. Let Θ

denote the (arbitrary) choice set. Given then an order continuous Banach lattice Y , we let

4

F : Θ×U → Y be the objective function of our optimization problem. For each parameter

u ∈ U , we define the value function as:

V (u) = supθ∈Θ

F (θ, u). (3)

If u is a Fréchet differentiability point for F (θ, · ) or V , we denote the Fréchet differen-

tials respectively with DuF (θ) and Du

V .

Following Milgrom and Segal (2002), we start by showing that the envelope formula

holds at any differentiability point of the value function. For any u ∈ U , we define the set

of optimal choices

Θ∗(u) = θ ∈ Θ : F (θ, u) = V (u).

We point out that Θ∗(u) can possibly be empty.

We can then extend Milgrom and Segal’s argument to our setting to prove the following:

Theorem 1 Let u ∈ U and θ ∈ Θ∗(u). Assume that there exists some r > 0 such that

V (x) ∈ Y for every x ∈ B(u, r)1. If both F (θ, · ) and V ( · ) admits directional derivative

in u along some direction y ∈ X, then DuF (θ)(y) = Du

V (y). If both F (θ, · ) and V ( · ) are

Fréchet-differentiable in u, then necessarily

DuF (θ) = Du

V .

Proof. Let h ∈ R and y ∈ X such that khykX ≤ r. Then:

F (θ, u+ hy)− F (θ, u) ≤ V (u+ hy)− V (u).

In particular, taking hn in R+, which decreases to 0 as n→ +∞, and dividing both sides

of the inequalities by hn, we obtain

F (θ, u+ hny)− F (θ, u)

hn≤ V (u+ hny)− V (u)

hn. (4)

If F (θ, · ) admits directional derivative along y, then according to (2), we have that

F (θ, u+ hny)− F (θ, u)

hn1B(u, r) denotes as usual the ball centered in u and with radius r

5

converges in Y -norm to DuF (θ)(y). Then there exists a a subsequence which converges in

order to the same limit (see Remark 4 in the Appendix). Analogously, if V has a derivative

along y, thenV (u+ hny)− V (u)

hn

will converge in Y -norm, and, up to a subsequence, in order, to DuV (y). Hence

DuF (θ)(y) ≤ Du

V (y). (5)

If we repeat the same argument with a sequence hn in R−, which increases to 0 as n→ +∞,

the inequality (4) is reversed as well as inequality (5), so we see that

DuF (θ)(y) = Du

V (y).

If F (θ, · ) and V are also Fréchet differentiable, then for all y ∈ X,

DuF (θ)(y) = Du

F (θ)(y) = DuV (y) = Du

V (y)

and the claim follows. ¤

Our aim now is to determine a set of sufficient conditions for the value function to

be Fréchet differentiable. As already observed by Milgrom and Siegel, the structure of

the choice set is not relevant to this aim, though several versions of the envelope theorem

exploit topological properties of the choice set and regularity of the objective function

in the choice variable. Following Milgrom and Siegel, the basic idea is to require that the

objective function satisfies some properties uniformly with respect to the choice parameter.

We start by fixing a value of the parameter u ∈ U at which we want to investigate the

Fréchet differentiability of the value function. Our first requirement is that there exists an

optimal choice for this parameter.

Assumption 1 The set Θ∗(u) is not empty.

The assumptions which follows extend Milgrom and Segal’s assumption on the objective

function. First of all we require a stronger condition than Fréchet differentiability.

Assumption 2 The objective function F (θ, · ) is Fréchet differentiable at u for every θ ∈

Θ. In particular, it is equidifferentiable, i.e.:

F (θ, u+ x)− F (θ, u) = DuF (θ)(x) + σ(θ, u, x) · kxkX

where |σ(θ, u, x)| ≤ Σ kxkX , for Σ ∈ Y , for all θ ∈ Θ, for x ∈ X such that u+ x ∈ U .

6

This assumption implies in particular that F (θ) is differentiable in u for any θ in the

choice set. In addition, we have that°°°F (θ, u+ x)− F (θ, u)−DuF (θ)(x)

°°°Y

kxkX

goes to 0 uniformly in θ as kxkX tends to 0. This is why our Assumption 2 is in some sense

the moral equivalent of the equidifferentiability imposed by Milgrom and Segal (2002).

Before introducing the next assumption, it is useful to recall some basic notation from

functional analysis. We denote with L(X,Y ) the vector space of all linear continuous

(bounded) operators from X to Y . When well-defined, the Fréchet differential DuF (θ) be-

longs to L(X,Y ). We can define a norm on this space as follows: for T ∈ L(X,Y ), we

set

kTkL = supx∈X,kxkX≤1

kT (x)kY .

The spirit of the next assumption is basically to require the Fréchet differential of F to

be norm bounded uniformly in θ.

Assumption 3 For every x ∈ U there exists a vector yx ∈ Y such that¯DuF (θ)(x)

¯≤ yx kxkX

for all θ ∈ Θ.

As a consequence,°°°Du

F (θ)(x)°°°Y≤ Mx for all θ ∈ Θ, where Mx = kyxkY kxkX , that is,

the family of operators³DuF (θ)

´θ∈Θ

is pointwise bounded. The Banach-Steinhaus theorem,

also known as the Uniform Boundedness Principle (see, for instance, Brezis (1983)), implies

then that the set³DuF (θ)

´θ∈Θ

is also norm bounded, namely

supθ∈Θ

kDuF (θ)kL < +∞.

In other words, there exists a constant Λ such that°°°DuF (θ)(x)

°°°Y≤ Λ kxkX (6)

for all θ ∈ Θ.

7

A first step in the search for sufficient conditions for the value function to be differ-

entiable at u is to show that V takes values in Y and, at the minimum, is continuous

at the differentiability points of the objective function. This is ensured by the previous

assumptions, as the next lemma proves.

Lemma 1 If Assumptions 1, 2, 3 hold, the value function V takes values in Y and is

continuous in u.

Proof. First, we notice that V (u) ∈ Y : indeed by Assumption 1, there exists some θ such

that V (u) = F (θ, u) ∈ Y .

For x ∈ U we have:

|V (x)− V (u)| =

¯supθ1

F (θ1, x)− supθ2

F (θ2, u)

¯≤ sup

θ|F (θ, x)− F (θ, u)|

≤ supθ

¯DuF (θ)(x− u)

¯+sup

θ|σ(θ, u, x− u)| · k x− ukX

where the last inequality is a consequence of Assumption 2. Taking the Y -norms of both

sides and exploiting the triangle inequality together with Assumption 2 and 3 (in particular,

inequality (6)), we obtain

kV (x)− V (u)kY < Λ k x− ukX + kΣkY · k x− uk2X .

This shows that V (x) ∈ Y and that V is continuous in u. ¤In order to obtain that V is differentiable, we will need to require the Fréchet differential

of F at u to be continuous (in some sense better specified below). To this end, we impose

the following assumption:

Assumption 4 (i) There exists some r > 0, such that B(u, r) ⊂ U and for all x ∈

B(u, r), the set Θ∗(x) is not empty;

(ii) for x ∈ B(u, r) and θx ∈ Θ∗(x), θ ∈ Θ∗(u), the Fréchet differential of F at u satisfies:

limkx−ukX→0

kDuF (θx)

−DuF (θ)kL = 0.

8

We do not require continuity of the Fréchet differential in the classical sense: in fact we do

not ask F to be differentiable in other points but u. Instead, we ask F (θ, · ) to be differential

at u for all θ (Assumption 2) and require continuity of the family of differentials (DuF (θ))θ∈Θ

as the choice θ approaches an optimal choice.

Exploiting this assumption, which naturally implies Assumption 1 together with As-

sumption 2 on the equidifferentiability of F , we are able to derive the following property,

which is useful to show the differentiability of the value function.

Lemma 2 Suppose that Assumptions 2,4 hold. Let let θ ∈ Θ∗(u). Moreover, for some

x ∈ X, such that kxkX ≤ r, let θu+x ∈ Θ∗(u+ x). Then

limkxkX→0

°°°F (θu+x, u+ x)− F (θu+x, u)−DuF (θ)(x)

°°°Y

kxkX= 0

Proof. We have, thanks to Assumption 2, that:

F (θu+x, u+ x)− F (θu+x, u) = DuF (θu+x)

(x) + σ(θu+x, u, x) · kxkX .

where |σ(θu+x, u, x)| ≤ Σ kxkX . Hence, the following inequalities hold:°°°F (θx, u+ x)− F (θx, u)−DuF (θ)(x)

°°°Y

≤°°°Du

F (θu+x)(x)−Du

F (θ)(x)°°°Y+ kσ(θu+x, u, x)kY · kxkX

≤°°°Du

F (θu+x)−Du

F (θ)

°°°L· kxkX + kΣkY kxk

2X .

Dividing by kxkX and taking the limit as kxkX → 0 we obtain the claim by using Assump-

tion 4 (ii). ¤

We are finally ready to prove our main theorem.

Theorem 2 If Assumptions 2, 3, 4 hold, then the value function V is Fréchet-differentiable

in u and

DuV = Du

F (θ)

for θ ∈ Θ∗(u).

9

Proof. Let θu ∈ Θ∗(u). Then,

V (u) = F (θu, u) ≥ F (θ, u) for any θ ∈ Θ.

Now, let x be such that kxkX ≤ r and take θu+x in Θ∗(u + x), which is not empty by

Assumption 4(i). Then

V (u+ x) = F (θu+x, u+ x) ≥ F (θ, u+ x) for any θ ∈ Θ.

In particular, V (u) ≥ F (θu+x, u) and V (u+ x) ≥ F (θu, u+ x). Thus, we can write:

F (θu, u+ x)− F (θu, u) ≤ V (u+ x)− V (u) ≤ F (θu+x, u+ x)− F (θu+x, u) .

Subtracting DuF (θu)

(x) and dividing by kxkX , we obtain the following inequalities:

F (θu, u+ x)− F (θu, u)−DuF (θu)

(x)

kxkX

≤V (u+ x)− V (u)−Du

F (θu)(x)

kxkX

≤F (θu+x, u+ x)− F (θu+x, u)−Du

F (θu)(x)

kxkX

Take now the limit as kxkX → 0. Since the first and the last term converge to 0 in Y -norm,

the middle term must converge to 0 as well. This implies that V is Fréchet-differentiable

in u and

DuV = Du

F (θ). ¤

3 Envelope results for general asset pricing models

We consider a frictionless security market in which J assets are traded over the investment

horizon T = 0, 1, . . . , T. Asset prices and cash-flows are denominated in units of the

single good consumed in the economy. We assume that investors can freely dispose of the

good. To describe the stochastic evolution of asset prices and cash-flows we take as given a

filtered probability space (Ω,F , P, FtTt=0),2 and denote by dj (t) the Ft−measurable cash

flow distributed by asset j at date t and by Sj (t) the Ft−measurable date t price of asset j

net of the current cash flow. Given p ∈ [1,+∞[, we assume that Sj (t) , dj (t) ∈ Lp(Ω,Ft, P )

2As usual, we assume that F is augmented with P−null sets, F0 is the trivial sigma-algebra ∅,Ω andFT = F .

10

for all t. Without loss of generality, we assume that the assets distribute no cash flow at

date 0 and a liquidating one at date T , that is dj (0) = Sj (T ) = 0 almost surely.

A dynamic investment strategy is a sequence θ = θ (t)T−1t=0 of J-dimensional, Ft−measurable

random variables, that is θ (t) = θ1 (t) , θ2 (t) , . . . , θJ (t), where θj (t) represents the posi-

tion (in number of units) in assets j taken at date t and liquidated at date t+1. We denote

by Vθ = Vθ (t)Tt=0 the value process of the dynamic investment strategy θ, namely Vθ (t)

is the date t value of a dynamic investment strategy, defined as the cost of establishing the

positions in the J assets at their net-of-cash-flow prices, if t precedes the last trading date,

and, at T , as the payoff from the final liquidation of θ. Formally:

Vθ (t) =

⎧⎨⎩ θ(t) · S(t) t < T

θ(T − 1) · d(T ) t = T .

At any date t, a dynamic investment strategy θ produces a cash flow xθ (t), generated

by the difference between the resources obtained from liquidating the positions taken at

t− 1 at the cum-cash flow prices S(t)+ d(t), and the cost to establish the new positions at

the net-of-cash flow prices S(t). The cash-flow xθ (t) is therefore related to the value Vθ (t)

as follows:

xθ (t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩−Vθ(0) t = 0

θ (t− 1) · [S(t) + d(t)]− Vθ(t) t = 1, . . . , T − 1

Vθ (T ) t = T .

(7)

Henceforth, we call the sequence xθ = xθ (t)Tt=0 the cash-flow process of θ.

Definition 3 We call admissible any dynamic investment strategy θ such that Vθ(t),

xθ(t) ∈ Lp(Ω,Ft, P ) for t = 0, 1, . . . , T . We denote with Θ the set of all admissible dynamic

investment strategies.

An agent in this market is identified by an initial endowment e0 ≥ 0 of the single con-

sumption good and a complete and transitive preference relation on the set C =TQt=0

Lp(Ω,Ft, P )

of consumption sequences c = (c(0), c(1), . . . , c(T )), with c(t) ∈ Lp(Ω,Ft, P ) for all t. In

choosing the optimal intertemporal consumption and asset allocation, each agent (e0,º)

in A faces the budget constraint

B(e0) = c ∈ C | c(0) ≤ xθ(0) + e0, c(t) ≤ xθ(t) ∀ t > 0 for some θ ∈ Θ .

11

In particular, we consider the class of agents whose preferences have a time-additive

von Neumann-Morgenstern representation, such that the period-utilities are allowed to

depend on the state ω. In details, we assume that the preference U(c) of an agent takes

the following form

U(c) =TXt=0

ZΩut(c(t, ω), ω)dP (ω) =

TXt=0

E [ut (c(t)] (8)

where for all t < T ,the period utilities ut : <×Ω→ < are assumed to satisfy the following

conditions:3

(i) for all t, the function ut(c, ω) : <×Ω→ < is measurable with respect to the product

σ-algebra B(<)⊗ Ft (where B(<) denotes the Borel σ-algebra)4;

(ii) for all c ∈ B(e0), the integrals in (8)RΩ ut(c(t, ω), ω)dP (ω) are well defined

5 and

either are finite or take the value −∞; as a consequence, U(c) < +∞ for all c ∈ B(e0).

(iii) for all t, the function ut( · , ω) : < → < is real-valued and strictly increasing for

almost every ω.

An optimal consumption-portfolio choice for such an agent is a couple (c∗, θ∗) ∈ C×Θ

such that c∗(0) ≤ xθ∗(0)+ e0, c∗(t) ≤ xθ∗(t) for t = 1, . . . , T and U(c∗) ≥ U(c) for all c ∈ C

such that c(0) ≤ xθ(0) + e0, c(t) ≤ xθ(t) for t = 1, . . . , T for some θ ∈ Θ. We make the

following assumption:

Condition 1 : There exists an optimal solution to the consumption-portfolio problem

for an agent with preferences as in (8) and initial endowment e0.

It can be easily shown, as a consequence of the strict monotonicity of the period-utilities,

that the constraints will be binding at the optimum, namely

c∗(0) = xθ∗(0) + e0

c∗(t) = xθ∗(t) for t = 1, . . . , T.

3For a discussion of period utilities that depend directly also on the state of nature ω see for instance

Berrier, Rogers and Tehranchi (2007) or Frittelli, Maggis (2011).4This condition guarantees that for every Ft-measurable random vector (c(t)), the function ut (c(t, ω), ω)

(defined on Ω with values in <) is Ft-measurable.5As usual for a random variable Z the integral

ΩZ(ω)dP (ω) is well defined and finite if both

ΩZ+(ω)dP (ω) < +∞ and

ΩZ−(ω)dP (ω) < +∞.We set

ΩZ(ω)dP (ω) = −∞ if

ΩZ−(ω)dP (ω) = +∞

andΩZ+(ω)dP (ω) < +∞.We set

ΩZ(ω)dP (ω) = +∞ if

ΩZ+(ω)dP (ω) = +∞ and

ΩZ−(ω)dP (ω) <

+∞. Otherwise the integral is not defined.

12

To any optimal consumption-portfolio choice (c∗, θ∗) for an agent with preferences as

in (8) and initial endowment e0, we associate the optimal intertemporal wealth W ∗ =

W ∗(t)Tt=0 generated by θ∗, that is

W ∗(t) =

⎧⎨⎩ e0 t = 0

θ∗(t− 1) · [S(t) + d(t)], t = 1, . . . , T.

Note that W ∗(t) = xθ∗(t) + Vθ∗(t) = c∗(t) + Vθ∗(t) for t = 1, . . . , T − 1.

Fix now t ∈ 0, 1, . . . , T − 1, and let W be an Ft- measurable random variable. We

define a random variable H(t,W ) which represents the maximum remaining utility (or

continuation utility) at time t for an agent whose current level of wealth is W :

H(t,W ) ≡ ess sup(c,θ)∈C×Θ

TPs=t

Et [us(c(s))]

s.t.

⎧⎨⎩ c(t) + Vθ(t) ≤W

c(s) ≤ xθ(s) s = t+ 1, . . . , T

(9)

for t = 0, 1, . . . , T , where Et[ · ] denotes the conditional expectation with respect to Ft.

We assume that the integrals E [us(c(s))] (and hence the conditional expectations in (9))

are well defined, and, for all consumptions satisfying the budget constraint at time t, are

either finite or take the value −∞ (in which case we set Et [us(c(s))] = −∞). In particular,

for W = W ∗(t), we have the maximum remaining utility H(t,W ∗(t)), given the optimal

wealth level and the optimal past consumption. We remark that the necessity to define

the maximum remaining utility H via an essential sup, instead of a normal sup over real

numbers, is due to the fact that we do not impose the information structure of the model

to be Markovian.

In Battauz et al. (2011, Proposition 1), it is shown that at the optimum H is well-

defined and finite, and that it satisfies the dynamic programming principle, that is:

H (t,W ∗(t)) = ut(c∗(t)) +Et [H (t+ 1,W

∗(t+ 1))] ; (10)

or, equivalently

H (t,W ∗(t)) =TXs=t

Et [us (c∗(s))] .

13

In what follow, we consider the time t as fixed: therefore, for the sake of simplicity, we

will let H(W ) = H(t,W ) for some W ∈ Lp(Ω,Ft, P ). Following now the approach and the

notation introduced in Section 2, we define the function:

F (θ,W ) = ut(W − Vθ(t)) +Et

"TX

s=t+1

us (xθ(s))

#.

Since all the period-utilities are not satiated, the constraints in (9) will be binding, so we

can write the optimization problem (9) as

H(W ) ≡ ess supθ∈Θt

F (θ,W )

where the choice set Θt is the set of admissible strategies at time t, namely the set of

sequences θ = θ (s)T−1s=t of J-dimensional, Fs−measurable random variables such that

Vθ(s), xθ(s) ∈ Lp(Ω,Fs, P ) for s = t, . . . , T . Note that Θt is a convex set. Assume that

F (θ,W ) takes value in L1(Ω,Ft, P ) in a neighborhood of the optimal wealth W (t). We

have thus reduced our initial problem to a problem of the form (3), where the parameter

W lies in the Banach space Lp(Ω,Ft, P ) and the objective function takes values in the

Banach lattice L1(Ω,Ft, P ).

As in the previous section, we denote with Θ∗(W ) the set of optimal choices given the

parameter W , namely the set of optimal admissible strategies (from time t up to time

T − 1), given the wealth level W at time t:

Θ∗(W ) = θ ∈ Θt : F (θ,W ) = H(W ) .

Remark 1 Note that if we fix a strategy θ ∈ Θt, and two wealth levelsW1,W2 ∈ Lp(Ω,Ft, P )

and define ci(t) =Wi − Vθ(t) for i = 1, 2 we have that

F (θ,W1)− F (θ,W2) = ut(W1 − Vθ(t))− ut(W2 − Vθ(t))

= ut(c1(t))− ut(c2(t)).

In particular, if ut is Fréchet-differentiable at some point c(t) =W − Vθ(t) ∈ Lp(Ω,Ft, P ),

the function F (θ, · ) is Fréchet-differentiable at W and

DWF (θ) = DW−Vθ(t)

ut = Dc(t)ut . (11)

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Corollary 1 Assume that the time t-period utility ut(c(t)) is Fréchet-differentiable at the

optimal consumption c∗(t) and that the time t value function H(W ) is Fréchet-differentiable

at the optimal wealth W ∗(t) = c∗(t)− Vθ∗(t). Then

Dc∗(t)ut = DW∗(t)

H .

Proof. The optimal strategy θ∗ belongs to the set Θ∗(W ∗(t)). From Remark 1, we deduce

that F (θ∗, · ) is Fréchet-differentiable in W ∗(t). The claim then follows from Theorem 1

and equality (11). ¤

Remark 2 The Fréchet differentials of H and u allow to define a notion of marginal

utilities of wealth and consumption respectively. Following the argument in Battauz et

al. (2011), given the Fréchet differential of H, one can define a linear and continuous

functional EH : Lp(Ω,Ft, P )→ < via

EH(Y ) = EhDW∗(t)H (Y )

ifor all Y ∈ Lp(Ω,Ft, P ). By the Riesz representation theorem, there exists a unique HW ∈

Lq(Ω,Ft, P ) such that

EH(Y ) = EhDW∗(t)H (Y )

i= E [HW Y ] for all Y ∈ Lp(Ω,Ft, P ).

We call HW the time t-marginal utility of optimal wealth. Analogously, we can

uniquely find a random variable uc ∈ Lq(Ω,Ft, P ) such that

EhDc∗(t)ut (Y )

i= E [uc Y ] for all Y ∈ Lp(Ω,Ft, P )

which we call time t-marginal utility of optimal consumption.

Then Corollary 1 implies that, if they exist, the time t- marginal utilities of optimal con-

sumption and of optimal wealth coincide because of the uniqueness of the Riesz represen-

tation.

In general, conditions are given on the utility functions us but not on the value func-

tion. It becomes important then to understand which assumptions on the period utilities

guarantee the Fréchet differentiability of the value function. Benveniste and Scheinkman

(1979) give sufficient condition for the case t = 0, when the choice set is convex and the

15

objective function is concave. Milgrom and Segal showed that this result can be seen as a

consequence of their envelope theorem. Our next step is to exploit the results in Section 2

to obtain an envelope condition for general asset pricing models in which the utility func-

tion is allowed to be state-dependent and the information structure is not required to be

markovian. To do so, we first translate the Assumptions introduced in the previous section

in terms of the period utility ut of our general asset pricing model.

Assumption 10 There exists an optimal consumption-portfolio choice for an agent whose

wealth at time t is W ∗(t).

Note in our general asset pricing model this assumption is clearly implied by Condition

1.

We define the set of admissible maximal consumptions at time t which can be obtained

with the wealth W ∗:

C∗(W ∗) = c ∈ Lp(Ω,Ft, P ) : c =W ∗ − Vθ(t) for some θ ∈ Θt.

Assumptions 2 and 3 become respectively a sort of equidifferentiability and uniform bound-

edness on the set of admissible consumptions. In particular, Assumption 20 implies that ut

is Fréchet-differentiable in c∗(t).

Assumption 20 For every X ∈ Lp(Ω,Ft, P ) with a sufficiently small norm:

ut(c+X)− ut(c) = Dut(c)(X) + σt(c,X) · kXkLp

with

esssupc∈C∗(W∗)

|σt(c,X)| ≤ Σ kXkLp ,

for some integrable random variable Σ

Assumption 30 For every X ∈ Lp(Ω,Ft, P ) there exists an integrable random variable

ΛX such that

esssupc∈C∗(W∗)

|Dut(c)(X)| < ΛX kXkLp .

Finally Assumption 4 requires the continuity of the Fréchet differential of u as an admis-

sible consumption approaches the optimal consumption c∗, provided that after perturbing

the optimal wealth the agent is still able to find an optimal consumption-portfolio pair and

that the optimal consumption is continuous as a function of the optimal wealth.

Assumption 40 There exists a neighborhood I∗ of W ∗ such that:

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(i) for each W ∈ I∗ the set Θ∗(W ) is not empty, namely there exists an optimal

consumption-portfolio choice for every level of wealth W ∈ I∗;

(ii) if W ∈ I∗ tends to W ∗ in Lp(Ω,Ft, P ), then the corresponding optimal consumption

at time t, cW (t) converges to c∗(t) in Lp(Ω,Ft, P );

(iii) the Fréchet differential Dut satisfies:

limkXkLp→0

kDut(c∗(t) +X)−Dut(c

∗(t))kL = 0.

Remark 3 It is not difficult to verify that if Assumption 40 holds for ut, then F satisfies

Assumption 4. Indeed, let (cW , θW ) be the optimal consumption-portfolio pair for some

W ∈ I∗: this means in particular that cW (t) + VθW (t) =W . Moreover

DF (θW ,W ∗)−DF (θ∗,W ∗) = Dut(W

∗ − VθW (t)))−Dut(W∗ − Vθ∗(t))

= Dut(c∗(t) + Vθ∗(t)− VθW (t))−Dut(c

∗(t)).

If we denote X = Vθ∗(t) − VθW (t) = (W ∗ −W ) − (cW (t) − c∗(t)), it is evident that, by

Assumption 40 (ii), if W tends toW ∗ in Lp then X tends to 0 in Lp. Therefore Assumption

40 (iii) implies Assumption 4 (ii).

We are now ready to obtain the envelope theorem for state-dependent utilities and for a

general (i.e. not required to be Markov) information structure as a corollary of our general

Theorem 2:

Corollary 2 If the time t period utility ut satisfies Assumptions 20, 30, 40, then the value

function H(t,W (t)) is Fréchet-differentiable in the optimal wealth W ∗(t) and

DH(t,W∗(t)) = Du(t, c

∗(t))

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4 Conclusions

In this paper we have extended the most general class of envelope results, i.e. those due

to Milgrom and Segal (2002), to the case in which the objective function takes values in

a general Banach Lattice, and not necessarily the real line. Employing the concept of

Fréchet-differentiability, our main results consists in identifying a set of assumptions under

which the value function is Fréchet-differentiable, and its Fréchet differential coincides with

the Fréchet differential of the objective function, seen as a function of the parameters.

We then apply our general result to the consumption-portfolio problem of an agent with

time additive but possibly state-dependent utility, in a context in which the information

structure is not required to be Markovian. In this setting, at any time t the value function

(maximum remaining utility) is in fact a random variable itself, and not just a real-valued

function defined on a set of state variables. To investigate if the value function for this

problem has a well-defined marginal utility of wealth, defined as the Fréchet differential of

the value seen as a function of wealth levels accumulated up to time t, we recognize that

the value function takes values in L1, the space of integrable random variables, and that

L1 is indeed a Banach lattice. This allows us to bring to full bearing our general results to

identify a set of conditions under which the marginal utility of wealth is well defined and

coincides with the marginal utility consumption, when the last one exists.

A Banach lattices

Definition 4 Let (X,≥) be an ordered vector space. We call X a Riesz space if it is also

a lattice, namely if each pair of elements x, y ∈ X has a supremum x ∨ y and an infimum

x ∧ y.

We denote as usual

x+ = x ∨ 0, x− = −x ∨ 0, |x| = x+ + x−.

A subset A of a Riesz space is order bounded from above if there is a vector u that dominates

each element of A. The definition of order bounded from below is analogous and it is clear

that A is order bounded from above if and only inf −A is order bounded from below.

Definition 5 A Riesz space is Archimedean if 0 ≤ nx ≤ y for all n ∈ N and some

y ∈ X+ implies x = 0.

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A Riesz space is called order complete if every nonempty susbset that is order bounded

from above has a supremum.

Every order complete Riesz space is Archimedean, but the converse is not true.

Definition 6 A net (xα) converges in order to some x ∈ X (xαo→ x) if there is a net

(yα) such that yα ↓ 0 and |xα − x| ≤ yα for each α.

A net can have at most one order limit.

A Riesz space can be equipped with a norm. A lattice norm k · k has the property that

|x| ≤ |y| implies kxk ≤ kyk.

Definition 7 A complete normed Riesz space is called a Banach lattice

Examples

1. The Euclidean space Rn with the Euclidean norm.

2. The space C(K) of all continuous functions on a compact space K with the sup norm

(it is Archimedean but not order complete).

3. The spaces Lp(μ) (1 ≤ p ≤ ∞) with the usual Lp-norm. Order convergence coincide

with μ-almost-sure convergence.

4. The Orlicz spaces

Definition 8 A lattice norm k · k on a Riesz space is order continuous if xα ↓ 0 implies

kxαk ↓ 0.

All reflexive Banach lattices are order continuous. This property is important because

a Banach lattice with order continuous norm is order complete. For instance, Lp(μ) is

order complete for 1 ≤ p <∞ but not for p =∞.

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Remark 4 In a Banach space, the norm convergence is equivalent to relative uniform

star convergence, namely a sequence xn converges in norm to x if and only if for every

subsequence xnk there exist a subsequencexnk(l) and an element y ∈ X such that |xnk(l)−

x| ≤ y/l for l = 1, 2, . . ..

In an Archimedean vector lattice, relative uniform convergence implies order conver-

gence. As a consequence, in an order continuous Banach lattice, if a sequence xn converges

in norm to x, then there exist a subsequence which is order convergent to x.

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