No-arbitrage concepts in topological vector lattices

We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA$_1$ and NA$_1$ may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a num\'{e}raire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs.


Introduction
Let (Ω, G , P) be a probability space, and let K 0 ⊂ L 0 (Ω, G , P) be a set of random variables, where we think of outcomes of trading strategies with initial wealth zero. Then an arbitrage opportunity is an element X ∈ K 0 such that P(X ≥ 0) = 1 and P(X > 0) > 0.
Therefore, No Arbitrage (NA) means that It is well-known that for concrete financial models it is easy to find mathematical conditions which are sufficient for NA (like the existence of an equivalent martingale measure), but typically these conditions fail to be necessary for NA. In order to overcome this problem, two approaches have been suggested in the literature: (1) We choose a subspace of L 0 , say L ∞ , and define the subset C ⊂ L ∞ as C := (K 0 − L 0 + ) ∩ L ∞ . Then NA can equivalently be written as C ∩ L 0 + = {0}, and we consider the stronger condition where the closure is taken with respect to some topology on L ∞ . If this topology is the norm topology on L ∞ , then we have the well-known concept of NFLVR, see [7]. It is well-known that for suitable semimartingale models in continuous time NFLVR is equivalent to the existence of an equivalent local martingale measure; see, for example, the papers [7,8], the textbook [9], and also the paper [16]. (2) Instead of considering the set K 0 of outcomes of trading strategies with initial wealth zero, we rather consider the outcomes (K α ) α>0 of trading strategies with positive, but arbitrary small initial wealth α. Then the appropriate concepts are NUPBR, NAA 1 and NA 1 . It is well-known that for suitable semimartingale models in continuous time these three conditions are equivalent, and that they are satisfied if and only if there exists an equivalent local martingale deflator; see [41], and also the earlier papers [6] and [23]. The goal of this paper is to provide a general mathematical framework for noarbitrage concepts which goes beyond the settings which have been considered in the literature so far. The idea is as follows. It is known that the space L 0 has rather poor topological properties. It fails to be a locally convex space, and its dual space is typically trivial. However, the space (L 0 , ≤) is an example of a topological vector lattice; indeed it is even a so-called Fréchet lattice. The properties of the space L 0 and in particular its positive cone L 0 + have already been studied in the literature, often with a focus to applications in finance; see, for example [3,43,11,45,22,30,24,27,25,26,13]. We also mention the related paper [4], where the equivalence between economic viability and no-arbitrage in the presence of Knigthian uncertainty has been studied, and the recent paper [31], where a reverse approach to model uncertainty is presented.
The observation that (L 0 , ≤) is a topological vector lattice motivates the general study of no-arbitrage concepts in topological vector lattices. For a topological vector lattice (V, ≤) we consider the positive cone Furthermore, let K 0 ⊂ V be a subset and let (K α ) α>0 be a family of subsets such that certain structural conditions are satisfied. In particular, K 0 is supposed to be a convex cone and for each α > 0 the set K α is convex; we refer to Section 3 for further details. Then NA simply means that Let us also indicate how the remaining above mentioned no-arbitrage concepts are defined: (1) Consider the convex cone K 0 . Let U ⊂ V be an ideal which is dense in V . Then (U, ≤) is also a topological vector lattice with positive cone U + = V + ∩ U . We define the subset C ⊂ U as Then NA is satisfied if and only if Let τ be a topology on U . Then we say that NFL τ holds if the stronger condition is fulfilled. In the particular case V = L 0 and U = L ∞ , we obtain the well-known concepts NFLVR, NFLBR and NFL; see [9] or [16].
(2) Consider the family (K α ) α>0 . We define the family (B α ) α>0 of convex and semi-solid subsets of V + as We may think of all nonnegative elements which are equal to or below the outcome of a trading strategy with initial value α. As we will show, then we have B α = αB for each α > 0. Therefore, rather than focusing on all outcomes of trading strategies with positive initial wealth, it suffices to concentrate on all outcomes of trading strategies with initial wealth one. Mathematically speaking, we may focus on the set B := B 1 rather than on the whole family (B α ) α>0 . We introduce the no-arbitrage concepts as follows: • NUPBR holds if B is topologically bounded.
• NAA 1 holds if B is sequentially bounded.
• NA 1 holds if p B (x) > 0 for all x ∈ V + \{0}, where p B is the Minkowski functional, which can also be interpreted as the minimal superreplication price. As we will show, in the particular case V = L 0 these concepts correspond to the well-known concepts used in the literature.
In this paper we will introduce all these no-arbitrage concepts formally for a topological vector lattice (V, ≤), show the connections between these concepts, and consider the particular situation where the topological vector lattice is the space (L 0 , ≤) of all random variables.
In particular, we will show that in a topological vector lattice the concepts NUPBR, NAA 1 and NA 1 are generally not equivalent. More precisely, the concepts NUPBR and NAA 1 are equivalent, and they are satisfied if and only if for every neighborhood of zero the Minkowski functional considered on V + is bounded from below by a positive constant outside this neighborhood; see Proposition 3. 16. In particular, the Minkowski functional has no zeros on V + \ {0}, and therefore NUPBR (or NAA 1 ) implies NA 1 , but we also see that the converse can generally not be true; see Example 2.14 for a counter example. However, in the particular case V = L 0 these concepts are known to be equivalent, and in Theorem 5.12 we will present further equivalent conditions, including the von Weizsäcker property and the Banach Saks property of the convex subset B.
Topological vector lattices also provide a suitable framework for abstract versions of the fundamental theorem of asset pricing (FTAP); in particular if the ideal U is a locally convex space or even a Banach function space. References about locally convex spaces with a view to applications in finance include [29,35,38,15,36,5,37,33,34]. In this paper we present several abstract versions of the FTAP, including abstract FTAPs on locally convex spaces (see Theorems 4.4 and 4.10), an abstract FTAP on spaces of continuous functions on compact sets (see Theorem 4.18), and abstract FTAPs on Banach function spaces (see Theorem 6.5 and Corollary 6.6). Our investigations show that those Banach function spaces, on which the FTAP is warranted in its classical form with a separating measure, are precisely σ-order continuous Banach function spaces. Later on, these findings are used for an abstract FTAP on L p -spaces (see Corollary 6.7) and for an extension of the well-known noarbitrage result in discrete time; see Theorem 6.14.
Furthermore, using our general theory we will derive results for no-arbitrage concepts in a market with semimartingales which does not need to have a numéraire, in particular for self-financing portfolios; see Theorem 7.25 and Propositions 7. 26-7.29.
The remainder of this paper is organized as follows. In Section 2 we present the required background about topological vector lattices. In Section 3 we introduce noarbitrage concepts in topological vector lattices. In Section 4 we present versions of the abstract FTAP on locally convex spaces. In Section 5 we review the no-arbitrage concepts in the particular situation where the topological vector lattice is the space L 0 of random variables. In Section 6 we present versions of the abstract FTAP on Banach function spaces, and discuss on which spaces the FTAP is warranted in its classical form with a separating measure. In Section 7 we consider a financial market with nonnegative semimartingales which does not need to have a numéraire, and derive consequences for the no-arbitrage concepts; in particular regarding selffinancing portfolios.

Topological vector lattices
In this section we provide the required background about topological vector lattices and some related results. For further details about topological vector lattices, we refer, for example, to [39,Chap. V].
Let V be an R-vector space. Furthermore, let ≤ be a binary relation over V which is reflexive, anti-symmetric and transitive; more precisely: • We have x ≤ x for all x ∈ V .
• If x ≤ y and y ≤ x, then we have x = y.
• If x ≤ y and y ≤ z, then we have x ≤ z.
Then (V, ≤) is called an ordered vector space if the following axioms are satisfied: (1) If x ≤ y, then we have x + z ≤ y + z for all x, y, z ∈ V .
Let V be a topological vector space such that (V, ≤) is an ordered vector space. Then we call (V, ≤) an ordered topological vector space if the positive cone is closed in V . A vector lattice (or a Riesz space) is an ordered vector space (V, ≤) such that the supremum x ∨ y and the infimum x ∧ y exist for all x, y ∈ V . We introduce further lattice operations. Namely, for x ∈ V we define the positive part x + := x ∨ 0, the negative part x − := −x ∨ 0, and the absolute value |x| := x ∨ (−x). Let (V, ≤) be a vector lattice. A subspace U ⊂ V is called a vector sublattice (or a Riesz subspace) of V if x ∨ y ∈ U for all x, y ∈ U . Then (U, ≤) is a vector lattice with positive cone U + = V + ∩ U .
A subset A ⊂ V is called solid if for all x ∈ A and y ∈ V with |y| ≤ |x| we have y ∈ A. A solid subspace U ⊂ V is called an ideal. Every ideal is a vector sublattice of V .
A topological vector space V is called locally solid if it has a zero neighborhood basis of solid sets. A vector lattice (V, ≤) is called a topological vector lattice if it is a Hausdorff topological vector space which is locally solid.
A topological vector space V is called completely metrizable if there is a metric d on V which induces the topology and for which the metric space (V, d) is complete. A Fréchet lattice is a completely metrizable topological vector lattice.
A vector lattice V equipped with a norm · such that (V, · ) is a Banach space and for all x, y ∈ V with |x| ≤ |y| we have x ≤ y is called a Banach lattice. Note that every Banach lattice is a topological vector lattice. A Banach lattice V is called order continuous if for every net (x i ) i∈I ⊂ V with x i ↓ 0 we have x i ↓ 0, and it is called σ-order continuous if for every sequence (x n ) n∈N ⊂ V with x n ↓ 0 we have x n ↓ 0. An element e ≥ 0 of a Banach lattice V is called a weak unit of For what follows, let (V, ≤) be a topological vector lattice. Recall that we have five lattice operations The following statements are true: (1) The lattice operations are continuous.
Proof. By statement 7.1 on page 234 in [39] the lattice operations are continuous. Furthermore, by statement 7.2 on page 235 in [39] the positive cone V + is closed, showing that (V, ≤) is an ordered topological vector space.
2.3. Lemma. If B ⊂ V is solid, then it is also circled.
(2) B is called semi-solid if for all x ∈ B and all y ∈ V + with y ≤ x we have y ∈ B.

2.5.
Lemma. If B ⊂ V + is semi-solid, then it is also semi-circled.
2.6. Lemma. Let K ⊂ V be a subset. We define the subset B ⊂ V + as Then the following statements are true: (1) B is semi-solid.
(2) If K is convex, then B is also convex.
Proof. Let x ∈ B and y ∈ V with 0 ≤ y ≤ x be arbitrary. Then we have y ∈ V + . Furthermore, we have Now, we assume that K is also convex. Let x, y ∈ B and λ ∈ [0, 1] be arbitrary.
showing that B is also convex.
Proof. By Lemma 2.6 the subset B is semi-solid. Furthermore, (U, ≤) is a topological vector lattice with positive cone U + because U ⊂ V be an ideal. Let x ∈ B be arbitrary. Then we have Since x ∈ V + and the lattice operations are continuous, this gives us and hence x ∈ C ∩ U + . 2.8. Lemma. For a subset B ⊂ V + the following statements are equivalent: (i) B is sequentially bounded.
Proof. (i) ⇒ (ii): This implication is obvious. (ii) ⇒ (i): Let (x n ) n∈N ⊂ B and (α n ) n∈N ⊂ R be sequences with α n → 0. There exist a decreasing sequence (β n ) n∈N ⊂ (0, ∞) with β n ↓ 0 and an index n 1 ∈ N such that |α n | ≤ β n for each n ≥ n 1 . (2.1) Indeed, since α n → 0 there is a subsequence (n k ) k∈N such that for each k ∈ N we have |α n | ≤ k −1 for all n ≥ n k .
We define the sequence (β n ) n∈N ⊂ (0, ∞) as Then we have (2.1). Now, let U ⊂ V be an arbitrary zero neighborhood. Since V is locally solid, we may assume that U is solid, and hence circled. By assumption there exists an index N ≥ n 1 such that β n x n ∈ U for all n ≥ N .
Since U is circled, by (2.1) we also have α n x n ∈ U for all n ≥ N , showing that α n x n → 0.

2.9.
Lemma. Let B ⊂ V + be a semi-circled subset. Then the following statements are true: (1) We have 0 ∈ B.
Proof. The proof is obvious, and therefore omitted.
2.10. Lemma. Let B ⊂ V + be a semi-circled subset. Then the following statements are true: Proof. The first three statements are obvious. Let x ∈ B and α ∈ R + be arbitrary. We may assume that α > 0 because otherwise the identity follows from the first statement. Since B is semi-circled, for each β > 0 we have αx ∈ βB if and only if x ∈ β α B, and for each γ > 0 we have x ∈ γB if and only if αx ∈ αγB. Therefore, we have Now assume that B is semi-solid, and let x, y ∈ B with x ≤ y be arbitrary. Then for each α > 0 with y ∈ αB we have x ∈ αB, and hence completing the proof.
For each α ≥ 0 we agree on the notation 2.11. Lemma. Let B ⊂ V + be a semi-circled subset. Then we have Proof. Let x ∈ B be arbitrary. Then we have p B (x) ≤ α if and only if Since B is semi-circled, by Lemma 2.9 this is the case if and only if x ∈ αB. This proves Since p B (x) ≥ 1 for all x ∈ V + \ B, this completes the proof.
2.12. Proposition. Let B ⊂ V + be a semi-circled subset. Then the following statements are equivalent: (i) B is topologically bounded.
(ii) B is sequentially bounded.
(iii) For each neighborhood U ⊂ V of zero there exists α ∈ (0, 1) such that In either case, we have p B (x) > 0 for all x ∈ V + \ {0}.
The subset B is topologically bounded if and only if for each neighborhood U of zero there exists α ∈ (0, 1) such that αB ⊂ U ∩ V + . Using Lemma 2.11 completes the proof. The additional statement is obvious.
Hence, in the situation of Proposition 2.12 for every neighborhood of zero the Minkowski functional p B considered on V + is bounded from below by a positive constant outside this neighborhood. In particular, it has no zeros on V + \ {0}.
The following example shows that a convex, semi-solid subset B ⊂ V + with α>0 αB = {0} does not need to be topologically bounded. 2.14. Example. Let V = ℓ 2 (N) be the space of all square-integrable sequences, equipped with the Hilbert space topology induced by the norm We agree to write x ≤ y if x k ≤ y k for all k ∈ N. Then (V, ≤) is a Banach lattice, and the positive cone is given by We define the sequence (f k ) k∈N as f k := k e k , where e k denotes the kth unit vector. Furthermore, we define the subset B ⊂ V + as the convex hull Then B is unbounded because f k → ∞ for k → ∞, and B consists of all linear combinations for some n ∈ N, where λ k ≥ 0 for k = 1, . . . , n and n k=1 λ k ≤ 1. From this representation we see that B is semi-solid. For each α > 0 the set αB consists of all x ∈ V + with representation (2.3) such that λ k ≥ 0 for k = 1, . . . , n and n k=1 λ k ≤ α. Let x ∈ B \ {0} with representation (2.3) be arbitrary. Since x = 0, we have λ > 0, where λ := n k=1 λ k , and hence x / ∈ αB for each α ∈ (0, λ). Consequently, we have α>0 αB = {0}.
However, surprisingly there are some examples of topological vector lattices V where every convex, semi-solid subset B ⊂ V + with α>0 αB = {0} is topologically bounded. As we will see in Section 5 later on, this is in particular the case if V = L 0 is the space of all random variables defined on some probability space.
Recall that a subset B ⊂ V + is unbounded if and only if there exist sequences (x n ) n∈N ⊂ B and (α n ) n∈N ⊂ (0, ∞) with α n ↓ 0 such that α n x n → 0. In the upcoming definition, we make a stronger assumption for unbounded subsets, which are convex and semi-solid.
2.15. Definition. The topological vector lattice (V, ≤) admits nontrivial minimal elements for unbounded, convex and semi-solid subsets of V + if for each unbounded, convex and semi-solid subset 2.16. Theorem. Suppose that (V, ≤) admits nontrivial minimal elements for unbounded, convex and semi-solid subsets of V + . Then for every convex, semi-solid subset B ⊂ V + the following statements are equivalent: Proof. By virtue of Propositions 2.12 and 2.13, we only need to prove the im- and hence the contradiction p B (x) = 0.
2.17. Proposition. Suppose that the topological vector lattice (V, ≤) is locally convex with a family (ρ i ) i∈I of seminorms satisfying the following two conditions: Then (V, ≤) admits nontrivial minimal elements for unbounded, convex and semisolid subsets of V + .
Proof. Let B ⊂ V + be an unbounded, convex and semi-solid subset. Then there exist sequences (x n ) n∈N ⊂ B and (α n ) n∈N ⊂ (0, ∞) with α n ↓ 0 such that α n x n → 0. Hence, there exists i ∈ I such that ρ i (α n x n ) → 0. Therefore, there exist ǫ > 0 and a subsequence (x n k ) k∈N such that ρ i (α n k x n k ) ≥ ǫ for each k ∈ N. Let f : I → R + be the function given by f (i) := ǫ and f (j) := 0 for all j ∈ I \ {i}. By assumption there exists 2.18. Remark. According to Proposition 2.17 the following examples of topological vector lattices (V, ≤) admit nontrivial minimal elements for unbounded, convex and semi-solid subsets of V + , which means that Theorem 2.16 applies: • The Euclidean space V = R n , equipped with the usual Euclidean topology.
• The space V = ℓ 0 (N) of all sequences, equipped with the topology of pointwise convergence.
equipped with the topology of pointwise convergence.
As we will see later on, the space V = L 0 is also such an example; see Proposition 5.5 below.

No-arbitrage concepts in topological vector lattices
In this section we introduce no-arbitrage concepts in topological vector lattices. Let (V, ≤) be a topological vector lattice. Furthermore, let K 0 ⊂ V be a subset. We may think of outcomes of trading strategies with initial value zero. Throughout this section, we make the following assumption.
3.1. Assumption. We assume that K 0 is a convex cone.
We define the subset B 0 ⊂ V + as The following auxiliary result is obvious.
3.3. Lemma. The following statements are equivalent:

3.4.
Lemma. The following statements are equivalent: By assumption we have x i = 0 for each i ∈ I, and hence x = 0.

3.5.
Definition. Let τ be a topology on U . We say that K 0 satisfies NFL τ (No Free Lunch with respect to τ ) if 3.6. Proposition. Let τ 1 and τ 2 be two topologies on U such that τ 1 ⊂ τ 2 . If K 0 satisfies NFL τ1 , then it also satisfies NFL τ2 .
Proof. By assumption we have C τ2 ⊂ C τ1 , hence the statement follows. Now, let τ be a topology on U .
Proof. This is an immediate consequence of Lemma 3.4.
3.8. Corollary. Suppose that C is closed in U with respect to τ . Then the following statements are equivalent: Proof. This is an immediate consequence of Lemma 3.4.
3.9. Corollary. Suppose that K 0 − V + is closed in V , and that σ ∩ U ⊂ τ , where σ denotes the topology on V . Then the following statements are equivalent: Consequently, the statement follows from Corollary 3.8. Now, let (K α ) α>0 be a family of subsets of V . We may think of outcomes of trading strategies with initial value α. Throughout this section we make the following assumption.
3.10. Assumption. We assume that Then for each α > 0 the set K α is convex, and the union is a convex cone. We define the family (B α ) α>0 of subsets of V + as We may think of all nonnegative elements which are equal to or below the outcome of a trading strategy with initial value α. By Lemma 2.6 for each α > 0 the set B α is convex and semi-solid. We set B := B 1 .
Consequently, it suffices to concentrate on all outcomes of trading strategies with initial wealth one rather than focusing on all outcomes of trading strategies with positive initial wealth, and for our upcoming no-arbitrage concepts it is enough to focus on the convex subset B.
3.12. Definition. We introduce the following concepts: 3.13. Remark. By Lemma 3.11 the following statements are equivalent: 3.14. Remark. By Lemma 2.8 the subset K 1 satisfies NAA 1 if and only if for each sequence (α n ) n∈N ⊂ (0, ∞) with α n ↓ 0 and every sequence (x n ) n∈N ⊂ V + with x n ∈ B αn for each n ∈ N we have x n → 0.
Hence p B (x) has the interpretation of the minimal superreplication price of x. Thus, K 1 satisfies NA 1 if and only if the superreplication price p B (x) is strictly positive for every strictly positive element x ∈ V + \ {0}.
3.16. Proposition. The following statements are equivalent: If the previous conditions are fulfilled, then K 1 satisfies NA 1 .
Proof. This is a direct consequence of Proposition 2.12.
3.17. Theorem. Suppose that (V, ≤) admits nontrivial minimal elements for unbounded, convex and semi-solid subsets of V + . Then the following statements are equivalent: Proof. This is a consequence of Theorem 2.16. Now, we consider K 0 and (K α ) α>0 together. The following remark provides a sufficient condition ensuring that Assumptions 3.1 and 3.10 are fulfilled.

3.19.
Proposition. The following statements are true: Proof. The first statement is a consequence of the representation (3.2) of the Minkowski functional p B . The second and the third statement are a consequence of Proposition 2.13 and Lemma 3.3.
3.20. Proposition. Let τ be a topology on U such that If K 0 satisfies NFL τ , then K 1 satisfies NA 1 .
Proof. By assumption we have Since the lattice operations are continuous, we obtain

Versions of the abstract fundamental theorem of asset pricing on locally convex spaces
In this section we present versions of the abstract FTAP on locally convex spaces, and later on an abstract FTAP on spaces of continuous functions on compact sets.
As in Section 3, let (V, ≤) be a topological vector lattice, and let K 0 ⊂ L 0 be a subset such that Assumption 3.1 is fulfilled; that is, K 0 is a convex cone. As already mentioned, we may think of the outcomes of trading strategies with initial value zero. Furthermore, let U ⊂ V be an ideal which is dense in V , and let τ be a topology on U . We assume that the topological vector lattice (U, ≤) is locally convex. Recall that the convex cone C ⊂ U is defined as We denote by U ′ the space of all continuous linear functionals with respect to τ .
showing that x ′ separates the sets C and U + . If x ′ is even strictly separating for C , then we have Proof. Let x ∈ U + \ {0} be arbitrary. By (4.1) we have x / ∈ C . Hence, by [1, Cor. 5.84] there exists a continuous linear functional x ′ ∈ U ′ such that x ′ (x) > 0 and x ′ (y) ≤ 0 for all y ∈ C . Let z ∈ U + be arbitrary. By (4.1) we have −z ∈ C , and hence x ′ (z) ≥ 0.

4.4.
Theorem (Abstract FTAP on locally convex spaces). The following statements are equivalent: Noting that C τ is a closed convex cone, by Lemma 4.3 there exists a separating Proof. This is an immediate consequence of Theorem 4.4.
The upcoming notion is inspired by the Halmos-Savage theorem; see, for example [12,Thm. 1.61]. In [15] this condition is called Lindelöf condition.

4.7.
Definition. The locally convex space (U, τ ) has the Halmos-Savage property if for every subset X ′ ⊂ U ′ + \{0} which is strictly positive separating for U + \{0} there is a countable subset Y ′ ⊂ X ′ which is strictly positive separating for U + \ {0}.
The upcoming definition is inspired by [37] Noting that C τ is a closed convex cone, there exists a strictly separating functional Of course, x ′ is also a strictly separating functional for C . 4.11. Corollary. Suppose that the locally convex space (U, τ ) has the Kreps-Yan property. If C is closed in U with respect to τ , then the following statements are equivalent: There exists a strictly separating functional x ′ ∈ U ′ ++ for C . Proof. This is a consequence of Theorem 4.10 and Corollary 3.8.

4.12.
Corollary. Suppose that the locally convex space (U, τ ) has the Kreps-Yan property, and that σ ∩ U ⊂ τ , where σ denotes the topology on V . If K 0 − V + is closed in V , then the following statements are equivalent: (i) K 0 satisfies NA. (ii) There exists a strictly separating functional x ′ ∈ U ′ ++ for C . Proof. This is a consequence of Theorem 4.10 and Corollary 3.9. Now, let us explain the connection to the framework considered in [29]; see also [38]. Suppose that K 0 ⊂ U is a subspace. 1 Let M ⊂ U be a subspace, and let π ∈ M ′ be a τ -continuous linear functional such that ker(π) = K 0 . We define the family (K α ) α>0 of subsets of M as Then we are in the framework of Section 3, and Assumption 3.10 is fulfilled. Indeed, for all a, b ∈ R + , α, β > 0 with aα + bβ > 0 and x ∈ K α , y ∈ K β we have x ′ (ax + by) = ax ′ (x) + bx ′ (y) = aα + bβ, and hence (3.1) is satisfied. We assume that π is positive; that is π(M + ) ⊂ R + .
Since π is positive, this proves the stated equivalence.
Hence, we assume π ∈ M ′ ++ . The market is called viable if there exists a strictly positive functional x ′ ∈ U ′ ++ such that x ′ | M = π. Our notion of NFL τ is equivalent to the notion of no free lunch in [29]. Using our previous results, we obtain the following self-contained proof for the statement of [29, Thm. 2]. 4.14. Theorem. Suppose that the market is viable. Then K 0 satisfies NFL τ .
Proof. By assumption there exists a strictly positive functional x ′ ∈ U ′ ++ such that x ′ | M = π. Since K 0 ⊂ U , we have C = K 0 − U + . Therefore, for each x ∈ C there are y ∈ K 0 and z ∈ U + such that x = y − z, and we obtain showing that x ′ is a separating functional for C . By Corollary 4.5 we deduce that K 0 satisfies NFL τ .

4.15.
Remark. It raises the question when the converse of Theorem 4.14 holds true. Sufficient conditions are provided in [29,Thm. 3]. As a consequence, which is in particular of interest for the upcoming Section 6, market viability and NFL τ are essentially equivalent.
For the rest of this section, let Ω be a compact metric space. We set V := U := C(Ω), and denote by τ the topology induced by the supremum norm. Equipped with this topology, the spaces V and U are Banach spaces. Proof. Let X ′ ⊂ U ′ + \ {0} be a subset which is strictly positive separating for U + \ {0}. By the Riesz representation theorem, for each x ′ ∈ X ′ there exists a regular Borel measure µ x ′ on (Ω, B(Ω)) such that Since x ′ ∈ U ′ + \ {0}, we have µ x ′ ≥ 0 and µ x ′ = 0. Furthermore, since Ω is compact, for each n ∈ N there exist an index m(n) ∈ N and finitely many elements x n,k ∈ Ω, k = 1, . . . , m(n) such that we have the covering where for ǫ > 0 and x ∈ Ω the set We define the countable subset Y ′ ⊂ X ′ as By the continuity of f there exists ǫ > 0 such that Let n ∈ N be such that 1 n ≤ ǫ 4 . There exists k ∈ {1, . . . , m(n)} such that x n,k ∈ B ǫ 2 (x 0 ). Indeed, suppose this is not true, and let k ∈ {1, . . . , m(n)} be arbitrary. Then for each y ∈ B 1 n (x n,k ) we have and hence d(x 0 , y) ≥ ǫ 4 , showing that x 0 / ∈ B 1 n (x n,k ), which contradicts the covering (4.2). Now, we obtain B 1 n (x n,k ) ⊂ B ǫ (x 0 ) because for each y ∈ B 1 n (x n,k ) we have d(y, x 0 ) ≤ d(y, x n,k ) Therefore, we obtain n (x n,k ). Consequently, choosing x ′ n,k ∈ Y ′ , by (4.3) and (4.4) we obtain showing that Y ′ is strictly positive separating for U + \ {0}.
Let G := B(Ω) be the Borel σ-algebra. A probability measure Q on (Ω, G ) is called a separating measure if E Q [X] ≤ 0 for all X ∈ C . Note that in the present situation the convex cone C ⊂ V is given by
Proof. This is a consequence of Lemma 4.16, Proposition 4.9, Theorem 4.10 and Lemma 4.17.

4.19.
Remark. From the perspective of financial modeling we can equip the space (Ω, G ) with a probability measure P. Theorem 4.18 tells us that K 0 satisfies NFL τ if and only if there exists a separating measure Q with full support. Note that this measure is not related to the physical probability measure P; in particular, it does not need to be equivalent to P, as it is the case for classical versions of the FTAP.

The space of random variables
In this section we will consider our abstract no-arbitrage concepts on the space of random variables, and review the concepts which are known in the literature. Let (Ω, G , P) be a probability space. We denote by V = L 0 (Ω, G , P) the space of all equivalence classes of real-valued random variables, in short V = L 0 . Here two random variables X and Y are identified if P(X = Y ) = 1. Furthermore, we write is a topological vector space, and convergence with respect to this metric is convergence in probability; that is, we have d(X n , X) → 0 if and only if X n P → X. For this statement see, for example Exercise A.8.9 on page 450 in [2]. The positive cone of L 0 is denoted by L 0 + . We define the sequence (α n ) n∈N ⊂ (0, ∞) as α n := P(Ω u ) 2 n+1 for each n ∈ N. Then we have α n ↓ 0, and there is a sequence (X n ) n∈N ⊂ B such that P(Ω u ∩ {X n < α −1 n }) < α n for each n ∈ N. We define the sequence (A n ) n∈N ⊂ G as A n := Ω u ∩ {X n ≥ α −1 n }. We set A := n∈N A n ∈ G and define X ∈ L 0 + as X := ½ A . Then we have 0 ≤ X = ½ A ≤ ½ An ≤ α n X n for each n ∈ N.
In particular, since B is semi-solid, we have X ∈ B. Furthermore, we have and hence P(A) > 0, showing that X ∈ B \ {0}.
As in Section 3, let K 0 ⊂ L 0 be a subset such that Assumption 3.1 is fulfilled; that is, K 0 is a convex cone. As already mentioned, we may think of outcomes of trading strategies with initial value zero. Then we can define the concept NA as in Section 3. In order to introduce further concepts in the present setting, we fix some p ∈ [1, ∞]. Note that the space L p is an ideal which is dense in L 0 . 5.6. Definition. We introduce the following concepts: (1) K 0 satisfies NFL p (No Free Lunch with respect to L p ) if it satisfies NFL τ1 , where τ 1 is the weak- * topology on L p with respect to L q and q ∈ [1, ∞] is such that 1 p + 1 q = 1. (2) K 0 satisfies NFLBR p (No Free Lunch with Bounded Risk with respect to L p ) if it satisfies NFL τ2 , where τ 2 is the sequential weak- * topology on L p with respect to L q and q ∈ [1, ∞] is such that 1 p + 1 q = 1. In case p = ∞ we agree to write NFLVR, NFLBR and NFL rather than NFLVR ∞ , NFLBR ∞ and NFL ∞ . These are the well-known no-arbitrage concepts which are widely used in the literature; see for example [9] or [16].
Recall that the convex cone C ⊂ L p is given by Suppose that C is closed in L p with respect to · L p . Then the following statements are equivalent: Proof. This is an immediate consequence of Corollary 3.8.

5.9.
Corollary. Suppose that K 0 −L 0 + is closed in L 0 . Then the following statements are equivalent: Proof. Since X n − X L p → 0 implies X n P → X, this is a consequence of Corollary 3.9. Now, let (K α ) α>0 be a family of subsets of L 0 + such that Assumption 3.10 is fulfilled. As already mentioned, we may think of the outcomes of trading strategies with initial value α. Recall that we had defined the family (B α ) α>0 of convex, semi-solid subsets of L 0 + as B α := (K α − L 0 + ) ∩ L 0 + , α > 0, and that we have set B := B 1 . In Definition 3.12 we had defined the concepts NUPBR, NAA 1 and NA 1 . Lemma 5.4 shows that NUPBR corresponds to the wellknown respective concept that is usually used in the finance literature. By Remark 3.15 the concept NA 1 corresponds to the respective concept that is usually used in the finance literature. The following result shows that also NAA 1 corresponds to the well-known respective concept that is usually used in the finance literature; cf. for example [17]. 5.10. Lemma. The following statements are equivalent: For each sequence (α n ) n∈N ⊂ (0, ∞) with α n ↓ 0 and every sequence (X n ) n∈N ⊂ L 0 + with X n ∈ B αn for each n ∈ N we have X n P → 0.
Proof. Let ǫ > 0 be arbitrary. We set Y n := X n /ǫ and β n := α n /ǫ for each n ∈ N. Then we have β n ↓ 0 and Y n ∈ B βn for each n ∈ N as well as Since ǫ > 0 was arbitrary, this shows X n P → 0.
For the proof of the upcoming Theorem 5.12 we require the following auxiliary result.
5.11. Lemma. Let (X n ) n∈N ⊂ L 0 + be a sequence such that for some ǫ > 0 we have P(X n ≥ n) ≥ ǫ for each n ∈ N.
Then for each subsequence (X n k ) k∈N there exists a sequence (a k ) k∈N ⊂ (0, ∞) with a k → ∞ such that Proof. Let k ∈ N be arbitrary. Then we have P X n l k ≥ n l k ≥ ǫ for all l = 1, . . . , k.
We say that the subset B is L 1 (Q)-bounded for some equivalent probability Furthermore, we say that the convex subset B has the Banach Saks property with respect to almost sure convergence (convergence in probability) if every sequence (X n ) n∈N ⊂ B has a subsequence (X n k ) k∈N which is almost surely Cesàro convergent (Cesàro convergent in probability) to a finite nonnegative random variable X ∈ L 0 + . Similarly, we say that the convex subset B has the permutation invariant (subsequence invariant) von Weizsäcker property with respect to almost sure convergence (convergence in probability) if for every sequence (X n ) n∈N ⊂ B there exist a subsequence (X n k ) k∈N and a finite nonnegative random variable X ∈ L 0 + such that for every permutation π : N → N the sequence (X n π(k) ) k∈N (for every further subsequence (n k l ) l∈N the sequence (X n k l ) l∈N ) is almost surely Cesàro convergent (Cesàro convergent in probability) to X.

5.12.
Theorem. The following statements are equivalent: (v) There exists an equivalent probability measure Q ≈ P such that B is L 1 (Q)bounded. (vi) There exists an equivalent probability measure Q ≈ P with bounded Radon-Nikodym derivative dQ dP such that B is L 1 (Q)-bounded. (i) ⇒ (vii): Let (X n ) n∈N ⊂ B be an arbitrary sequence. By the von Weizsäcker theorem as presented in the original paper [43] there exist a subsequence (X n k ) k∈N and a nonnegative random variable X : Ω → [0, ∞] such that for every every permutation π : N → N the sequence (X n π(k) ) k∈N is almost surely Cesàro convergent to X. Since B is bounded in probability, we have X < ∞ almost surely, that is X ∈ L 0 + ; see, for example [42,Cor. 2.12]. (i) ⇒ (ix): Let (X n ) n∈N ⊂ B be an arbitrary sequence. By the von Weizsäcker theorem as presented in [18,Thm. 5.2.3] there exist a subsequence (X n k ) k∈N and a nonnegative random variable X : Ω → [0, ∞] such that for every further subsequence (n k l ) l∈N the sequence (X n k l ) l∈N is almost surely Cesàro convergent to X. Since B is bounded in probability, we have X < ∞ almost surely, that is X ∈ L 0 + ; see, for example [42,Cor. 2.12]. (xii) ⇒ (i): Suppose that B is not bounded in probability. Then there are ǫ > 0 and a sequence (X n ) n∈N ⊂ B such that P(X n ≥ n) ≥ ǫ for each n ∈ N.
By assumption there exist a subsequence (X n k ) k∈N and a nonnegative random variable X ∈ L 0 + such thatX n k P → X, wherē X n k := 1 k k l=1 X n l for each k ∈ N.
By Lemma 5.11 there exists a sequence (a k ) k∈N ⊂ (0, ∞) with a k → ∞ such that SinceX n k P → X and a k → ∞, there exists an index k 0 ∈ N such that Note that for each k ∈ N we have Therefore, for all k ≥ k 0 we have Since a k → ∞, we obtain P(X = ∞) > 0, which contradicts X ∈ L 0 + . (i) ⇔ (xiii): This equivalence is a consequence of Prohorov's theorem.

5.13.
Remark. If the convex subset B is closed, then it is bounded in probability (which means that K 1 satisfies NUPBR) if and only if it is convexly compact; see [45] for further details. Now, we consider K 0 and (K α ) α>0 together. 5.14. Proposition. The following statements are true: (1) We have α>0 B α = {x ∈ V + : p B (x) = 0}.
Proof. This is an immediate consequence of Proposition 3.19.
Recall that the convex cone C ⊂ L p is given by If K 0 satisfies NFLVR p , then K 1 satisfies NA 1 .
Proof. This is an immediate consequence of Proposition 3.20.

The fundamental theorem of asset pricing on Banach function spaces
In this section we provide a version of the abstract FTAP on Banach function spaces. First, let us briefly recall some properties of Banach function spaces.
Let (Ω, G , µ) be a complete σ-finite measure space. A Banach space U ⊂ V = L 0 = L 0 (Ω, G , µ) is called a Banach function space (or Köthe function space or Banach ideal space) if the following conditions are fulfilled: (1) For all f ∈ L 0 and g ∈ U such that |f | ≤ |g| we have f ∈ U and f U ≤ g U .
(2) We have ½ A ∈ U for all A ∈ G with µ(A) < ∞.
Every Banach function space is a Banach lattice. Furthermore, every order continuous Banach lattice with a weak unit is order isometric to a Banach function space contained in L 0 (Ω, G , P) for some probability space (Ω, G , P); see [32, Prop. 1.b.14] for further details. Note that every Banach function space U is an ideal which is dense in V . Indeed, by the definition above, a Banach function space space U ⊂ V is solid subspace, and it is dense in V because every measurable function can be approximated by a sequence of simple functions. The Köthe dual space U * of a Banach function space U is defined as the space of all g ∈ L 0 such that f g ∈ L 1 for all f ∈ U . The Köthe dual U * equipped with the norm is also a Banach function space. A linear functional x ′ ∈ U ′ is called an integral if there exists g ∈ U * such that x ′ (f ) = Ω f g dµ for all f ∈ U .
A linear functional x ′ ∈ U ′ is called order continuous if x ′ (f n ) → 0 for every sequence (f n ) n∈N ⊂ U such that f n ↓ 0 almost everywhere. We denote by U ′ c ⊂ U ′ the subspace of all order continuous functionals. A linear functional x ′ ∈ U ′ is an integral if and only if it is order continuous; see [32, p. 29]. The linear operator T ∈ L(U * , U ′ c ) given by (T g)f := Ω f g dµ for all g ∈ U * and f ∈ U (6.1) is an isometric isomorphism. 6.1. Lemma. [32, p. 29] We have U ′ = U ′ c if and only if U is σ-order continuous. From now on, let (Ω, G , P) be a complete probability space, and let U ⊂ V = L 0 = L 0 (Ω, G , P) be a Banach function space. Then the isometric isomorphism T ∈ L(U * , U ′ c ) defined in (6.1) is given by Let U * ++ be the set of all Y ∈ U * + such that P(Y > 0) = 1. 6.2. Lemma. Let x ′ ∈ U ′ c and Y ∈ U * be such that x ′ = T Y . Then the following statements are true: Then we have Y ≥ 0, and hence x ′ (X) = E[XY ] ≥ 0 for all X ∈ U + , showing that x ′ ∈ U ′ + . Conversely, assume that x ′ ∈ U ′ + . Then we have E[XY ] = x ′ (X) ≥ 0 for all X ∈ U + . Let A := {Y < 0} ∈ G . We claim that P(A) = 0. Suppose, on the contrary, that P(A) > 0, and set X := ½ A . Then we obtain the contradiction E[XY ] = E[½ A Y ] < 0. Therefore, we have Y ∈ U * + . Now, suppose that Y ∈ U * ++ . Then we have P(Y > 0) = 1. Therefore, for each X ∈ U + \ {0} we obtain XY = 0, and hence We claim that P(A) = 0. Suppose, on the contrary, that P(A) > 0, and set X := ½ A ∈ U + \ {0}. Then we obtain the contradiction E[XY ] = E[½ A Y ] = 0. Therefore, we have Y ∈ U * ++ . 6.3. Lemma. For each continuous linear functional x ′ ∈ U ′ the following statements are equivalent: There exist an equivalent probability measure Q ≈ P on (Ω, G ) with Radon-Nikodym derivative dQ dP ∈ U * ++ and a constant c ∈ (0, ∞) such that we have c ∈ (0, ∞). Since P(Y > 0) = 1, there exists an equivalent probability measure Q ≈ P such that Y c = dQ dP . Therefore, we have dQ dP ∈ U * ++ and (6.2).
showing that x ′ ∈ U ′ c . Furthermore, since Z ∈ U * ++ , by Lemma 6.2 we have x ′ ∈ U ′ ++ . As in Section 3, let K 0 ⊂ L 0 be a subset such that Assumption 3.1 is fulfilled; that is K 0 is a convex cone. Recall that the convex cone C ⊂ U is given by We denote by S (C ) the set of all strictly separating functionals for C ; see Definition 4.2.
6.4. Definition. A probability measure Q on (Ω, G ) is called a separating measure for C if each X ∈ C is quasi-integrable and we have Let τ be the topology induced by the norm of U . 6.5. Theorem (Abstract FTAP on Banach function spaces). The following statements are true: (1) If a separating measure Q ≈ P for C with Radon-Nikodym derivative dQ dP ∈ U * ++ exists, then K 0 satisfies NFL τ . (2) If the topology τ is stronger than the topology induced by convergence in probability, and a separating measure Q ≈ P for C exists, then K 0 satisfies NFL τ . (3) K 0 satisfies NFL τ if and only if S (C ) = ∅, and in this case, the following statements are equivalent: (i) There exists a separating measure Q ≈ P for C with Radon-Nikodym derivative dQ dP ∈ U * ++ .
Proof. According to [37,Thm. 1] the Banach function space U has the Kreps-Yan property. Therefore, statements (1) and (3) are a consequence of Theorem 4.10 and Lemma 6.3. In order to prove statement (2), let X ∈ C τ ∩ U + be arbitrary. Then there exists a sequence (X n ) n∈N ⊂ C such that X n − X U → 0. By assumption we have X n P → X. Hence, there is a subsequence (X n k ) k∈N such that P-almost surely X n k → X for k → ∞. Since Q ≈ P, we also have Q-almost surely X n k → X for k → ∞. Since Q is a separating measure for C , we have E Q [X n ] ≤ 0 for all n ∈ N. By Fatou's lemma we obtain Since X ∈ U + , we deduce that X = 0, proving that K 0 satisfies NFL τ . By Lemma 6.1 those Banach function spaces, on which the FTAP is warranted in its classical form with a separating measure, are exactly σ-order continuous Banach function spaces. More precisely, we have the following result. 6.6. Corollary (Abstract FTAP on σ-order continuous Banach function spaces). Suppose that the Banach function space U is σ-order continuous. Then the following statements are equivalent: There exists a separating measure Q ≈ P for C with Radon-Nikodym derivative dQ dP ∈ U * ++ . If, moreover, the topology τ is stronger than the topology induced by convergence in probability, then the following statements are equivalent: There exists a separating measure Q ≈ P for C . (iii) There exists a separating measure Q ≈ P for C with Radon-Nikodym derivative dQ dP ∈ U * ++ . Proof. By Lemma 6.1 the assumption that U is σ-order continuous just means that U ′ c = U ′ . Therefore, the result is an immediate consequence of Theorem 6.5. As a consequence, we obtain the following result for L p -spaces. Note that for U = L p the convex cone C ⊂ L p is given by Corollary (Abstract FTAP on L p -spaces). Let p ∈ [1, ∞) be arbitrary. If U = L p , then the following statements are equivalent: (i) K 0 satisfies NFLVR p . (ii) There exists a separating measure Q ≈ P for C . (iii) There exists a separating measure Q ≈ P for C with Radon-Nikodym derivative dQ dP ∈ L q ++ , where q ∈ (1, ∞] is such that 1 p + 1 q = 1. Proof. Taking into account the Riesz representation theorem, the Banach function space U = L p is σ-order continuous with Köthe dual space U * = L q . Furthermore, the topology induced by · L p is stronger than the topology induced by convergence in probability. Hence, the statement is a consequence of Corollary 6.6. 6.8. Remark. Note that Corollary 6.7 is in accordance with [38,Thm. 1.4], which is a reformulation of [29,Thm. 3] for L p -spaces. 6.9. Corollary. Suppose that C is closed in L p with respect to · L p for some p ∈ [1, ∞). Then the following statements are equivalent: (i) K 0 satisfies NA. (ii) There exists a separating measure Q ≈ P for C . (iii) There exists a separating measure Q ≈ P for C with Radon-Nikodym derivative dQ dP ∈ L q ++ , where q ∈ (1, ∞] is such that 1 p + 1 q = 1.
Proof. This is a consequence of Corollaries 5.8 and 6.7.
In the following result we consider U = L 1 . Then the convex cone C ⊂ L 1 is given by Then the following statements are equivalent: (i) K 0 satisfies NA. (ii) There exists a separating measure Q ≈ P for C . (iii) There exists a separating measure Q ≈ P for C with bounded Radon-Nikodym derivative dQ dP ∈ L ∞ ++ . Proof. This is a consequence of Corollaries 5.9 and 6.7 with p = 1 and q = ∞. Now, let us consider some examples of Banach function spaces which are not σ-order continuous, and where, as a consequence, not every continuous linear functional is an integral.
6.11. Example (Space of bounded functions). Let U = L ∞ . By the Yosida-Hewitt theorem (see [44]) every continuous linear functional x ′ ∈ (L ∞ ) ′ is of the form with Y ∈ L 1 and an absolutely continuous finitely additive signed measure µ ≪ P on (Ω, G ) which is singular in the sense that for some sequence (A k ) k∈N ⊂ G we have µ(A k ) ↓ 0 for k → ∞ and µ(Ω \ A k ) = 0 for all k ∈ N.
with Y ∈ LĀ and an absolutely continuous finitely additive signed measure µ ≪ P such that for some f ∈ N A with f ≥ 0 and f NA ≤ 1 we have the property that for all B ∈ G with |µ|(B) = 0 it follows that f ½ B NA = 1.
6.13. Remark. Let us have a closer look at the situation U = L ∞ . Suppose that the convex cone K 0 satisfies NFLVR. By Theorem 6.5 we have S (C ) = ∅. However, it is not clear whether a separating measure exists because continuous linear functionals are generally of the form (6.3). Now, assume that the convex cone K 0 has a structure as in [16]. Then, surprisingly, a separating measure Q ≈ P for C exists, see [16, Thm. 1.1 and Thm. 1.2]; and hence, by Theorem 6.5 we even have For the rest of this section, we consider the discrete time setting. Let (Ω, F , F, P) be a filtered probability space with discrete filtration F = (F t ) t=0,...,T for some T ∈ N. Let X = {X 1 , . . . , X d } be a discounted market consisting of d ∈ N assets X i = (X i t ) t=0,...,T for i = 1, . . . , d. We assume that X i ≥ 0 for each i = 1, . . . , d. Consider the convex cone where every predictable process ξ (that is ξ t is F t−1 -measurable for each t = 1, . . . , T ) is called a strategy. As usual, we say that an equivalent probability measure Q ≈ P is an equivalent martingale measure (EMM) for X if X 1 , . . . , X d are Q-martingales. The following result extends the well-known no-arbitrage result in discrete time (see, for example [12] or [19]) by additionally providing a characterization in terms of separating measures. Now, the convex cone C ⊂ L 1 is given by 14. Theorem (FTAP for discrete time models). The following statements are equivalent: There exists an EMM Q ≈ P for X. (iii) There exists an EMM Q ≈ P for X with bounded Radon-Nikodym derivative dQ dP ∈ L ∞ ++ . (iv) There exists a separating measure Q ≈ P for C .
(v) There exists a separating measure Q ≈ P for C with bounded Radon-Nikodym derivative dQ dP ∈ L ∞ ++ . Proof.

Financial market with semimartingales
In this section we consider a financial market with nonnegative semimartingales which does not need to have a numéraire. We will derive consequences for the no-arbitrage concepts considered so far; in particular, regarding self-financing portfolios.
Let (Ω, F , (F t ) t∈R+ , P) be a stochastic basis satisfying the usual conditions, see [14,Def. I.1.3]. Furthermore, we assume that F 0 is P-trivial. Then every F 0measurable random variable is P-almost surely constant. Let L be the space of all equivalence classes of adapted, càdlàg processes X : Ω × R + → R, where two processes X and Y are identified if X and Y are indistinguishable, that is if almost all paths of X and Y coincide; see [14, I.1.10]. Let (K α ) α≥0 be a family of subsets of L such that for each α ≥ 0 and each X ∈ K α we have X 0 = α. Throughout this section, we make the following assumptions. 7.1. Assumption. We assume that K 0 is a convex cone.
The following remark provides a sufficient condition ensuring that Assumptions 7.1 and 7.2 are fulfilled.
The following example shows that the framework considered in [17, Appendix A] is contained in our present setting. 7.4. Example. Let X ⊂ L be a convex set of processes such that X 0 = 0 and X ≥ −1 for each X ∈ X. We define the family (K α ) α≥0 as Then Assumptions 7.1 and 7.2 are fulfilled. Indeed, the set K 0 is a convex cone. Let a, b ∈ R + , α, β > 0 with aα + bβ > 0 and X ∈ K α , Y ∈ K β be arbitrary. Then there are Z, W ∈ X such that X = α(1 + Z) and Y = β(1 + W ). Since X is convex, we obtain showing that (7.1) is fulfilled.
As we will see, in all examples which we consider in this section later on, relation (7.2) from Remark 7.3 will be satisfied. Now, let T ∈ (0, ∞) be a deterministic terminal time. We define the family (K α ) α≥0 of subsets of L 0 = L 0 (Ω, F T , P) as Then we are in the framework of Section 5. The next result shows that Assumptions 3.1 and 3.10 are fulfilled. 7.5. Lemma. The following statements are true: (1) K 0 is a convex cone.
As in Section 3, we define the family (B α ) α≥0 of convex, semi-solid subsets of L 0 + as B α := (K α − L 0 + ) ∩ L 0 + , α ≥ 0, and we set B := B 1 . Furthermore, we define the convex cone C ⊂ L ∞ as C := (K 0 − L 0 + ) ∩ L ∞ . Now, we consider particular examples for the family of processes (K α ) α≥0 . Let I = ∅ be an arbitrary index set, and let (S i ) i∈I be a family of semimartingales.
We assume that S i ≥ 0 for each i ∈ I. We define the market S := {S i : i ∈ I}. For an R d -valued semimartingale X we denote by L(X) the set of all X-integrable processes in the sense of vector integration; see [40] or [14,Sec. III.6]. For δ ∈ L(X) we denote by δ · X the stochastic integral according to [40]. For a finite set F ⊂ I we define the multi-dimensional semimartingale S F := (S i ) i∈F . 7.6. Definition. We call a process δ = (δ i ) i∈I a strategy for S if the set F ⊂ I given by F := {i ∈ I : δ i = 0} is finite and we have δ F ∈ L(S F ). 7.7. Definition. We denote by ∆(S) the set of all strategies δ for S. 7.8. Definition. For a strategy δ ∈ ∆(S) we set δ · S := δ F · S F , where F ⊂ I denotes the finite set from Definition 7.6. 7.9. Theorem. [40,Thm. 4.3] Let δ 1 , δ 2 ∈ ∆(S) and α 1 , α 2 ∈ R be arbitrary. Then we have 7.10. Definition. For α ∈ R and a strategy δ ∈ ∆(S) we define the integral process I α,δ := α + δ · S. 7.11. Definition. For a strategy δ ∈ ∆(S) we define the portfolio S δ := δ · S, where we use the short-hand notation δ · S := i∈F δ i S i with F ⊂ I denoting the finite set from Definition 7.6. 7.12. Definition. A strategy δ ∈ ∆(S) and the corresponding portfolio S δ are called self-financing for S if S δ = S δ 0 + δ · S. 7.13. Definition. We denote by ∆ sf (S) the set of all self-financing strategies for S.
The following auxiliary result is obvious. 7.14. Lemma. For a strategy δ ∈ ∆(S) the following statements are equivalent: Recall that a process X is called admissible if X ≥ −a for some constant a ∈ R + . 7.15. Definition. We introduce the following families: (1) We define the family of all integral processes (I α (S)) α≥0 as I α (S) := {I α,δ : δ ∈ ∆(S)}, α ≥ 0.

7.16.
Remark. Consider the particular case where S i ≡ 1 for some i ∈ I. Then the market S can be interpreted as discounted price processes of risky assets with respect to some savings account, and the families (I α (S)) α≥0 , (I adm α (S)) α≥0 , (I + α (S)) α≥0 can be regarded as wealth processes in this case.
For each i ∈ I we denote by e i ∈ ∆(S) the strategy with components (1) For each i ∈ I we have e i ∈ ∆ sf (S).
Proof. This is a consequence of Proposition 5.15 and Lemma 7.23.
We emphasize that the previous results are proven in a rather straightforward manner, only relying on our previous results about topological vector lattices and well-known results from stochastic analysis.