Superreplication under Volatility Uncertainty for Measurable Claims

We establish the duality-formula for the superreplication price in a setting of volatility uncertainty which includes the example of"random G-expectation."In contrast to previous results, the contingent claim is not assumed to be quasi-continuous.


Introduction
This note is concerned with superreplication-pricing in a setting of volatility uncertainty. We see the canonical process B on the space Ω of continuous paths as the stock price process and formalize this uncertainty via a set P of (non-equivalent) martingale laws on Ω. Given a contingent claim ξ measurable at time T > 0, we are interested in determining the minimal initial capital x ∈ R for which there exists a trading strategy H whose terminal gain x + T 0 H u dB u exceeds ξ P -a.s., simultaneously for all P ∈ P. The aim is to show that under suitable assumptions, this minimal capital is given by x = sup P ∈P E P [ξ]. We prove this duality formula for Borelmeasurable (and, more generally, upper semianalytic) claims ξ and a model P where the possible values of the volatility are determined by a set-valued process. Such a model of a "random G-expectation" was first introduced in [9], as an extension of the "G-expectation" of [14,15].
The duality formula under volatility uncertainty has been established for several cases and through different approaches: [6] used ideas from capacity theory, [16,20,22] used an approximation by Markovian control problems, and [19,12] used a method discussed below. The main difference between our results and the previous ones is that we do not impose continuity assumptions on the claim ξ (as a function of ω ∈ Ω), so that we retrieve the level of generality that is standard in Mathematical Finance.
The main difficulty in our endeavor is to construct the superreplicating strategy H. We adopt the approach of [19] and [12], which can be outlined as follows: (i) Construct the conditional (nonlinear) expectation E t (ξ) related to P and show the tower property E s (E t (ξ)) = E s (ξ) for s ≤ t.
(ii) Check that the right limit Y t := E t+ (ξ) exists and defines a supermartingale under each P ∈ P (in a suitable filtration).
(iii) For every P ∈ P, show that the martingale part in the Doob-Meyer decomposition of Y is of the form H P dB. Using that H P can be expressed via the quadratic (co)variation processes of Y and B, deduce that there exists a universal process H coinciding with H P under each P , and check that H is the desired strategy.
Step (iii) can be accomplished by ensuring that each P ∈ P has the predictable representation property. To this end-and for some details of Step (ii) that we shall skip for the moment- [19] introduced the set of Brownian martingale laws with positive volatility, which we shall denote by P S : if P is chosen as a subset of P S , then every P ∈ P has the representation property (cf. Lemma 4.1) and Step (iii) is feasible.
Step (i) is the main reason why previous results required continuity assumptions on ξ. Recently, it was shown in [13] that the theory of analytic sets can be used to carry out Step (i) when ξ is merely Borel-measurable (or only upper semianalytic), provided that the set of measures satisfies a condition of measurability and invariance, called Condition (A) below (cf. Proposition 2.2). It was also shown in [13] that this condition is satisfied for a model of random G-expectation where the measures are chosen from the set of all (not necessarily Brownian) martingale laws. Thus, to follow the approach outlined above, we formulate a similar model using elements of P S and show that Condition (A) is again satisfied. This essentially boils down to proving that the set P S itself satisfies Condition (A), which is our main technical result (Theorem 2.4). Using this fact, we can go through the approach outlined above and establish our duality result (Theorem 2.3 and Corollary 2.6). As an aside of independent interest, Theorem 2.4 yields a rigorous proof for a dynamic programming principle with a fairly general reward functional (cf. Remark 2.7).
The remainder of this paper is organized as follows. In Section 2, we first describe our setup and notation in detail and we recall the relevant facts from [13]; then, we state our main results. Theorem 2.4 is proved in Section 3, and Section 4 concludes with the proof of Theorem 2.3.

Notation
We fix the dimension d ∈ N and let Ω = {ω ∈ C([0, ∞); R d ) : ω 0 = 0} be the canonical space of continuous paths equipped with the topology of locally uniform convergence. Moreover, let F = B(Ω) be its Borel σ-algebra. We denote by B := (B t ) t≥0 the canonical process B t (ω) = ω t , by P 0 the Wiener measure and by F := (F t ) t≥0 the (raw) filtration generated by B. Furthermore, we denote by P(Ω) the set of all probability measures on Ω, equipped with the topology of weak convergence.
We recall that a subset of a Polish space is called analytic if it is the image of a Borel subset of another Polish space under a Borel map. Moreover, an Rvalued function f is called upper semianalytic if {f > c} is analytic for each c ∈ R; in particular, any Borel-measurable function is upper semianalytic. (See [1, Chapter 7] for background.) Finally, the universal completion of a σ-field A is given by A * := ∩ P A P , where A P denotes the completion with respect to P and the intersection is taken over all probability measures on A.

Preliminaries
We formalize volatility uncertainty via a set of local martingale laws with different volatilities. To this end, we denote by S the set of all symmetric d × d-matrices and by S >0 its subset of strictly positive definite matrices. The set P S ⊂ P(Ω) consists of all local martingale laws of the form where α ranges over all F-progressively measurable processes with values in S >0 satisfying T 0 |α s | ds < ∞ P 0 -a.s. for every T ∈ R + . (We denote by | · | the Euclidean norm in any dimension.) In other words, these are all laws of stochastic integrals of a Brownian motion, where the integrand is strictly positive and adapted to the Brownian motion. The set P S was introduced in [19] and its elements have several nice properties; in particular, they have the predictable representation property which plays an important role in the proof of the duality result below (see also Section 4). We intend to model "uncertainty" via a subset P ⊂ P(Ω) (below, each P ∈ P will be a possible scenario for the volatility). However, for technical reasons, we make a detour and consider an entire family of subsets of P(Ω), indexed by (s, ω) ∈ R + × Ω, whose elements at s = 0 coincide with P. As illustrated in Example 2.1 below, this family is of purely auxiliary nature.
We remark that (A2) and (A3) imply that the family {P(s, ω)} is essentially determined by the set P. As mentioned above, in applications, P will be the primary object and we shall simply write down a corresponding family {P(s, ω)} such that P = P(0, ω) and such that Condition (A) is satisfied. To illustrate this, let us state a model where the possible values of the volatility are described by a set-valued process D and which will be the main application of our results. This model was first introduced in [9] and further studied in [13]; it generalizes the G-expectation of [14,15] which corresponds to the case where D is a (deterministic) compact convex set.
We consider a set-valued process D : Ω × R + → 2 S ; i.e., D t (ω) is a set of matrices for each (t, ω). We assume that D is progressively graph-measurable: for every t ∈ R + , We want a set P consisting of all P ∈ P S under which the volatility takes values in D P -a.s. To this end, we introduce the auxiliary family {P(s, ω)}: given (s, ω) ∈ R + × Ω, we define P(s, ω) to be the collection of all P ∈ P S such that In particular, P := P(0, ω) then consists, as desired, of all P ∈ P S such that d B u /du ∈ D u holds P × du-a.e. We shall see in Corollary 2.6 that Condition (A) is satisfied in this example.
The following is the main result of [13]; it is stated with the conventions sup ∅ = −∞ and E P [ξ] := −∞ if E P [ξ + ] = E P [ξ − ] = +∞, and ess sup P denotes the essential supremum under P . Proposition 2.2. Suppose that {P(s, ω)} satisfies Condition (A) and that P = ∅. Let σ ≤ τ be stopping times and let ξ : Ω → R be an upper semianalytic function. Then the function s. for all P ∈ P. (2.5)

Main Results
Some more notation is needed to state our duality result. In what follows, the set P determined by the family {P(s, ω)} will be a subset of P S . We shall use a finite time horizon T ∈ R + and the filtration G = (G t ) 0≤t≤T , where G t := F * t+ ∨ N P for t < T and G T := F * T ∨ N P . Here F * t+ is the right limit of the universal completion of F t and N P is the collection of sets which are (F T , P )-null for all P ∈ P. We remark that when P ⊂ P S , the filtration G is not much larger that the original filtration F, in the sense that G is contained in the P -augmentation of F for any P ∈ P; cf. Lemma 4.1.
Let H be a G-predictable process taking values in R d and such that s. for all P ∈ P. Then H is called an admissible trading strategy if 1 H dB is a P -supermartingale for all P ∈ P; as usual, this is to rule out doubling strategies. We denote by H the set of all admissible trading strategies.
The assumption that P ⊂ P S will be essential for our proof, which is stated in Section 4. In order to have nontrivial examples where the previous theorem applies, it is essential to know that the set P S (seen as a constant family P(s, ω) ≡ P S ) satisfies itself Condition (A). This fact is our main technical result.
The proof is stated Section 3. It is easy to see that if two families satisfy Condition (A), then so does their intersection. In particular, we have: The following is the main application of our results. Remark 2.7. In view of (2.4), Theorem 2.4 yields the dynamic programming principle for the optimal control problem sup α E P 0 [ξ(X α )] with a very general reward functional ξ, where X α = · 0 α 1/2 s dB s . We remark that the arguments in the proof of Theorem 2.4 could be extended to other control problems; for instance, the situation where the state process X α is defined as the solution of a stochastic functional/differential equation as in [11].

Proof of Theorem 2.4
In this section, we prove that P S (i.e., the constant family P(s, ω) ≡ P S ) satisfies Condition (A). Up to some minor differences in notation, property (A2) was already shown in [19,Lemma 4.1], so we focus on (A1) and (A3).
Let us fix some more notation. We denote by E[ · ] the expectation under P 0 ; more generally, any notion related to Ω that implicitly refers to a measure will be understood to refer to the Wiener measure P 0 . Unless otherwise stated, any topological space is endowed with its Borel σ-field. As usual, L 0 (Ω; R) denotes the set of equivalence classes of random variables on Ω, endowed with the topology of convergence in measure. Moreover, we denote by Ω×R + the product space; here the measure is P 0 ×dt by default, where dt is the Lebesgue measure. The basic space in this section is L 0 (Ω; S), the set of equivalence classes of S-valued processes that are product-measurable. We endow L 0 (Ω; S) (and its subspaces) with the topology of local convergence in measure; that is, the metric

Proof of (A1)
The aim of this subsection is to show that P S ⊂ P(Ω) is analytic. To this end, we shall show that P S ⊂ P(Ω) is the image of a Borel space (i.e., a Borel subset of a Polish space) under a Borel map; this implies the claim by [1, Proposition 7.40, p. 165]. Indeed, let L 0 prog (Ω; S) ⊂ L 0 (Ω; S) be the subset of F-progressively measurable processes and Moreover, we denote by the map which associates to α the corresponding law. Then P S is the image therefore, the claimed property (A1) follows from the two subsequent lemmas.
Proof. We start by noting that L 0 (Ω; S) is Polish. Indeed, as R + and Ω are separable metric spaces, we have that L 2 (Ω × [0, T ]; S) is separable for all T ∈ N (e.g., [7, Section 6.15, p. 92]). A density argument and the definition (3.1) then show that L 0 (Ω; S) is again separable. On the other hand, the completeness of S is inherited by L 0 (Ω; S); see, e.g., [3,Corollary 3]. Since L 0 prog (Ω; S) ⊂ L 0 (Ω; S) is closed, it is again a Polish space. Next, we show that L 1 loc (Ω; S) is a Borel subset of L 0 prog (Ω; S). We first observe that Therefore, it suffices to show that for fixed T ∈ N, is Borel. Indeed, this is the composition of the function The latter is Borel because it is, by monotone convergence, the pointwise limit of the maps which are continuous for fixed n ∈ N due to the elementary estimate This completes the proof that L 1 loc (Ω; S) is a Borel subset of L 0 prog (Ω; S). To deduce the same property for L 1 loc (Ω; S >0 ), note that is upper semicontinuous and we conclude that Proof. Consider first, for fixed n ∈ N, the mapping Φ n defined by where π n is the projection onto the ball of radius n around the origin in S. It follows from a direct extension of the dominated convergence theorem for stochastic integrals [17, Theorem IV.32, p. 176] that α → · 0 π n (α 1/2 s ) dB s is continuous for the topology of uniform convergence on compacts in probability ("ucp"), and hence that Φ n is continuous for the topology of weak convergence. In particular, Φ n is Borel. On the other hand, a second application of dominated convergence shows that · 0 π n (α 1/2 s ) dB s → · 0 α 1/2 s dB s ucp as n → ∞ for all α ∈ L 1 loc (Ω; S >0 ) and hence that Φ(α) = lim n Φ n (α) in P(Ω) for all α. Therefore, Φ is again Borel.

Proof of (A3)
Given a stopping time τ , a measure P ∈ P S and an F τ -measurable kernel ν : Ω → P(Ω) with ν(ω) ∈ P S for P -a.e. ω ∈ Ω, our aim is to show that defines an element of P S . That is, we need to show thatP = Pᾱ for somē α ∈ L 1 loc (Ω; S >0 ). In the case where ν has only countably many values, this can be accomplished by explicitly writing down an appropriate processᾱ, as was shown already in [19]. The present setup requires general kernels and a measurable selection proof. Roughly speaking, up to time τ ,ᾱ is given by the integrand α determining P , whereas after τ , it is given by the integrand of ν(ω), for a suitable ω. In Step 1 below, we state precisely the requirement forᾱ; in Step 2, we construct a measurable selector for the integrand of ν(ω); finally, in Step 3, we show how to construct the required processᾱ from this selector.
To this end, consider the set We have seen in Lemma 3.1 that L 1 loc (Ω; S >0 ) is a Borel space. On the other hand, we have from Lemma 3.2 that α → P α is Borel, and ν is Borel by assumption. Hence, A is a Borel subset of Ω × L 1 loc (Ω; S >0 ). As a result, we can use the Jankov-von Neumann theorem [1, Proposition 7.49, p. 182] to obtain an analytically measurable map φ from the Ω-projection of A to L 1 loc (Ω; S >0 ) whose graph is contained in A; that is, Since φ is, in particular, universally measurable, and since ν(·) ∈ P S P -a.s., we can alter φ on a P -nullset to obtain a Borel-measurable map φ : Ω → L 1 loc (Ω; S >0 ) such that P φ(·) = ν(·) P -a.s.
Step 3. It remains to constructα ∈ L 0 prog (Ω; S) as postulated in Step 1. While the map φ constructed in Step 2 will eventually yield the processesα ω defined in Step 1, we note that φ is a map into a space of processes and so we still have to glue its values into an actual process. This is simple when there are only countably many values; therefore, following a construction of [18], we use an approximation argument.

Proof of Theorem 2.3
We note that one inequality in Theorem 2.3 is trivial: if x ∈ R and there exists H ∈ H such that x + T 0 H dB ≥ ξ, the supermartingale property stated in the definition of H implies that x ≥ E P [ξ] for all P ∈ P. Hence, our aim in this section is to show that there exists H ∈ H such that sup P ∈P E P [ξ] + T 0 H u dB u ≥ ξ P -a.s. for all P ∈ P. (4.1) The line of argument (see also the Introduction) is similar as in [19] or [12]; hence, we shall be brief. We first recall the following known result (e.g., [ We recall our assumption that sup P ∈P E P [|ξ|] < ∞ and that ξ is G Tmeasurable. We also recall from Proposition 2.2 that the random variable is G t -measurable for all t ∈ R + . Moreover, we note that E T (ξ) = ξ P -a.s. for all P ∈ P. Indeed, for any fixed P ∈ P, Lemma 4.1 implies that we can find an F T -measurable function ξ ′ which is equal to ξ outside a P -nullset N ∈ F T , and now the definition of E T (ξ) and Galmarino's test show that Step 1. We fix t and show that sup P ∈P E P [|E t (ξ)|] < ∞. Note that |ξ| need not be upper semianalytic, so that the claim does not follows directly from (2.4). Hence, we make a small detour and first observe that P is stable in the following sense: if P ∈ P, Λ ∈ F t and P 1 , P 2 ∈ P(t; P ) (notation from Proposition 2.2), the measureP defined bȳ is again an element of P. Indeed, this follows from (A2) and (A3) as for the kernel ν(ω, dω ′ ) = P t,ω 1 (dω ′ ) 1 Λ (ω) + P t,ω 2 (dω ′ ) 1 Λ c (ω). Following a standard argument, this stability implies that for any P ∈ P, there exist P n ∈ P(t; P ) such that E Pn [|ξ| |F t ] ր ess sup P P ′ ∈P(t;P ) E P ′ [|ξ| |F t ] P -a.s. Since (2.5), applied with τ = T , yields that E P [|E t (ξ)|] = E P ess sup P P ′ ∈P(t;P ) E P ′ [ξ|F t ] ≤ E P ess sup P P ′ ∈P(t;P ) monotone convergence then allows us to conclude that Step 2. We show that the right limit Y t := E t+ (ξ) defines a (G, P )supermartingale for all P ∈ P. Indeed, Step 1 and (2.5) show that E t (ξ) is an (F * , P )-supermartingale for all P ∈ P. The standard modification theorem for supermartingales [5,Theorem VI.2] then yields that Y is well defined P -a.s. and that Y is a (G, P )-supermartingale for all P ∈ P, where the second conclusion uses Lemma 4.1. We omit the details; they are similar as in the proof of [12,Proposition 4.5].
For later use, let us also establish the inequality Y 0 ≤ sup P ′ ∈P E P ′ [ξ] P -a.s. for all P ∈ P. (4.2) Indeed, let P ∈ P. Then [5,Theorem VI.2] shows that where, of course, we have E P [Y 0 |F 0 ] = E P [Y 0 ] P -a.s. since F 0 = {∅, Ω}. However, as Y 0 is G 0 -measurable and G 0 is P -a.s. trivial by Lemma 4.1, we also have that Y 0 = E P [Y 0 ] P -a.s. In view of the definition of E 0 (ξ), the inequality (4.2) follows.
Step 3. Next, we construct the process H ∈ H. In view of Step 2, we can fix P ∈ P and consider the Doob-Meyer decomposition Y = Y 0 + M P − K P under P , in the filtration F P . By Lemma 4.1, the local martingale M P can be represented as an integral, M P = H P dB, for some F P -predictable integrand H P . The crucial observation (due to [19]) is that this process can be described via d Y, B = H P d B , and that, as the quadratic covariation processes can be constructed pathwise by Bichteler's integral [2, Theorem 7.14], this relation allows to define a process H such that H = H P P × dt-a.e. for all P ∈ P. More precisely, since Y is G-adapted, we see by going through the arguments in the proof of [12,Proposition 4.11] that H can be obtained as a G-predictable process in our setting. To conclude that H ∈ H, note that for every P ∈ P, the local martingale H dB is P -a.s. bounded from below by the martingale E P [ξ|G]; hence, on the compact [0, T ], it is a supermartingale as a consequence of Fatou's lemma. Summing up, we have found H ∈ H such that Y 0 + T 0 H u dB u ≥ Y T = E T + (ξ) = ξ P -a.s. for all P ∈ P, and in view of (4.2), this implies (4.1).