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Separability Vs. Robustness of Orlicz Spaces: Financial and Economic Perspectives

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Separability Vs. Robustness of Orlicz Spaces: Financial and Economic Perspectives SEPARABILITY VS. ROBUSTNESS OF ORLICZ SPACES: FINANCIAL AND ECONOMIC PERSPECTIVES FELIX-BENEDIKT LIEBRICH AND MAX NENDEL Abstract. We investigate robust Orlicz spaces as a generalisation of robust Lp-spaces. Two construc- tions of such spaces are distinguished, a top-down approach and a bottom-up approach. We show that separability of robust Orlicz spaces or their subspaces has very strong implications in terms of the domi- natedness of the set of priors and the lack of order completeness. Our results have subtle implications for the field of robust finance. For instance, norm closures of bounded continuous functions with respect to the worst-case Lp-norm, as considered in the G-framework, lead to spaces which are lattice isomorphic to a sublattice of a classical L1-space lacking, however, any form of order completeness. We further show that the topological spanning power of options is always limited under nondominated uncertainty. Keywords: Orlicz space, model uncertainty, nonlinear expectation, capacity, G-Framework, Dedekind completeness AMS 2020 Subject Classification: 46E30; 28A12; 60G65 1. Introduction Since the beginning of this century, the simultaneous consideration of families of prior distributions instead of a single probability measure has become of fundamental importance for the risk assessment of financial positions. In this context, one often speaks of model uncertainty or ambiguity, where the uncertainty is modeled by a set P of probability measures. Especially after the subprime mortgage crisis, the desire for mathematical models based on nondominated families of priors arose: no single reference probability measure can be chosen which determines whether an event is deemed certain or negligible. Beside the seminal contribution of [9], the to date most prominent example of a model involving nondominated uncertainty is a Brownian motion with uncertain volatility, the so-called G- Brownian motion, cf. [37].1 The latter is intimately related to the theory of second-order backward stochastic differential equations, cf. [13]. An extensive strand of literature has formed around this model. On another note, there has been renewed interest in the role of Orlicz spaces in mathematical finance in the past few years. They have appeared, e.g., as canonical model spaces for risk measures, premium principles, and utility maximisation problems, see [5, 6, 12, 20] and many others. The present manuscript investigates robust Orlicz spaces in a setting of potentially nondominated sets P of probability measures and the role of separability in financial and economic applications. A priori, the construction of robust Orlicz spaces can follow two conceivable paths which often lead to the same p result if a probability space is underlying, i.e., P = fPg. The classical L (P)-space, for p 2 [1; 1), shall serve as illustration. It can be obtained either by a top-down approach, considering the maximal set p 1=p of all equivalence classes of real-valued measurable functions with finite norm k · kLp(P) := EP[j · j ] ; or, equivalently, by proceeding in a bottom-up manner, closing a smaller test space of, say, bounded random variables w.r.t. the norm k · kLp(P). If the underlying state space is topological, one has even more degrees of freedom and may select suitable spaces of continuous functions as a test space. Both approaches lead to a Banach lattice which naturally carries the P-almost-sure order, and this turns out The authors thank Julian H¨olzermann, Fabio Maccheroni, Frank Riedel, and Johannes Wiesel for helpful comments and suggestions related to this work. The second author gratefully acknowledges financial support of the German Research Foundation via CRC 1283. 1 More precisely, the resulting measures are mutually singular. 1 2 FELIX-BENEDIKT LIEBRICH AND MAX NENDEL p to be L (P) in both cases. Morally speaking, the reason for this equivalence is that k · kLp(P) is not very robust and rather insensitive to the tail behaviour of a given random variable. In general, the two paths tend to diverge substantially for (robust) Orlicz spaces, while both turn out to have their economic merits. The present note may be understood as a comparison of the two approaches against the backdrop of financial applications. Throughout, we consider a fixed measurable space (Ω; F), a nonempty set of probability measures (priors) P on (Ω; F), and a family Φ = (φP)P2P of Orlicz functions, possibly varying with the prior. On the quotient space L0(P) of all real-valued measurable functions on (Ω; F) up to P-quasi-sure (P-q.s.) equality, we consider the robust Luxemburg norm 0 kXk Φ := sup kXk φ 2 [0; 1]; for X 2 L (P); (1.1) L (P) L P (P) P2P where k · k φ denotes the Luxemburg seminorm for φ under the probability measure 2 P. L P (P) P P The (top-down) robust Orlicz space LΦ(P) is then defined to be the space of all X 2 L0(P) with kXkLΦ(P) < 1. In the field of robust finance, special cases of robust Orlicz spaces have been studied, e.g., in the G-Framework [35, 46] and over general measurable spaces [21, 26, 27, 30, 36]. However, defining Lp- spaces by a worst-case approach over a set of priors instead of a single prior stems back at least to [40]. Noteworthy are [42, 43], where a general Orlicz function is considered instead of the Lp-case. We essentially work in their framework. A more detailed discussion of these references is provided in Remark 3.7. Another top-down approach to robust Orlicz spaces that appears in the literature { cf. [34, 45] and [11, Example 2.6] { is given by the space Φ 0 L (P) := X 2 L (P) 8 P 2 P 9 α > 0 : EP[φP(αjXj)] < 1 0 = X 2 L (P) 8 2 P : kXk φ < 1 ; P L P (P) the \intersection" of the individual Orlicz spaces. We discuss this construction in Section 3.2 and show in Proposition 3.12 and Theorem 3.13 that, in many situations, LΦ(P) = LΦ(P): These uniform-boundedness-type results prove the equivalence of both constructions in terms of the extension of the resulting spaces in L0(P). 2. Main results and related literature For better orientation of the reader, we summarise below the implications of our results for the field of robust finance and discuss further related literature. A more detailed discussion can be found in Section 5. 2.1. Function spaces in the G-Framework. In order to combine analytic tractability and uncer- tainty modelled by nondominated priors, a rich strand of literature has pursued the bottom-up con- struction of robust Orlicz spaces (though not in the generality of our definition above). For G-Brownian p p motion, in particular, closures C of the space Cb of bounded continuous functions under robust L - norms k · kLp(P) for a nonempty set of priors P have become a frequent choice for commodity spaces or spaces of contingent claims, cf. [4, 16, 24]. We shall cover such closures in Section 5.1. The analytic properties of these spaces have been studied in, e.g., [3, 15], and a complete stochastic calculus has been developed for them, cf. [38]. There are also preliminary studies of their order structure against the backdrop of financial applications, cf. [7] and the discussion in [14]. Apart from this, little is known about Cp as Banach lattice. We shall fill this gap and prove in Section 5.1 that Cp is usually separable. SEPARABILITY VS. ROBUSTNESS OF ORLICZ SPACES 3 As such, it illustrates a general conflict between separability and nondominatedness of the underlying priors { or robustness of the function space in question { that the present paper aims to explore. In that regard, we would like to draw attention to two results of our discussion: Theorem 3.8 states that every separable subspace of a robust Orlicz space is order isomorphic to a subspace of a classical L1-space over the same underlying measurable space. As a consequence, its elements are dominated by a single probability measure, and the P-q.s. order collapses to an almost sure order { even for nondominated sets P of priors. We thus identify separability to be a property that leads to a collapse of quasi-sure orders, thereby providing a generalisation of and a different angle for the result of [7]. Hence, even though separable spaces appear naturally in financial applications (see also Section 2.2 below), it seems questionable in which sense they are actually \robust". In a similar spirit, our second main result concerns the Dedekind σ-completeness of sublattices of robust Orlicz spaces, i.e., the existence of least upper bounds for countable bounded subsets. Dedekind σ-completeness is necessary for the existence of \essential suprema", cf. [14], and therefore derives its importance from the widespread use of essential suprema in financial applications, cf. [19]. In the context of volatility uncertainty, the existence of essential suprema is studied as \feasibility of aggregations" in [45] and the literature based thereon. In Theorem 4.7, we state that typically at most one separable Dedekind σ-complete Banach sublattice exists that generates the σ-algebra, and if it exists, the family of priors P is already dominated with uniformly integrable densities. We thereby qualify that what prevents nondominated models from being dominated is the lack of all order completeness properties for separable Banach sublattices that generate the σ-algebra. p One therefore concludes that the robust closure C of Cb is too similar to the original space Cb both in terms of its order completeness properties and in terms of dominatedness. 2.2. Spanning power of options.
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