3Optimality and equilibrium

Consider an investor whose preferences can be expressed in terms of expected utility. In Section 3.1, we discuss the problem of constructing a portfolio which maximizes the expected utility of the resulting payoff. The existence of an optimal solution is equivalent to the absence of arbitrage opportunities. This leads to an alternative proof of the “fundamental theorem of asset pricing”, and to a specific choice of an equivalent martingale measure defined in terms of marginal utility. Section 3.2 contains a detailed case study describing the interplay between exponential utility and relative entropy. In Section 3.3, the optimization problem is formulated for general contingent claims. Typically, optimal profiles will be nonlinear functions of a given market portfolio, and this is one source of the demand for financial derivatives. Section 3.6 introduces the idea of market equilibrium. Prices of risky assets will no longer be given in advance; they will be derived as equilibrium prices in a microeconomic setting, where different agents demand contingent claims in accordance with their preferences and with their budget constraints.

3.1Portfolio optimization and the absence of arbitrage

Let us consider the one-period market model of Section 1.1 in which d + 1 assets are priced at time 0 and at time 1. Prices at time 0 are given by the price system

prices at time 1 are modeled by the price vector

consisting of nonnegative random variables Si defined on some probability space (Ω,F, P). The 0th asset models a riskless bond, and so we assume that

for some constant r > −1. At time t = 0, an investor chooses a portfolio

where ξ i represents the amount of shares of the ith asset. Such a portfolio requires an initial investment · and yields at time 1 the random payoff · S.

Consider a risk-averse economic agent whose preferences are described in terms of a utility function , and who wishes to invest a given amount w into the financial market. Recall from Definition 2.35 that a real-valued function is called a utility function if it is continuous, strictly increasing, and strictly concave. A rational choice of the investor’s portfolio = (ξ0, ξ) will be based on the expected utility

of the payoff · S at time 1, where the portfolio satisfies the budget constraint

Thus, the problem is to maximize the expected utility (3.1) among all portfolios ∈ ℝd+1 which satisfy the budget constraint (3.2). Here we make the implicit assumption that the payoff · S is P-a.s. contained in the domain of definition of the utility function .

In a first step, we remove the constraint (3.2) by considering instead of (3.1) the expected utility of the discounted net gain

earned by a portfolio = (ξ0, ξ). Here Y is the d-dimensional random vector with components

For any portfolio with · < w, adding the risk-free investment w − · would lead to the strictly better portfolio (ξ0 + w − · , ξ). Thus, we can focus on portfolios which satisfy · = w, and then the payoff is an affine function of the discounted net gain:

Moreover, for any ξ ∈ ℝd there exists a unique numéraire component ξ0 ∈ ℝ such that the portfolio := (ξ0, ξ) satisfies · = w.

Let u denote the following transformation of our original utility function :

Note that u is again a utility function, and that CARA and (shifted) HARA utility functions are transformed into utility functions in the same class.

Clearly, the original utility maximization problem with budget constraint (3.2) is equivalent to the problem of maximizing the expected utility E[ u(ξ·Y) ] over all ξ ∈ ℝd such that ξ · Y is contained in a certain domain D.

Assumption 3.1. We assume one of the following two cases:

(a) D = ℝ. In this case, we will admit all portfolios ξ ∈ ℝd, but we assume that u is bounded from above.

(b) D = [a,∞) for some a < 0. In this case, we only consider portfolios which satisfy the constraint

and we assume that the expected utility generated by such portfolios is finite, i.e.,

Remark 3.2. Part (a) of this assumption is clearly satisfied in the case of an exponential utility function u(x) = 1 − e−αx. Domains of the form D = [a,∞) appear, for example, in the case of (shifted) HARA utility functions u(x) = log(x − b) for b < a and for c ≤ a and 0 < γ < 1. The integrability assumption in (b) holds if E[ |Y| ] < ∞, because any concave function is bounded from above by an affine function.

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In order to simplify notations, let us denote by

the set of admissible portfolios for D. Clearly, S(D) = ℝd if D = ℝ. Our aim is to find some ξ∗ ∈ S(D) which is optimal in the sense that it maximizes the expected utility E[ u(ξ · Y) ] over all ξ ∈ S(D). In this case, ξ∗ will be an optimal strategy for investments into the risky assets. Complementing ξ∗ with a suitable numéraire component ξ0 yields a portfolio ∗ = (ξ0, ξ∗) which maximizes the expected utility E[ ( · S) ] under the budget constraint · = w. Our first result in this section will relate the existence of such an optimal portfolio to the absence of arbitrage opportunities.

Theorem 3.3. Suppose that the utility function u : D → ℝ satisfies Assumption 3.1. Then there exists a maximizer of the expected utility

if and only if the market model is arbitrage-free. Moreover, there exists at most one maximizer if the market model is nonredundant in the sense of Definition 1.15.

Proof. The uniqueness part of the assertion follows immediately from the strict concavity of the function for nonredundant market models. As to existence, we may assume without loss of generality that our model is nonredundant. If the nonredundance condition (1.9) does not hold, then we define a linear space N ⊂ ℝd by

Clearly, Y takes P-a.s. values in the orthogonal complement N ⊥ of N. Moreover, the no-arbitrage condition (1.3) holds for all ξ ∈ ℝd if and only if it is satisfied for all ξ ∈ N ⊥. By identifying N ⊥ with some ℝn, we arrive at a situation in which the nonredundance condition (1.9) is satisfied and where we may apply our result for nonredundant market models.

If the model admits arbitrage opportunities, then a maximizer ξ∗ of the expected utility E[ u(ξ · Y) ] cannot exist: Adding to ξ∗ some nonzero η ∈ ℝd for which η · Y ≥ 0 P-a.s., which exists by Lemma 1.4, would yield a contradiction to the optimality of ξ∗, because then

From now on, we assume that the market model is arbitrage-free. Let us first consider the case in which D = [a,∞) for some a ∈ (−∞, 0). Then S(D) is compact. In order to prove this claim, suppose by way of contradiction that (ξn) is a sequence in S(D) such that |ξn| → ∞. By choosing a subsequence if necessary, we may assume that ηn := ξn/|ξn| converges to some unit vector η ∈ ℝd. Clearly,

and so nonredundance implies that η := (−π · η, η) is an arbitrage-opportunity.

In the next step, we show that our assumptions guarantee the continuity of the function

which, in view of the compactness of S(D), will imply the existence of a maximizer of the expected utility. To this end, it suffices to construct an integrable random variable which dominates u(ξ · Y) for all ξ ∈ S(D). Define η ∈ ℝd by

Then, η · S ≥ ξ · S for ξ ∈ S(D), and hence

Note that η · Y is bounded from below by −π · η and that there exists some α ∈ (0, 1] such that απ · η < |a|.Hence αη ∈ S(D), and so E[ u(αη·Y) ] < ∞.Nowwe apply Lemma 3.4 below. Taking first b := απ· η and then b := (minξ∈S(D) π · ξʹ)−, we obtain that

This concludes the proof of the theorem in the case D = [a,∞).

Let us now turn to the case of a utility function on D = ℝ which is bounded from above. We will reduce the assertion to a general existence criterion for minimizers of lower semicontinuous convex functions on ℝd, given in Lemma 3.5 below. It will be applied to the convex function h(ξ) := −E[ u(ξ · Y) ]. Note that h(0) < ∞ by assumption. We must show that h is lower semicontinuous. Take a sequence (ξn)n∈N in ℝd converging to some ξ. By part (a) of Assumption 3.1, the random variables −u(ξn ·Y) are uniformly bounded from below, and so we may apply Fatou’s lemma:

Thus, h is lower semicontinuous.

By our nonredundance assumption, h is strictly convex and admits at most one minimizer. We claim that the absence of arbitrage opportunities is equivalent to the following condition:

This is just the condition (3.4) required in Lemma 3.5. It follows from (1.3) and (1.9) that a nonredundant market model is arbitrage-free if and only if each nonzero ξ ∈ ℝd satisfies P[ξ · Y < 0] > 0. Since the utility function u is strictly increasing and concave, the set {ξ · Y < 0} can be described as

The probability of the right-hand set is strictly positive if and only if

because u is bounded from above. This observation proves that the absence of arbitrage opportunities is equivalent to the condition (3.3) and completes the proof.

Lemma 3.4. If D = [a,∞), b < |a|, 0 < α ≤ 1, and X is a nonnegative random variable, then

Proof. As in (A.5) in the proof of Proposition A.7 we obtain that, on {X > 0},

Multiplying by X shows that u(X) can be dominated by a multiple of u(αX − b) plus some constant.

Lemma 3.5. Suppose h : ℝd → ℝ ∪ {+∞} is a convex and lower semicontinuous function with h(0) < ∞. Then h attains its infimum provided that

Moreover, if h is strictly convex on {h < ∞}, then also the converse implication holds: the existence of a minimizer implies (3.4).

Proof. First suppose that (3.4) holds. We will show below that for c > inf h the level sets {x | h(x) ≤ c} of h are bounded and hence compact, since they are closed due to lower semicontinuity. Once the compactness of the level sets is established, it follows that the set

of minimizers of h is nonempty as an intersection of decreasing and nonempty compact sets.

Suppose c > inf h is such that the level set {h ≤ c} is not compact, and take a sequence (xn) in {h ≤ c} such that |xn|→ ∞. By passing to a subsequence if necessary, we may assume that xn/|xn| converges to some nonzero ξ. For any α > 0,

Thus, we arrive at a contradiction to condition (3.4). This completes the proof of the existence of a minimizer under assumption (3.4).

In order to prove the converse implication, suppose that the strictly convex function h has a minimizer x∗ but that there exists a nonzero ξ ∈ ℝd violating (3.4), i.e., there exists a sequence (αn)n∈N and some c < ∞ such that αn ↑ ∞ but h(αnξ) ≤ c for all n. Let

where λn is such that |x∗ − xn| = 1, which is possible for all large enough n. By the compactness of the Euclidean unit sphere centered in x∗, we may assume that xn converges to some x. Then necessarily |x − x∗| = 1. As αnξ diverges, we must have that λn → 1. By using our assumption that h(αnξ) is bounded, we obtain

Hence, x is another minimizer of h besides x∗, contradicting the strict convexity of h. Thus, (3.4) must hold if the strictly convex function h takes on its infimum.

Remark 3.6. Note that the proof of Theorem 3.3 under Assumption 3.1 (a) did not use the fact that the components of Y are bounded from below. The result remains true for arbitrary Y.

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We turn now to a characterization of the solution ξ∗ of our utility maximization problem for continuously differentiable utility functions.

Proposition 3.7. Let u be a continuously differentiable utility function on D such that E[ u(ξ · Y) ] is finite for all ξ ∈ S(D). Suppose that ξ∗ is a solution of the utility maximization problem, and that one of the following two sets of conditions is satisfied:

–u is defined on D = ℝ and is bounded from above.

–u is defined on D = [a,∞), and ξ∗ is an interior point of S(D).

Then

and the following “first-order condition” holds:

Proof. For ξ ∈ S(D) and ε ∈ (0, 1] let ξε := εξ + (1 − ε)ξ∗, and define

The concavity of u implies that Δε ≥ Δδ for ε ≤ δ, and so

Note that our assumptions imply that u(ξ · Y) ∈ L1(P) for all ξ ∈ S(D). In particular, we have Δ1 ∈ L1(P), so that monotone convergence and the optimality of ξ∗ yield that

In particular, the expectation on the right-hand side of (3.6) is finite.

Both sets of assumptions imply that ξ∗ is an interior point of S(D). Hence, for η ∈ ℝd with |η| sufficiently small, we still have ξ := η + ξ∗ ∈ D. Then we deduce from (3.6) that

Replacing η by −η shows that the expectation must vanish.

Remark 3.8. Let us comment on the assumption that the optimal ξ∗ is an interior point of S(D):

(a) If the nonredundance condition (1.9) is not satisfied, then either each or none of the solutions to the utility maximization problem is contained in the interior of S(D). This can be seen by using the reduction argument given at the beginning of the proof of Theorem 3.3.

(b) Note that ξ · Y is bounded from below by −π · ξ in case ξ has only nonnegative components. Thus, the interior of S(D) is always nonempty.

(c) As shown by the following example, the optimal ξ∗ need not be contained in the interior of S(D) and, in this case, the first-order condition (3.5) will generally fail.

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Example 3.9. Take r = 0, d = 1, and let S1 be integrable but unbounded. We choose D = [a,∞) with a := −π1, and we assume that P[ S1 ≤ ε ] > 0 for all ε > 0. Then S(D) = [0, 1]. If 0 < E[ S1 ] < π1 then Example 2.40 shows that the optimal investment is given by ξ∗ = 0, and so ξ∗ lies at the boundary of S(D). Thus, if u is sufficiently smooth,

The intuitive reason for this failure of the first-order condition is that taking a short position in the asset would be optimal as soon as E [ S1 ] < π1. This choice, however, is ruled out by the constraint ξ ∈ S(D).

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Proposition 3.7 yields a formula for the density of a particular equivalent risk-neutral measure. Recall that P∗ is risk-neutral if and only if E∗[ Y ] = 0.

Corollary 3.10. Suppose that the market model is arbitrage-free and that the assumptions of Proposition 3.7 are satisfied for a utility function u : D → ℝ and an associated maximizer ξ∗ of the expected utility E[u(ξ · Y)]. Then

defines an equivalent risk-neutral measure.

Proof. Proposition 3.7 states that uʹ(ξ∗ ·Y)Y is integrable with respect to P and that its expectation vanishes. Hence, we may conclude that P∗ is an equivalent risk-neutral measure if we can show that P∗ is well-defined by (3.7), i.e., if uʹ(ξ∗ · Y) ∈ L1(P). Let

which is finite by our assumption that u is continuously differentiable on all of D. Thus,

and the right-hand side has a finite expectation.

Remark 3.11. Corollary 3.10 yields an independent and constructive proof of the “fundamental theorem of asset pricing” in the form of Theorem 1.7: Suppose that the model is arbitrage-free. If Y is P-a.s. bounded, then so is u(ξ∗ · Y), and the measure P∗ of (3.7) is an equivalent risk-neutral measure with a bounded density dP∗/dP. If Y is unbounded, then we may consider the bounded random vector

which also satisfies the no-arbitrage condition (1.3). Let ∗ be a maximizer of the expected utility E[ u(ξ · ) ]. Then an equivalent risk-neutral measure P∗ is defined through the bounded density

where c is an appropriate normalizing constant.

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Example 3.12. Consider the exponential utility function

with constant absolute risk aversion α > 0. The requirement that E[ u(ξ · Y) ] is finite is equivalent to the condition

If ξ∗ is a maximizer of the expected utility, then the density of the equivalent risk neutral measure P∗ in (3.7) takes the particular form

In fact, P∗ is independent of α since ξ∗ maximizes the expected utility 1 − E [ e−αξ·Y ] if and only if λ∗ := −αξ∗ is a minimizer of the moment generating function

of Y. In Corollary 3.25 below, the measure P∗ will be characterized by the fact that it minimizes the relative entropy with respect to P among the risk-neutral measures in P; see Definition 3.20 below.

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3.2Exponential utility and relative entropy

In this section we give a more detailed study of the problem of portfolio optimization with respect to a CARA utility function

for α > 0. As in the previous Section 3.1, the problem is to maximize the expected utility

of the discounted net gain ξ · Y earned by an investment into risky assets. The key assumption for this problem is that

Recall from Example 3.12 that the maximization of E[ u(ξ · Y) ] is reduced to the minimization of the moment generating function

which does not depend on the risk aversion α. The key assumption (3.8) is equivalent to the condition that

Throughout this section, we will always assume that (3.9) holds. But we will not need the assumption that Y is bounded from below (which in our financial market model follows from assuming that asset prices are nonnegative); all results remain true for general random vectors Y; see also Remarks 1.9 and 3.6.

Lemma 3.13. The condition (3.9) is equivalent to

In order to show that the ith factor on the right is finite, take λ ∈ ℝd such that λi = αcd and λj = 0 for j ≠ i. With this choice,

which is finite by (3.9).

Definition 3.14. The exponential family of P with respect to Y is the set of measures

defined via

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Example 3.15. Suppose that the risky asset S1 has under P a Poisson distribution with parameter α > 0, i.e., S1 takes values in {0, 1, . . . } and satisfies

Then (3.9) is satisfied for Y := S1 − π1, and S1 has under P λ a Poisson distribution with parameter eλα. Hence, the exponential family of P generates the family of all Poisson distributions.

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Example 3.16. Let Y have a standard normal distribution N (0, 1). Then (3.9) is satisfied, and the distribution of Y under Pλ is equal to the normal distribution N (λ, 1) with mean λ and variance 1.

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Remark 3.17. Two parameters λ and λʹ in ℝd determine the same element in the exponential family of P if and only if (λ − λʹ) · Y = 0 P-almost surely. It follows that the mapping

is injective provided that the nonredundance condition holds in the form

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In the sequel, we will be interested in the barycenters of the members of the exponential family of P with respect to Y. These barycenters will be denoted by

The next lemma shows that m(λ) can be obtained as the gradient of the logarithmic moment generating function.

Lemma 3.18. Z is a smooth function on ℝd, and the gradient of log Z at λ is the expectation of Y under Pλ:

Moreover, the Hessian of log Z at λ equals the covariance matrix (covPλ (Yi , Yj))i,j of Y under the measure Pλ:

In particular, log Z is convex.

Proof. Observe that

Hence, Lemma 3.13 and Lebesgue’s dominated convergence theorem justify the interchanging of differentiation and integration (see the “differentiation lemma” in [21], §16, for details). The result thus follows from a short computation.

The following corollary summarizes the results we have obtained so far. Recall from Section 1.5 the notion of the convex hull Γ(ν) of the support of a measure ν on ℝd and the definition of the relative interior ri C of a convex set C.

Corollary 3.19. Denote by μ := P ◦ Y−1 the distribution of Y under P. Then the function

takes on its maximum if and only if m0 is contained in the relative interior of the convex hull of the support of μ, i.e., if and only if

In this case, any maximizer λ∗ satisfies

In particular, the set {m(λ) | λ ∈ ℝd} coincides with ri Γ(μ). Moreover, if the nonredun-dance condition (3.10) holds, then there exists at most one maximizer λ∗.

Proof. Taking := Y − m0 reduces the problem to the situation where m0 = 0. Applying Theorem 3.3 with the utility function u(z) = 1 −e−z shows that the existence of a maximizer λ∗ of − logZ is equivalent to the absence of arbitrage opportunities. Corollary 3.10 states that m(λ∗) = 0 and that 0 belongs to M(μ), where M(μ) was defined in Lemma 1.44. An application of Theorem 1.49 completes the proof.

It will turn out that the maximization problem of the previous corollary is closely related to the following concept.

Definition 3.20. The relative entropy of a probability measure Q with respect to P is defined as

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Remark 3.21. Jensen’s inequality applied to the strictly convex function h(x) = x log x yields

with equality if and only if Q = P.

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Example 3.22. Let Ω be a finite set and F be its power set. Every probability Q on (Ω,F) is absolutely continuous with respect to the uniform distribution P. Let us denote Q(ω) := Q[ {ω} ]. Then,

The quantity

is usually called the entropy of Q. Observe that H (P) = log |Ω|, so that

Since the left-hand side is nonnegative by (3.11), the uniform distribution P has maximal entropy among all probability distributions on (Ω,F).

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Example 3.23. Let μ = N (m, σ2) denote the normal distribution with mean m and variance σ2 on ℝ. Then, for = N( , 2)

and hence

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The following result shows that Pλ is the unique minimizer of the relative entropy H (Q|P) among all probability measures Q with EQ[ Y ] = E λ[ Y ].

Proposition 3.24. Let m0 ∈ ℝd be given. Then, for any probability measure Q on (Ω,F) such that EQ[ Y ] = m0,

If, moreover, λ0 ∈ ℝd is such that m(λ0) = m0, then λ0 realizes the supremum on the right-hand side of (3.12) and

In particular, Pλ0 is the unique minimizer of the relative entropy H (Q|P) within the class of all probability measures Q on (Ω,F) such that EQ[ Y ] = m0.

Proof. Let Q be a probability measure on (Ω,F) such that EQ[ Y ] = m0.We show first that for all λ ∈ ℝd

To this end, note that both sides of (3.13) are infinite if Q P. Otherwise

and taking logarithms and integrating with respect to Q yields (3.13).

Since H (Q|Pλ)≥ 0 according to (3.11), we get from (3.13) that

for all λ ∈ ℝd and all measures Q such that EQ[ Y ] = m0. Moreover, equality holds in (3.14) if and only if H (Q|Pλ) = 0, which is equivalent to Q = Pλ. In this case, λ must be such that m(λ) = m0. In particular, for any such λ,

Thus, if λ0 is such that m(λ0) = m0, then λ0 maximizes the right-hand side of (3.14), and Pλ0 minimizes the relative entropy on the set

But the relative entropy H (Q|P) is a strictly convex functional of Q, and so it can have at most one minimizer in the convex set M0. Thus, any λ with m(λ) = m0 must induce the same measure Pλ0 .

Taking m0 = 0 in the preceding theorem yields a special equivalent risk-neutral measure in our financial market model, namely the entropy-minimizing risk neutral measure. Sometimes it is also called the Esscher transform of P. Recall our assumption (3.9).

Corollary 3.25. Suppose the market model is arbitrage-free. Then there exists a unique equivalent risk-neutral measure P∗ ∈ P which minimizes the relative entropy H(|P) over all ∈ P. The density of P∗ is of the form

where λ∗ denotes a minimizer of the moment generating function E[ eλ·Y ] of Y.

Proof. This follows immediately from Corollary 3.19 and Proposition 3.24.

By combining Proposition 3.24 with Remark 3.17, we obtain the following corollary. It clarifies the question of uniqueness in the representation of points in the relative interior of Γ(P ◦ Y−1) as barycenters of the exponential family.

Corollary 3.26. If the nonredundance condition (3.10) holds, then

is a bijective mapping from ℝd to ri Γ(P ◦ Y−1).

Remark 3.27. It follows from Corollary 3.19 and Proposition 3.24 that for all m ∈ ri Γ(P ◦ Y−1)

Here, the right-hand side is the Fenchel–Legendre transform of the convex function log Z evaluated at m ∈ ℝd.

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The following theorem shows that the variational principle (3.15) remains true for all m ∈ ℝd, if we replace “min” and “max” by “inf” and “sup”.

Theorem 3.28. For m ∈ ℝd,

The proof of this theorem relies on the following two general lemmas.

Lemma 3.29. For any probability measure Q,

The second supremum is attained by

Proof. We first show ≥ in (3.16). To this end, we may assume that H (Q|P) < ∞. For Z with eZ ∈ L1(P) let PZ be defined by

Then PZ is equivalent to P and

Integrating with respect to Q gives

Since H (Q|PZ) ≥ 0 by (3.11), we have proved that H (Q|P) is larger than or equal to both suprema on the right of (3.16).

To prove the reverse inequality, consider first the case Q P. Take Zn := nA where A is such that Q[ A ] > 0 and P[ A ] = 0. Then, as n ↑ ∞,

Now suppose that Q P with density φ = dQ/dP. Then Z := log φ satisfies eZ ∈ L1(P) and

For the first identity we use an approximation argument. Let Zn = (−n) ∨ (log φ) ∧ n∈ L∞(P), where a∨b := max{a, b} and a∧b := min{a, b}. We split the expectation E[ eZn ] according to the two sets {φ ≥ 1} and {φ < 1}. Using monotone convergence for the first integral and dominated convergence for the second yields

Since x log x ≥ −1/e, we have φZn ≥ −1/e uniformly in n, and Fatou’s lemma yields

Putting both facts together shows

and the inequality ≤ in (3.16) follows.

Remark 3.30. The preceding lemma shows that the relative entropy is monotone with respect to an increase of the underlying σ-algebra: Let P and Q be two probability measures on a measurable space (Ω,F), and denote by H (Q|P) their relative entropy. Suppose that F0 is a σ-field such that F0 ⊂ F and denote by H0(Q|P) the relative entropy of Q with respect to P considered as probability measures on the smaller space (Ω,F0). Then the relation L∞(Ω,F0, P) ⊂ L∞(Ω,F, P) implies

in general this inequality is strict.

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Lemma 3.31. For all α ≥ 0, the set

is weakly sequentially compact in L1(Ω,F, P).

Proof. Let Lp := Lp(Ω,F, P). The set of all P-densities,

is clearly convex and closed in L1. Hence, this set is also weakly closed in L1 by Theorem A.63. Moreover, Lemma 3.29 states that for φ ∈ D

In particular,

is a weakly lower semicontinuous functional on D, and so Φα is weakly closed. In addition, Φα is bounded in L1 and uniformly integrable, due to the criterion of de la Vallée Poussin; see, e.g., Lemma 3 in §6 of Chapter II of [262]. Applying the Dunford–Pettis theorem and the Eberlein–Šmulian theorem as stated in Appendix A.7 concludes the proof.

Proof of Theorem 3.28. In view of Proposition 3.24, it remains to prove that

for those m which do not belong to ri Γ(μ), where μ := P ◦ Y−1. Since the right-hand side of (3.17) is just the Fenchel– Legendre transform at m of the convex function log Z, we denote it by (log Z)∗(m).

First, we consider the case in which m is not contained in the closure Γ(μ) of the convex hull of the support of μ. The separating hyperplane theorem in the form of Proposition A.5 yields some ξ ∈ ℝd such that

By taking λn := nξ, it follows that

Hence, the right-hand side of (3.17) is infinite if m ∈ Γ(μ).

It remains to prove (3.17) for m ∈ Γ(μ) ri Γ(μ) with (log Z)∗(m) < ∞. Recall from (1.27) that ri Γ(μ) = ri Γ(μ). Pick some m1 ∈ ri Γ(μ) and let

Then mn ∈ ri Γ(μ) by (1.26). By the convexity of (log Z)∗, we have

We also know that to each mn there corresponds a λn ∈ ℝd such that

From (3.18) and (3.19) we conclude that

In particular, H (Pλn |P) is uniformly bounded in n, and Lemma 3.31 implies that – after passing to a suitable subsequence if necessary – the densities dPλn /dP converge weakly in L1(Ω,F, P) to a density φ. Let dP∞ = φ dP. By the weak lower semicontinuity of

which follows from Lemma 3.29, we conclude that H (P∞|P) ≤ (log Z)∗(m).

The theorem will be proved once we can show that E∞[ Y ] = m. To this end, let γ := supn(log Z)∗(mn), which is a finite nonnegative number by (3.18). Note that

is such that eZ ∈ L1(P), due to condition (3.9) and Lemma 3.13. Taking this Z on the right-hand side of (3.16) yields

By taking α large so that γ/α < ε/2 for some given ε > 0, and by choosing c such that

we obtain that

But

since dPλn /dP converges to dP∞/dP weakly in L1(P). Taking ε ↓ 0 yields

as desired.