1.5Geometric characterization of arbitrage-free models

The “fundamental theorem of asset pricing” in the form of Theorem 1.7 states that a market model is arbitrage-free if and only if the origin is contained in the set

where Y = (Y1, . . . , Yd) is the random vector of discounted net gains defined in (1.2). The aim of this section is to give a geometric description of the set M b(Y, P) as well as of the larger set

To this end, it will be convenient to work with the distribution

of Y with respect to P. That is, μ is a Borel probability measure on ℝd such that

If ν is a Borel probability measure on ℝd such that ∫ |y| ν(dy) < ∞, we will call ∫ y ν(dy) its barycenter.

Lemma 1.44. We have

and

Proof. If ν ≈ μ is a Borel probability measure on ℝd, then the Radon–Nikodym derivative of ν with respect to μ evaluated at the random variable Y defines a probability measure Q ≈ P on (Ω,F):

Clearly, EQ[ Y ] = ∫ y ν(dy). This shows that M(μ) ⊆ M(Y, P) and Mb(μ) ⊆ Mb(Y, P).

Conversely, if is a given probability measure on (Ω,F) which is equivalent to P, then the Radon–Nikodym theorem in Appendix A.2 shows that the distribution := ◦ Y−1 must be equivalent to μ, whence M(Y, P) ⊆ M(μ). Moreover, it follows from Proposition A.15 that the density d/dμ is bounded if d/dP is bounded, and so Mb(Y, P) ⊆ Mb(μ) also follows.

By the preceding lemma, the characterization of the two sets M b(Y, P) and M (Y, P) is reduced to a problem for Borel probability measures on ℝd. Here and in the sequel, wedo not need the fact that μ is the distribution of the lower bounded random vector Y of discounted net gains; our results are true for arbitrary μ such that see also Remark 1.9.

Proposition 1.45. For every Borel probability measure ν on ℝd there exists a smallest closed set S ⊂ ℝd such that ν(Sc) = 0. This set is called the support of ν, it will be denoted by supp ν, and it can be characterized as follows: supp ν is the unique closed set S that satisfies ν(Sc) = 0 and ν(G ∩ S) > 0 for any open set G ⊂ ℝd with G ∩ S ≠ ∅.

Proof. For x ∈ ℝd and r > 0 let Br(x) := {y ∈ ℝd | |x − y| < r} denote the open ball with radius r and center x. We define

Then U is open and satisfies ν(U) = 0, due to the σ-additivity of ν and the fact that the definition of U involves a union of countably many balls with ν-measure zero. Now we claim that Uc is the smallest closed set S ⊂ ℝd such that ν(Sc) = 0. Indeed, the complement of any such set S is open and can be written as follows:

Every ball Br(x) on the right-hand side satisfies ν(Br(x)) = 0. Therefore, Sc ⊂ U and hence Uc ⊂ S.

Now we turn to the proof of the alternative characterization of supp ν. Let G be an open set with G ∩supp ν ≠ ∅ and assume by way of contradiction that ν(G∩supp ν) = 0. Then 0 = ν(G∩supp ν)+ν(G∩(supp ν)c) = ν(G), and so it follows that supp ν ⊂ Gc, which is a contradiction to our assumption G ∩ supp ν ≠ ∅. Finally, we show that if S1, S2 are closed sets with and the property that ν(G∩Si) > 0 for open sets G with G∩Si ≠ ∅, then S1 = S2. Indeed, taking yields that and in turn that S1 ⊂ S2, for otherwise we would have the contradiction Reversing the roles of S1 and S2 yields S2 ⊂ S1 and hence S1 = S2.

We denote by

the convex hull of the support of μ . That is, Γ(μ) is the smallest convex set containing supp μ (see Appendix A.1 for a definition and properties of the convex hull).

Example 1.46. Take d = 1, and consider the measure

Clearly, the support of μ is equal to {−1, +1} and so Γ(μ) = [−1, +1]. A measure ν is equivalent to μ if and only if

for some α ∈ (0, 1). Hence, Mb(μ) = M(μ) = (−1, +1).

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The previous example gives the correct intuition, namely that one always has the inclusions

But while the first inclusion will turn out to be an identity, the second inclusion is usually strict. Characterizing M(μ) in terms of Γ(μ) will involve the following concept.

Definition 1.47. The relative interior of a convex set C ⊂ ℝd is the set of all points x ∈ C such that for all y ∈ C there exists some ε > 0 with

The relative interior of C is denoted ri C.

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The preceding definition can be rephrased as follows: A point x belongs to the relative interior of a convex set C ⊂ ℝd if the straight line joining x and any other point y ∈ C can be extended within C beyond the point x. That is, there exists λ > 1 such that λx + (1 − λ)y ∈ C. By letting ε = λ − 1 one sees the equivalence of the preceding formulation and the one in Definition 1.47.

Remark 1.48. Let C be a nonempty convex subset of ℝd, and consider the affine hull aff C spanned by C, i.e., the smallest affine set which contains C. If we identify aff C with some ℝn, then the relative interior of C is equal to the topological interior of C, considered as a subset of aff C ℝn. In particular, each nonempty convex set has nonempty relative interior. See, for instance, §6 of [232] for proofs.

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Exercise 1.5.1. Let C be a nonempty convex subset of ℝd and denote by C its closure. Use Remark 1.48 to prove the following statements.

(a) The relative interior ri C is equal to the topological interior of C if the latter is nonempty.

(b) For x ∈ ri C,

In particular, ri C is convex. Moreover, show that the operations of taking the closure or the relative interior of a convex set C are consistent with each other:

Deduce in particular that ri C is nonempty.

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Exercise 1.5.2. Show that Γ(μ) has nonempty interior if the nonredundance condition (1.8) is satisfied.

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The following exercise shows that in the preceding exercise the nonredundance condition (1.8) cannot be replaced by the corresponding condition (1.9) for the random vector Y.

Exercise 1.5.3. On Ω = {ω1, ω2},we consider a financial market model with r = 0 and two risky assets with prices

(a) Show that this model satisfies (1.9) but not (1.8).

(b) Find Γ(μ) and show that it has empty interior.

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After these preparations, we can now state the announced geometric characterization of the set Mb(μ). Note that the proof of this characterization relies on the “fundamental theorem of asset pricing” in the form of Theorem 1.7.

Theorem 1.49. The set of all barycenters of probability measures ν ≈ μ coincides with the relative interior of the convex hull of the support of μ. More precisely,

Proof. In a first step, we show the inclusion ri Γ(μ) ⊆ M b(μ). Suppose we are given m ∈ ri Γ(μ). Let denote the translated measure

where A + m := {x + m | x ∈ A}. Then Mb() = Mb(μ) − m, and analogous identities hold for M () and Γ(). It follows that there is no loss of generality in assuming that m = 0, i.e., we must show that 0 ∈ Mb(μ) if 0 ∈ ri Γ(μ).

We claim that 0 ∈ ri Γ(μ) implies the following “no-arbitrage” condition:

If (1.28) is false, then we can find some ξ ∈ ℝd such that ξ · y ≥ 0 for μ-a.e. y but μ({y | ξ · y > δ} ) > 0 for some δ > 0. In this case, the support of μ is contained in the closed set {y | ξ · y ≥ 0} but not in the hyperplane {y | ξ · y = 0}. We conclude that ξ · y ≥ 0 for all y ∈ supp μ and that there exists at least one y∗ ∈ supp μ such that ξ · y∗ > 0. In particular, y∗ ∈ Γ(μ) so that our assumption m = 0 ∈ ri Γ(μ) implies the existence of some ε > 0 such that −εy∗ ∈ Γ(μ). By Proposition A.4, −εy∗ can be represented as a convex combination

of certain y1, . . . , yn ∈ supp μ. It follows that

in contradiction to our assumption that ξ · y ≥ 0 for all y ∈ supp μ. Hence, (1.28) must be true.

Due to (1.28),we can now apply the “fundamental theorem of asset pricing” in the form of Theorem 1.7 to Ω := ℝd, P := μ, and to the random variable Y(y) := y. It yields a probability measure μ∗ ≈ μ whose density dμ∗ /dμ is bounded and which satisfies and This proves the inclusion ri Γ(μ) ⊆ Mb(μ).

Clearly, Mb(μ) ⊂ M (μ). So the theorem will be proved if we can show the inclusion M (μ) ⊂ ri Γ(μ). To this end, suppose by way of contradiction that ν ≈ μ is such that

Again, we may assume without loss of generality that m = 0. We now apply the separating hyperplane theorem in the form of Proposition A.5 to C := ri Γ(μ), which is a nonempty convex set by part (b) of Exercise 1.5.1. We thus obtain some ξ ∈ ℝd such that ξ · y ≥ 0 for all y ∈ ri Γ(μ) and ξ · y∗ > 0 for at least one y∗ ∈ ri Γ(μ). We deduce from (1.26) that ξ · y ≥ 0 holds also for all y ∈ Γ(μ). Moreover, we conclude as above that ξ · y0 must be strictly positive for at least one y0 ∈ supp μ. Hence, the open set {y | ξ · y > 0} has a nonempty intersection with supp μ, and so Proposition 1.45 yields that

By the equivalence of μ and ν, (1.29) is also true for ν instead of μ, and so

in contradiction to our assumption that m = 0. We conclude that M (μ) ⊂ ri Γ(μ).

Remark 1.50. Theorem 1.49 does not extend to the set

Already the simple case serves as a counterexample, because here (μ) = [−1, +1] while ri Γ(μ) = (−1, +1). In this case, we have an identity between (μ) and Γ(μ). But also this identity fails in general as can be seen by considering the normalized Lebesgue measure λ on [−1, +1]. For this choice one finds (λ) = (−1, +1) but Γ(λ) = [−1, +1].

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From Theorem 1.49 we obtain the following geometric characterization of the absence of arbitrage.

Corollary 1.51. Let μ be the distribution of the discounted price vector S/(1 + r) of the risky assets. Then the market model is arbitrage-free if and only if the price system π belongs to the relative interior ri Γ(μ) of the convex hull of the support of μ.

Exercise 1.5.4. Prove Corollary 1.51.

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1.6Contingent initial data

The idea of hedging contingent claims develops its full power only in a dynamic setting in which trading may occur at several times. The corresponding discrete-time theory is presented in Chapter 5. The introduction of additional trading periods requires more sophisticated techniques than those we have used so far. In this section we will introduce some of these techniques in an extended version of our previous market model in which initial prices, and hence strategies, are contingent on scenarios. In this context, we are going to characterize the absence of arbitrage strategies. The results will be used as building blocks in the multiperiod setting of Part II; their study can be postponed until Chapter 5.

Suppose that we are given a σ-algebra F0 ⊂ F which specifies the information that is available to an investor at time t = 0. The prices for our d + 1 assets at time 0 will be modeled as nonnegative F0-measurable random variables Thus, the price system of our previous discussion is replaced by the vector

The portfolio chosen by an investor at time t = 0 will also depend on the information available at time 0. Thus, we assume that

is an F0-measurable random vector. The asset prices observed at time t = 1 will be denoted by

They are modeled as nonnegative random variables which are measurable with respect to a σ-algebra F1 such that F0 ⊂ F1 ⊂ F. The σ-algebra F1 describes the information available at time 1, and in this section we can assume that F = F1.

A riskless bond could be included by taking and by assuming measurable and P-a.s. strictly positive. However, in the sequel it will be sufficient to assume that measurable, measurable, and that

Under this condition, we can take the 0th asset as numéraire, and we denote by

the discounted asset prices and by

the vector of the discounted net gains.

Definition 1.52. An arbitrage opportunity is a portfolio such that P-a.s., and

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By our assumption (1.30), any arbitrage opportunity = (ξ0, ξ) satisfies

In fact, the existence of a d-dimensional F0-measurable random vector ξ with (1.31) is equivalent to the existence of an arbitrage opportunity. This can be seen as in Lemma 1.4.

The space of discounted net gains which can be generated by some portfolio is given by

Here, L0(Ω,F0, P;ℝd) denotes the space of ℝd-valued random variables which are P-a.s. finite and F0-measurable modulo the equivalence relation (A.35) of coincidence up to P-null sets. The spaces Lp(Ω,F0, P;ℝd) for p > 0 are defined in the same manner. We denote by the cone of all nonnegative elements in the space With this notation, the absence of arbitrage opportunities is equivalent to the condition

We will denote by

the convex cone of all Z ∈ L0 which can be written as the difference of some ξ ·Y ∈ K and some

The following definition involves the notion of the conditional expectation

of a random variable Z with respect to a probability measure Q, given the σ-algebra F0 ⊂ F; see Appendix A.2. If Z = (Z1, . . . , Zn) is a random vector, then EQ[ Z | F0 ] is shorthand for the random vector with components E Q[ Zi | F0 ], i = 1, . . . , n.

Definition 1.53. A probability measure Q satisfying

and

is called a risk-neutral measure or martingale measure. We denote by P the set of all risk-neutral measures P∗ which are equivalent to P.

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Remark 1.54. The definition of a martingale measure Q means that for each asset i = 0, . . . , d, the discounted price process (X)t=0,1 is a martingale under Q with respect to the σ-fields (Ft)t=0,1. The systematic discussion of martingales in a multi-period setting will begin in Section 5.2. The martingale aspect will be crucial for the theory of dynamic hedging in Part II.

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As the main result of this section, we can now state an extension of the “fundamental theorem of asset pricing” in Theorem 1.7 to our present setting. In the context of Section 1.2, where F0 = {∅, Ω}, the following arguments simplify considerably, and they yield an alternative proof of Theorem 1.7, in which the separation argument in ℝd is replaced by a separation argument in L1.

Theorem 1.55. The following conditions are equivalent:

(c) There exists a measure P∗ ∈ P with a bounded density dP∗/dP.

Proof. (d)⇒(a): Suppose by way of contradiction that there exist both a P∗ ∈ P and some ξ ∈ L0(Ω,F0, P;ℝd) with nonzero payoff For large enough c > 0, will be bounded, and the payoff ξ(c) · Y will still be nonzero and in However,

which is the desired contradiction. Here we have used the boundedness of ξ(c) to guarantee the integrability of ξ(c) ·Y and, the validity of the formula E∗[ ξ(c) ·Y | F0 ] = ξ(c) · E∗[ Y | F0 ].

(a)⇔(b): It is obvious that (a) is necessary for (b). In order to prove sufficiency, suppose that we are given some Then there exists a random variable U ≥ 0 and a random vector ξ ∈ L0(Ω,F0, P;ℝd) such that

This implies that ξ · Y ≥ U ≥ 0, which, according to condition (a), can only happen if ξ · Y = 0. Hence, also U = 0 and in turn Z = 0.

(b)⇒(c): This is the difficult part of the proof. The assertion will follow by combining Lemmas 1.57, 1.58, 1.60, and 1.68.

Remark 1.56. If Ω is discrete, or if there exists a decomposition of Ω in countable many atoms of (Ω,F0, P), then the martingale measure P∗ can be constructed by applying the result of Theorem 1.7 separately on each atom. In the general case, the idea of patching together conditional martingale measures would involve subtle arguments of measurable selection; see [70]. Here we present a different approach which is based on separation arguments in L1(P). It is essentially due to W. Schachermayer [244]; our version uses in addition arguments by Y. Kabanov and C. Stricker [170].

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We start with the following simple lemma, which helps taking care of the integrability condition in Definition 1.53.

Lemma 1.57. For the proof of the implication (b)⇒(c) in Theorem 1.55, we may assume without loss of generality that

Proof. Define a probability measure by

where c is chosen such that the right-hand side integrates to 1. Clearly, (1.32) holds for . Moreover, condition (b) of Theorem 1.55 is satisfied by P if and only if it is satisfied by the equivalent measure . If P∗ ∈ P is such that the density dP∗/d is bounded, then so is the density

Therefore, the implication (b)⇒(c) holds for P if and only if it holds for .

From now on, we will always assume (1.32). Our goal is to construct a suitable Z ∈ L∞ such that

defines an equivalent risk-neutral measure P∗. The following simple lemma gives a criterion for this purpose, involving the convex cone

Lemma 1.58. Suppose Z ∈ L∞ is such that

for some constant c ≥ 0. Then:

(a) E[Z W ]≤ 0 for all W ∈ C , i.e., we can take c = 0.

(b) Z ≥ 0 P-a.s.

(c) If Z does not vanish P-a.s., then

defines a risk-neutral measure Q P.

Proof. (a): Note that C is a cone, i.e., W ∈ C implies that αW ∈ C for all α ≥ 0. This property excludes the possibility that E [ ZW ] > 0 for some W ∈ C if (1.33) holds.

(b): C contains the function W := −{ Z<0}. Hence, by part (a),

where Z − := max{−Z, 0} = −(0∧Z) denotes the negative part of the random variable Z.

(c): For all ξ ∈ L∞(Ω,F0, P;ℝd) and α ∈ ℝwe have αξ ·Y ∈ C by our integrability assumption (1.32). Thus, a similar argument as in the proof of (a) yields E[ Z ξ · Y ] = 0. Since ξ is bounded, we may conclude that

As ξ is arbitrary, this yields E [ ZY | F0 ] = 0 P-almost surely. Proposition A.16 now implies

which concludes the proof.

In view of the preceding lemma, the construction of risk-neutral measures Q P with bounded density is reduced to the construction of elements of the set

In the following lemma, we will construct such elements by applying a separation argument suggested by the condition

which follows from condition (b) of Theorem 1.55. This separation argument needs the additional assumption that C is closed in L1. Showing that this assumption is indeed satisfied in our situation will be one of the key steps in our proof; see Lemma 1.68 below.

Lemma 1.59. Assume that C is closed in L1 and satisfies Then for each nonzero there exists some Z ∈ Z such that E[ FZ ] > 0.

Proof. Let B := {F} so that B ∩ C = ∅. Since the set C is nonempty, convex and closed in L1, we may apply the Hahn–Banach separation theorem in the form of Theorem A.60 to obtain a continuous linear functional on L1 such that

Since the dual space of L1 can be identified with L∞, there exists some Z ∈ L∞ such that (F) = E[ FZ ] for all F ∈ L1. By scaling , we may assume without loss of generality that Z∞ ≤ 1. By construction, Z satisfies the assumptions of Lemma 1.58, and so Z ∈ Z . Moreover, E[ FZ ] = (F) > 0 since the constant function W ≡ 0 is contained in C .

We will now use an exhaustion argument to conclude that Z contains a strictly positive element Z ∗ under the assumptions of Lemma 1.59. After normalization, Z ∗ will then serve as the density of our desired risk-neutral measure P∗ ∈ P.

Lemma 1.60. Under the assumptions of Lemma 1.59, there exists Z∗ ∈ Z with Z∗ > 0 P-a.s.

Proof. As a first step, we claim that Z is countably convex: If (αk)k∈N is a sequence of nonnegative real numbers summing up to 1, and if Z(k) ∈ Z for all k, then

Indeed, we have 0 ≤ Z ≤ 1, P[ Z > 0 ] > 0, and for W ∈ C

so that Lebesgue’s dominated convergence theorem implies that

For the second step, let

We choose Z(n) ∈ Z such that P[ Z(n) > 0] → c. Then

by step one, and

Hence P[Z∗ > 0] = c.

In the final step, we show that c = 1. Then Z ∗ will be as desired. Suppose by way of contradiction that P[Z∗ = 0] > 0, so that W := {Z∗=0} is a nonzero element of Lemma 1.59 yields some Z ∈ Z with E[ WZ ] > 0. Hence,

and so

in contradiction to the maximality of P[Z∗ > 0].

Thus, we have completed the proof of the implication (b)⇒(c) of Theorem 1.55 up to the requirement that C is closed in L1. Let us pause here in order to state general versions of two of the arguments we have used so far. The first is known as the Halmos–Savage theorem.

Theorem 1.61. Let Q be a set of probability measures which are all absolutely continuous with respect to a given measure P. Suppose moreover that Q ≈ P in the sense that Q[ A ] = 0 for all Q ∈ Q implies that P[ A ] = 0. Then there exists a countable subfamily which satisfies Q ≈ P. In particular, there exists an equivalent measure in Q as soon as Q is countably convex in the following sense: If (αk)k∈N is a sequence of nonnegative real numbers summing up to 1, and if Qk ∈ Qfor all k, then

Exercise 1.6.1. Prove Theorem 1.61 by modifying the exhaustion argument used in the proof of Lemma 1.60.

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An inspection of Lemmas 1.58, 1.59, and 1.60 shows that the particular structure of was only used for part (c) of Lemma 1.58. All other arguments relied only on the fact that C is a closed convex cone in L1 that contains all bounded negative functions and no nontrivial positive function. Thus, we have in fact proved the following Kreps–Yan theorem, which was obtained independently in [277] and [192].

Theorem 1.62. Suppose C is a closed convex cone in L1 satisfying

Then there exists Z ∈ L∞ such that Z > 0 P-a.s. and E[W Z] ≤ 0 for all W ∈ C .

Let us now turn to the closedness of our set The following example illustrates that we cannot expect C to be closed without assuming the absence of arbitrage opportunities.

Example 1.63. Let P be the Lebesgue measure on the Borel field F1 of Ω = [0, 1], and take F0 = {∅, Ω} and Y(ω) = ω. This choice clearly violates the no-arbitrage condition, i.e., we have The convex set is a proper subset of L1. More precisely, C does not contain any function F ∈ L1 with F ≥ 1: If we could represent F as ξ · Y − U for a nonnegative function U, then it would follow that

which is impossible for any ξ. However, as we show next, the closure of C in L1 coincides with the full space L1. In particular, C cannot be closed. Let F ∈ L1 be arbitrary, and observe that

converges to F in L1 as n ↑ ∞. Moreover, each F n belongs to C as

Consequently, F is contained in the L1-closure of C.

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In the special case F0 = {∅, Ω}, we can directly go on to the proof that C is closed, using a simplified version of Lemma 1.68 below. In this way, we obtain an alternative proof of Theorem 1.7. In the general case we need some preparation. Let us first prove a “randomized” version of the Bolzano–Weierstraß theorem. It yields a simple construction of a measurable selection of a convergent subsequence of a given sequence in L0(Ω,F0, P;ℝd).

Lemma 1.64. Let (ξn) be a sequence in L0(Ω,F0, P;ℝd) with lim infn |ξn| < ∞. Then there exists ξ ∈ L0(Ω,F0, P;ℝd) and a strictly increasing sequence (σm) of F0-measurable integer-valued random variables such that

Proof. Let Λ(ω) := lim infn |ξn(ω)|, and define σm := m on the P-null set {Λ =∞}. On we let and we define F0-measurable 10random indices by

We use recursion on i = 1, . . , d to define the ith component ξ i of the limit ξ and to extract a new subsequence of random indices. Let

which is already defined if i = 1. This ξ i can be used in the construction of Let and, 1for im = 2,3, . . . ,

Then yields the desired sequence of random indices.

It may happen that

although ξ and are two different portfolios in L0(Ω,F0, P;ℝd).

Remark 1.65. We could exclude this possibility by the following assumption of nonredundance:

Under this assumption, we can immediately move on to the final step in Lemma 1.68.

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Without assumption (1.34), it will be convenient to have a suitable linear space N⊥ of “reference portfolios” which are uniquely determined by their payoff. The construction of N⊥ is the purpose of the following lemma. We will assume that the spaces L0 and L0(Ω,F0, P;ℝd) are endowed with the topology of convergence in P-measure, which is generated by the metric d of (A.36).

Lemma 1.66. Define two linear subspaces N and N⊥ of L0(Ω,F0, P;ℝd) by

(a) Both N and N⊥ are closed in L0(Ω,F0, P;ℝd) and, in the following sense, invariant under the multiplication with scalar functions g ∈ L0(Ω,F0, P): If η ∈ N and ξ ∈ N⊥, then gη ∈ N and gξ ∈ N⊥.

(b) If ξ ∈ N⊥ and ξ · Y = 0 P-a.s., then ξ = 0, i.e., N ∩ N⊥ = {0}.

(c) Every ξ ∈ L0(Ω,F0, P;ℝd) has a unique decomposition ξ = η + ξ⊥, where η ∈ N and ξ⊥ ∈ N⊥.

Remark 1.67. For the proof of this lemma, we will use a projection argument in Hilbert space. Let us sketch a more probabilistic construction of the decomposition ξ = η+ξ⊥. Take a regular conditional distribution of Y given F0, i.e., a stochastic kernel K from (Ω,F0) to ℝd such that K(ω, A) = P[ Y ∈ A | F0 ](ω) for all Borel sets A ⊂ ℝd and P-a.e. ω (see, e.g., §44 of [20]). If one defines ξ⊥(ω) as the orthogonal projection of ξ(ω) onto the linear hull L(ω) of the support of the measure K (ω, ·), then η := ξ−ξ⊥ satisfies η·Y = 0 P-a.s., and any with the same property must be P-a.s. perpendicular to L(ω). However, carrying out the details of this construction involves certain measurability problems; this is why we use the projection argument below.

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Proof of Lemma 1.66. (a): The closedness of N and N⊥ follows immediately from the metrizability of L0(Ω,F0, P;ℝd) (see Appendix A.7) and the fact that every sequence which converges in measure has an almost-surely converging subsequence. The invariance under the multiplication with F0-measurable scalar functions is obvious.

(b): Suppose that ξ ∈ N ∩ N⊥. Then taking η := ξ in the definition of N⊥ yields ξ · ξ = |ξ|2 = 0 P-a.s.

(c): Any given ξ ∈ L0(Ω,F0, P;ℝd) can be written as

where ei denotes the ith Euclidean unit vector, and where ξ i(ω) is the ith component of ξ(ω). Consider ei as a constant element of L0(Ω,F0, P;ℝd), and suppose that we can decompose ei as

Since by part (a) both N and N⊥ are invariant under the multiplication with F0-measurable functions, we can then obtain the desired decomposition of ξ by letting

Uniqueness of the decomposition follows from N ∩ N⊥ = {0}.

It remains to construct the decomposition (1.35) of ei. The constant ei is an element of the space H := L2(Ω,F0, P;ℝd), which becomes a Hilbert space if endowed with the natural inner product

Observe that both N ∩ H and N⊥ ∩ H are closed subspaces of H, because convergence in H implies convergence in L0(Ω,F0, P;ℝd). Therefore, we can define the corresponding orthogonal projections

Thus, letting will be the desired decomposition (1.35), once we know that ei = π0(ei) + π⊥(ei). To prove this, we need only show that ζ := ei − π0(ei) is contained in N⊥. We assume by way of contradiction that ζ is not contained in N⊥ ∩ H. Then there exists some η ∈ N such that P[ ζ · η > 0] > 0. Clearly,

is contained in N∩H for each c > 0. But if c is large enough, then 0 < E[ ·ζ ] = (, ζ )H, which contradicts the fact that ζ is by construction orthogonal to N ∩ H.

After these preparations, we can now complete the proof of Theorem 1.55 by showing the closedness of This is an immediate consequence of the following lemma, since convergence in L1 implies convergence in L0, i.e., convergence in P-measure. Recall that we have already proved the equivalence of the conditions (a) and (b) in Theorem 1.55.

Lemma 1.68. Ifthenis closed in L0.

Proof. Suppose converges in L0 to some W as n ↑ ∞. By passing to a suitable subsequence, we may assume without loss of generality that Wn → W P-almost surely. We can write Wn = ξn · Y − Un for ξn ∈ N⊥ and

In a first step, we will prove the assertion given the fact that

which will be established afterwards. Assuming (1.36), Lemma 1.64 yields F0-measurable integer-valued random variables σ1 < σ2 < · · · and some ξ ∈ L0(Ω,F0, P;ℝd) such that P-a.s. ξσn → ξ. It follows that

so that and Thus, is closed in L0 if (1.36) is satisfied.

Let us now show that A := {lim infn |ξn| = +∞} satisfies P[ A ] = 0 as claimed in (1.36). Let

where e1 is the unit vector (1, 0, . . . , 0). Using Lemma 1.64 on the sequence (ζn) yields F0-measurable integer-valued random variables τ1 < τ2 < · · · and some ζ ∈ L0(Ω,F0, P;ℝd) such that P-a.s. ζτn → ζ . The convergence of (Wn) implies that

Hence, our assumption K ∩ L0+ = {0} yields (A ζ) · Y = 0. Below we will show that A ζ ∈ N⊥, so that

On the other hand, the fact that |ζn| = 1 P-a.s. implies that |ζ | = 1 P-a.s., which can only be consistent with (1.38) if P[ A ] = 0.

It remains to show that ζ ∈ N⊥. To this end, we first observe that each ζτn A belongs to N⊥ since, for each η ∈ N,

The closedness of N⊥ implies ζ ∈ N⊥, and A ∈ F0 yields A ζ ∈ N⊥.

If in the proof of Lemma 1.68 Wn = ξn ·Y for all n, then U = 0 in (1.37), and W = limn Wn is itself contained in K . We thus get the following lemma, which will be useful in Chapter 5.

Lemma 1.69. Suppose thatThen K is closed in L0.

In fact, it is possible to show that K is always closed in L0; see [268], [244]. But this stronger result will not be needed here.

As an alternative to the randomized Bolzano–Weierstraß theorem in Lemma 1.64, we can use the following variant of Komlós’ principle of subsequences. It yields a convergent sequence of convex combinations of a sequence in L0(Ω,F0, P;ℝd), and this will be needed later on. Recall from Appendix A.1 the notion of the convex hull

of a subset A of a linear space, which in our case will be L0(Ω,F0, P;ℝd).

Lemma 1.70. Let (ξn) be a sequence in L0(Ω,F0, P;ℝd) such that supn |ξn| < ∞ P-almost surely. Then there exists a sequence of convex combinations

which converges P-almost surely to some η ∈ L0(Ω,F0, P;ℝd).

Proof. We can assume without loss of generality that supn |ξn| ≤ 1 P-a.s.; otherwise we consider the sequence n := ξn/supn |ξn|. Then (ξn) is a bounded sequence in the Hilbert space H := L2(Ω,F0, P;ℝd). Since the closed unit ball in H is weakly compact, the sequence (ξn) has an accumulation point η ∈ H; here, weak sequential compactness follows from the Banach–Alaoglu theorem in the form of Theorem A.66 and the fact that the dual H ʹ of the Hilbert space H is isomorphic to H itself. For each n, the accumulation point η belongs to the L2-closure Cn of conv{ξn , ξn+1, . . . }, due to the fact that a closed convex set in H is also weakly closed; see Theorem A.63. Thus, we can find ηn ∈ conv{ξn , ξn+1, . . . } such that

This sequence (ηn) converges P-a.s. to η.

Remark 1.71. The original result by Komlós [184] is more precise. It states that for any bounded sequence (ξn) in L1(Ω,F, P;ℝd) there is a subsequence (ξnk ) which satisfies a strong law of large numbers, i.e.,

exists P-almost surely; see also [275].

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