2.4Stochastic dominance

So far, we have considered preference relations on distributions defined in terms of a fixed utility function u. In this section, we focus on the question whether one distribution is preferred over another, regardless of the choice of a particular utility function.

For simplicity, we take S = ℝ as the set of possible payoffs. Let M be the set of all μ ∈ M1(ℝ) with well-defined and finite expectation

Recall from Definition 2.35 that a utility function on ℝ is a strictly concave and strictly increasing function u : ℝ → ℝ. Since each concave function u is dominated by an affine function, the existence of m(μ) implies the existence of the integral as an extended real number in [−∞,∞).

Definition 2.55. Let ν and μ be lotteries in M. We say that the lottery μ is uniformly preferred over ν and we write

if

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Thus, μ ≽uni ν holds if and only if every risk-averse agent will prefer μ over ν, regardless of which utility function the agent is actually using. In this sense, μ ≽uni ν expresses a uniform preference for μ over ν. Sometimes, ≽uni is also called second order stochastic dominance; the notion of first order stochastic dominance will be introduced in Definition 2.67.

Remark 2.56. The binary relation ≽uni is a partial order on M, i.e., ≽uni satisfies the following three properties:

–Reflexivity: μ ≽uni μ for all μ ∈ M.

–Transitivity: μ ≽uni ν and ν ≽uni λ imply μ ≽uni λ.

–Antisymmetry: μ ≽uni ν and ν ≽uni μ imply μ = ν.

The first two properties are obvious, the third is derived in Remark 2.58. Moreover, ≽uni is monotone and risk-averse in the sense that

Note, however, that ≽uni is not a weak preference relation in the sense of Definition 2.2, since it is not complete in the sense of Remark 2.3 (a).

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In the following theorem, we will give a number of equivalent formulations of the statement μ ≽uni ν. One of them needs the notion of a stochastic kernel on ℝ. This is a mapping

such that is measurable for each fixed Borel set A ⊂ ℝ. See Appendix A.3 for the notion of a quantile function, which will be used in condition (e).

Theorem 2.57. For any pair μ, ν ∈ M the following conditions are equivalent.

(a) μ ≽uni ν.

(b) dν for all increasing concave functions f .

(c) For all c ∈ ℝ

(d) If F μ and F ν denote the distribution functions of μ and ν, then

(e) If qμ and qν are quantile functions for μ and ν, then

(f) There exists a probability space (Ω,F, P) with random variables X μ and Xν having respective distributions μ and ν such that

(g) There exists a stochastic kernel Q(x, dy) on ℝ such that Q(x, ·) ∈ M and m(Q(x, ·)) ≤ x for all x and such that ν = μQ, where μQ denotes the measure

Below we will show the following implications between the conditions of the theorem:

The difficult part is the proof that (b) implies (f). It will be deferred to Section 2.6, where we will prove a multidimensional variant of this result; cf. Theorem 2.94.

Proof of (2.23). (e)⇔(d): This follows from Lemma A.26 and Proposition A.9 (b).

(d)⇔(c): By Fubini’s theorem,

(c)⇔(b): Condition (b) implies (c) because f (x) := −(c − x)+ is concave and increasing. In order to prove the converse assertion, we take an increasing concave function f and let h := −f . Then h is convex and decreasing, and its increasing right-hand derivative can be regarded as a “distribution function” of a nonnegative Radon measure γ on ℝ,

see Appendix A.1. By Exercise 1.3.3 (a),

Using hʹ(b) ≤ 0, Fubini’s theorem, and condition (c), we obtain that

Taking b ↑ ∞yields ∫ f dμ ≥ ∫ f dν. Indeed, the convex decreasing function h decays at most linearly, and the existence of first moments for μ and ν implies that bμ((b, ∞)) → 0 and bν((b,∞)) → 0 for b ↑ ∞.

(a)⇔(b): That (b) implies (a) is obvious. For the proof of the converse implication, choose any utility function u0 for which both and are finite. For instance, one can take

Then, for f concave and increasing and for α ∈ [0, 1),

is a utility function. Hence,

(f)⇒(g): By considering the joint distribution of Xμ and Xν, we may reduce our setting to the situation in which Ω = ℝ2 and where Xμ and Xν are the respective projections on the first and second coordinates, i.e., for ω = (x, y) ∈ Ω = ℝ2 we have Xμ(ω) = x and Xν(ω) = y. Let Q(x, dy) be a regular conditional distribution of Xν given Xμ, i.e., a stochastic kernel on ℝ such that

for all Borel sets A ⊆ ℝ and for P-a.e. ω ∈ Ω (see, e.g., Theorem 44.3 of [20] for an existence proof). Clearly, ν = μQ. Condition (f) implies that

Hence, Q satisfies

By modifying Q on a μ-null set (e.g., by putting Q(x, ·) := δx there), this inequality can be achieved for all x ∈ ℝ.

(g)⇒(a): Let u be a utility function. Jensen’s inequality applied to the measure Q(x, dy) implies

Hence,

completing the proof of the set of implications (2.23).

Remark 2.58. Let us note some consequences of the preceding theorem. First, taking in condition (b) the increasing concave function f (x) = x yields

i.e., the expectation m(·) is increasing with respect to ≽uni.

Next, suppose that μ and ν are such that

Then we have both μ ≽uni ν and ν ≽uni μ, and condition (d) of the theorem implies that the respective distribution functions satisfy

Differentiating with respect to c gives the identity F μ = F ν and in turn μ = ν. It follows that a measure μ ∈ M is uniquely determined by the integrals for all c ∈ ℝ and that ≽uni is antisymmetric.

◊The following proposition characterizes the partial order ≽uni considered on the set of all normal distributions N (m, σ2). Recall that the standard normal distribution N (0, 1) is defined by its density function

The corresponding distribution function is usually denoted

More generally, the normal distribution N(m, σ2) with mean m ∈ ℝ and variance σ2 > 0 is given by the density function

Proposition 2.59. For two normal distributions, we have N(m, σ2) ≽uni N( , 2) if and only if both m ≥ and σ2 ≤ 2 hold.

Proof. In order to prove necessity, note that N (m, σ2) ≽uni N ( , 2) implies that

Hence, for α > 0,

which gives m ≥ by letting α ↓ 0 and σ2 ≤ 2 for α ↑ ∞.

We show sufficiency first in the case m = = 0. Note that the distribution function of N(0, σ2) is given by Φ(x/σ). Since φʹ(x) = −xφ(x),

Note that interchanging differentiation and integration is justified by dominated convergence. Thus, we have shown that Φ(x/σ) dx is strictly increasing for all c, and N (0, σ2) ≽uni N (0, 2) follows from part (d) of Theorem 2.57.

Now we turn to the case of arbitrary expectations m and . Let u be a utility function. Then

because m ≥ . Since is again a utility function, we obtain from the preceding step of the proof that

and N(m, σ2) ≽uni N( , 2) follows.

Remark 2.60. Let us indicate an alternative proof for the sufficiency part of Proposition 2.59 that uses condition (g) instead of (d) in Theorem 2.57. To this end, we define a stochastic kernel by Q(x, ·) := N(x + − m, ô2), where ô2 := 2 − σ2 > 0. Then m(Q(x, ·)) = x + − m ≤ x and

where * denotes convolution. Hence, N (m, σ2) ≽uni N ( , 2) follows.

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The following corollary investigates the relation μ ≽uni ν for lotteries with the same expectation. A multidimensional version of this result will be given in Corollary 2.95 below.

Corollary 2.61. For all μ, ν ∈ M the following conditions are equivalent.

(a) μ ≽uni ν and m(μ) = m(ν).

(b) for all (not necessarily increasing) concave functions f .

(c) m(μ) ≥ m(ν) and

(d) There exists a probability space (Ω,F, P) with random variables X μ and Xν having respective distributions μ and ν such that

(e) There exists a “mean-preserving spread” Q, i.e., a stochastic kernel on ℝ such that m(Q(x, ·)) = x for all x ∈ S, such that ν = μQ.

Proof. (a)⇒(e): Condition (g) of Theorem 2.57 yields a stochastic kernel Q such that ν = μQ and mQ(x, ·)≤ x. Due to the assumption m(μ) = m(ν), Q must satisfy mQ(x, ·)= x at least for μ-a.e. x. By modifying Q on the μ-null set where mQ(x, ·)< x (e.g. by putting Q(x, ·) := δx there), we obtain a kernel as needed for condition (e).

(e)⇒(b): Since

by Jensen’s inequality, we obtain

(b)⇒(c): Just take the concave functions f (x) = −(x − c)+, and f (x) = x.

(c)⇒(a): Note that

The finiteness of m(μ) implies that c μ(−∞, c] → 0 as c ↓ −∞. Hence, we deduce from the second condition in (c) that m(μ) ≤ m(ν), i.e., the two expectations are in fact identical. Now we can apply the following “put-call parity” (compare also (1.11))

to see that our condition (c) implies the third condition of Theorem 2.57 and, thus, μ ≽uni ν.

(d)⇔(a): Condition (d) implies both m(μ) = m(ν) and condition (f) of Theorem 2.57, and this implies our condition (a). Conversely, assume that (a) holds. Then Theorem 2.57 provides random variables Xμ and Xν having the respective distributions μ and ν such that E[ Xν | Xμ ] ≤ Xμ. Since Xμ and Xν have the same mean, this inequality must in fact be an almost-sure equality, and we obtain condition (d).

Let us denote by

the variance of a lottery μ ∈ M.

Exercise 2.4.1. Let μ and ν be two lotteries in M such that m(μ) = m(ν) and μ ≽uni ν. Show that var(μ) ≤ var(ν).

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In the financial context, comparisons of portfolios with known payoff distributions often use a mean-variance approach based on the relation

For normal distributions μ and ν, we have seen that the relation μ≽mvν is equivalent to μ ≽uni ν. Beyond this special case, the equivalence typically fails as illustrated by the following example and by Proposition 2.65 below.

Example 2.62. Let μ be the uniform distribution on the interval [−1, 1], so that m(μ) = 0 and var(μ) = 1/3. For ν we take ν = pδ−1/2 + (1 − p)δ2. With the choice of p = 4/5 we obtain m(ν) = 0 and 1 = var(ν) > var(μ). However,

so μ ≽uni ν does not hold.

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Remark 2.63. Let μ and ν be two lotteries in M. We will write μ ≽con ν if

Note that μ ≽con ν implies that m(μ) = m(ν), because both f (x) = x and (x) = −x are concave. Corollary 2.61 shows that ≽con coincides with our uniform partial order ≽uni if we compare two measures which have the same mean. The partial order ≽con is sometimes called concave stochastic order. It was proposed in [237] and [238] to express the view that μ is less risky than ν. The inverse relation μ ≽bal ν defined by

is sometimes called balayage order or convex stochastic order.

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The following class of asset distributions is widely used in Finance.

Definition 2.64. A real-valued random variable Y on some probability space (Ω,F,P) is called log-normally distributed with parameters α ∈ ℝ and σ ≥ 0 if it can be written as

where X has a standard normal law N (0, 1).

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Clearly, any log-normally distributed random variable Y on (Ω,F, P) takes P-a.s. strictly positive values. Recall from above the standard notations φ and Φ for the density and the distribution function of the standard normal law N (0, 1). We obtain from (2.26) the distribution function

and the density

of the log-normally distributed random variable Y. Its pth moment is given by the formula

In particular, the law μ of Y has the expectation

and the variance

Proposition 2.65. Let μ and be two log-normal distributions with parameters (α, σ) and (, ), respectively. Then μ ≽uni holds if and only if σ2 ≤ 2 and

Proof. First suppose that σ2 ≤ 2 and m(μ) ≥ m(). We define a kernel Q(x, ·) as the law of x · exp(λ + βZ) where Z is a standard normal random variable. Now suppose that μ is represented by (2.26) with X independent of Z, and let f denote a bounded measurable function. It follows that

where

is also N(0, 1)-distributed. Thus, μQ is a log-normal distribution with parameters By taking and λ := − α, we can represent as = μQ. With this parameter choice,

We have thus m(Q(x, ·)) ≤ x for all x, and so μ ≽uni follows from condition (g) of Theorem 2.57.

As to the converse implication, the inequality m(μ) ≥ m() is already clear. To prove σ2 ≤ 2, let ν := μ◦log−1 and := ◦log−1 so that ν = N(α, σ2) and = N(, 2). For ε > 0we define the concave increasing function fε(x) := log(ε+x). If u is a concave increasing function on ℝ, the function u ◦ fε is a concave and increasing function on [0,∞). It can be extended linearly to obtain a concave increasing function vε on the full real line. Therefore,

Consequently, ν ≽uni and Proposition 2.59 yields σ2 ≤ 2.

Remark 2.66. The inequality (2.28) shows that if ν = N(α, σ2), = N(, 2) and μ and denote the images of ν and under the map then μ ≽uni implies ν ≽uni . However, the converse implication “ ν ≽uni ⇒ μ ≽uni ” fails, as can be seen by increasing until m() > m(μ).

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Because of its relation to the analysis of the Black–Scholes formula for option prices, we will now sketch a second proof of Proposition 2.65.

Second proof of Proposition 2.65. Let

for a standard normally distributed random variable X. Then

see Example 5.58 in Chapter 5. Calculating the derivative of this expectation with respect to σ > 0, one finds that

see (5.49) in Chapter 5. The law μm,σ of Ym,σ satisfies m(μm,σ) = m for all σ > 0. Condition (c) of Corollary 2.61 implies that μm,σ is decreasing in σ > 0 with respect to ≽uni and hence also with respect to ≽con, i.e., μm,σ ≽con μm,σ if and only if σ ≤ .

For two different expectations m and , simply use the monotonicity of the function u(y) := (y − c)+ to conclude

provided that m ≥ and 0 < σ ≤ .

The partial order ≽uni was defined in terms of integrals against increasing concave functions. By taking the larger class of all concave functions as integrands, we arrived at the partial order ≽con defined by (2.24) and characterized in Corollary 2.61. In the remainder of this section, we will briefly discuss the partial order of stochastic dominance, which is induced by increasing instead of concave functions:

Definition 2.67. Let μ and ν be two arbitrary probability measures on ℝ. We say that μ stochastically dominates ν and we write μ ≽mon ν if

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Stochastic dominance is sometimes also called first order stochastic dominance. It is indeed a partial order on M1(ℝ): Reflexivity and transitivity are obvious, and antisymmetry follows, e.g., from the equivalence (a)⇔(b) below. As will be shown by the following theorem, the relation μ ≽mon ν means that the distribution μ is “higher” than the distribution ν. In our one-dimensional situation, we can provide a complete proof of this fact by using elementary properties of distribution functions. The general version of this result, given in Theorem 2.96, will require different techniques.

Theorem 2.68. For μ, ν ∈ M1(ℝ) the following conditions are equivalent.

(a) μ ≽mon ν.

(b) The distribution functions of μ and ν satisfy F μ(x) ≤ F ν(x) for all x.

(c) Any pair of quantile functions for μ and ν satisfies qμ(t) ≥ qν(t) for a.e. t ∈ (0, 1).

(d) There exists a probability space (Ω,F, P) with random variables Xμ and Xν with distributions μ and ν such that Xμ ≥ Xν P-a.s.

(e) There exists a stochastic kernel Q(x, dy) on ℝ such that Q(x, (−∞, x]) = 1 and such that ν = μQ.

(f) for all increasing, bounded, and measurable functions f .

In particular, μ ≽mon ν implies μ ≽uni ν.

Proof. (a)⇒(b): Note that F μ(x) = μ(−∞, x] can be written as

For each x we can construct a sequence of increasing continuous functions with values in [0, 1] that increase to (x,∞) . Hence,

(b)⇔(c): This follows from the definition of a quantile function and from Lemma A.21.

(c)⇒(d): Let (Ω,F, P) be a probability space supporting a random variable U with a uniform distribution on (0, 1). Then Xμ := qμ(U) and Xν := qν(U) satisfy Xμ ≥ Xν P-almost surely. Moreover, it follows from Lemma A.23 that they have the distributions μ and ν.

(d)⇒(e): This is proved as in Theorem 2.57 by using regular conditional distributions.

(e)⇒(f): Condition (e) implies that x ≥ y for Q(x, ·)-a.e. y. Hence, if f is bounded, measurable, and increasing, then

Therefore,

(f)⇒(a): Obvious.

Finally, due to the equivalence (a)⇔(b) above and the equivalence (a)⇔(d) in Theorem 2.57, μ ≽mon ν implies μ ≽uni ν.

Remark 2.69. It is clear from conditions (d) or (e) of Theorem 2.68 that the set of bounded, increasing, and continuous functions in Definition 2.67 can be replaced by the set of all increasing functions for which the two integrals make sense. Thus, μ ≽mon ν for μ, ν ∈ M implies μ ≽uni ν, and in particular m(μ) ≥ m(ν). Moreover, condition (d) shows that μ ≽mon ν together with m(μ) = m(ν) implies μ = ν.

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