2 Literature
The articles closest in spirit to ours are [
1,
18,
26]. Acciaio et al. [
1] consider an object related to the adapted Wasserstein distance in continuous time in connection with utility maximisation, enlargement of filtrations and optimal stopping. Glanzer et al. [
26] prove a deviation inequality for the so-called nested distance in a discrete-time framework,
3 and consider acceptability pricing over an ambiguity set described through the nested distance. Bion-Nadal and Talay [
18] study via PDE arguments a continuous-time optimisation problem which is related to the adapted Wasserstein distance.
The concept of causal couplings, and optimal transport over causal couplings, has been recently popularised by Lassalle [
41], although precursors can be found in the works by Yamada and Watanabe [
55] and Rüschendorf [
52]. This notion is central to the recent articles by Acciaio et al. [
1] and Backhoff-Veraguas et al. [
10,
8,
9].
The idea of strengthening weak convergence of measures in order to account for a temporal evolution has some history. Indeed, several authors have independently introduced different approaches to address this challenge. The seminal unpublished work by Aldous [
2] introduces the notion of extended weak convergence for the study of stability of optimal stopping problems. The principal idea is not to compare the laws of processes directly, but rather the laws of the corresponding prediction processes. Independently, Hellwig [
29] introduces the information topology for the stability of equilibrium problems in economics. Roughly, two probability measures on a product
\(X_{1}\times \cdots \times X_{N}\) of finitely many spaces are considered to be close if for each
\(t \leq N\), the projections onto the first
\(t\) coordinates as well as the corresponding conditional (regular) disintegrations are close. Unrelated to these developments, Pflug and Pichler [
46,
47,
48] have introduced nested distances for the stability of stochastic programming in discrete time. The nested distance is the obvious role model for the adapted Wasserstein distances considered in this article, and (as mentioned above) for a fixed number of time steps and
\(p \geq 1\), they are obviously equivalent. Yet another idea to account for the temporal evolution of processes would be to symmetrise the causal transport costs
\(\mathcal{W}_{c}({\mathbb{P}},{\mathbb{Q}})\) defined by Lassalle [
41] by taking the maximum or sum of
\(\mathcal{W}^{2}_{c}({\mathbb{P}},{\mathbb{Q}})\) and
\(\mathcal{W}^{2}_{c}({\mathbb{Q}},{\mathbb{P}})\); this was pointed out by Soumik Pal.
In parallel work [
6], the four authors of the present article investigate the relations between these concepts in detail. Remarkably, in (finite) discrete time,
all of the concepts mentioned above (adapted Wasserstein distances, extended weak convergence, information topology, nested distances, symmetrised causal transport costs) define the
same topology. As noted above, this “weak adapted topology” refines the usual weak topology (properly for
\(T\geq 2\); see also Remark
5.2). The articles [
8,
6,
23] investigate basic properties of this topology; e.g., the weak adapted topology is Polish [
8, Sect. 5], and sets are totally bounded with respect to the adapted Wasserstein distance/nested distance if and only if they are totally bounded with respect to the usual Wasserstein distance [
6, Lemma 1.6]. For recent applications of these concepts to optimal transport and probabilistic variants thereof, we refer to Backhoff-Veraguas et al. [
11,
12] and Wiesel [
54].
In contrast, fundamental topological properties of the above-mentioned concepts in the continuous-time case seem to be much less understood and, as far as the authors are concerned, pose an interesting challenge for future research. Specifically, it is not clear to us whether the topology associated to the adapted Wasserstein distance is Polish in the continuous-time case. In a similar vein, we expect that results analogous to those of the present article should apply in the case of càdlàg paths, but such an extension is beyond the scope of our current understanding of adapted Wasserstein distances.
The question of stability in mathematical finance has been studied from different perspectives over the years. Notably, starting with the articles of Lyons [
42] and Avellaneda et al. [
5], the area of robust finance has mainly focused on extremal models and hedging strategies which dominate the payoff for every model in a specified class. Following the publication of Hobson’s seminal article [
32], connections with the Skorokhod embedding problem have been a driving force of the field; see the surveys of Hobson [
34] and Obłój [
44]. Recently, this has been complemented by techniques coming from (martingale) optimal transport; early papers which advance this viewpoint include Hobson [
35], Beiglböck et al. [
15,
16], Galichon et al. [
25], Bouchard and Nutz [
19], Dolinsky and Soner [
22], Campi et al. [
20], and Beiglböck and Siorpaes [
14]. The literature on “local” misspecification of volatility in a sense more closely related to the present article appears more sparse. El Karoui et al. [
24] establish in a stochastic volatility framework that if the misspecified volatility dominates the true volatility, then the misspecified price of call options dominates the real price; see also the elegant account of Hobson [
33]. More recently, the question of pricing and hedging under uncertainty about the volatility of a reference local volatility model is studied by Herrmann et al. [
31] (see also Herrmann and Muhle-Karbe [
30]). Less plausible models are penalised through a mean-square distance to the volatility of the reference model, and the authors obtain explicit formulas for prices and hedging strategies in a limit for small uncertainty aversion. Becherer and Kentia [
13] derive worst-case good-deal bounds under model ambiguity which concerns drift as well as volatility. Indeed, discussions with Dirk Becherer motivated us to consider also models with drift in our results on stability of superhedging. The behaviour of the superhedging price in a ball (with respect to various notions of distance) around a reference model is studied in depth by Obłój and Wiesel [
45] for a
\(d\)-dimensional asset and one time period.
A notable implication of our work is that it yields a coherent way to measure model uncertainty (in the sense of Cont’s influential article [
21]): Fix a subset
\(M_{0}\) of the set
\(M\) of all consistent models, i.e., martingale measures which are consistent with benchmark instruments whose price can be observed on the market. Given
\(M_{0}\), the model uncertainty associated to a derivative
\(f\) can be gauged through
$$ \rho _{M_{0}}(f):=\sup \{ \mathbb{E}_{\mathbb{Q}} f :\mathbb{Q}\in M_{0} \} - \inf \{\mathbb{E}_{\mathbb{Q}} f : \mathbb{Q}\in M_{0} \}. $$
The worst-case approach typically pursued in robust finance then yields
\(\rho _{M_{0}}(f)\) for
\(M_{0}=M\), but it appears equally natural to take
\(M_{0}\) to be an infinitesimal ball around a reference model. This approach is being pursued by Bartl, Drapeau, Obłój, Wiesel and one of the present authors in a one-period framework. Our results indicate that an adapted Wasserstein distance provides a way to extend this to a multi-period setup, and we intend to pursue this further in future work.
On a different note, much work has been done regarding the convergence of discrete-time models to their continuous-time analogues. Due to the vastness of this literature, we refer the reader to the book by Prigent [
51] for references. Finally, in more recent times and starting from the works of Kardaras and Žitković, the stability of utility maximisation has been studied in Kardaras and Žitković [
37], Larsen [
39], Larsen and Žitković [
40], Mocha and Westray [
43] and Weston [
53], among others.
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