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In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $L^p$-spaces in mind as a typical application. We show that the basic results from linear $C_0$-semigroup theory extend to the convex case. We prove that the generator of a convex $C_0$-semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup; a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of $C_0$-semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations., The manuscript has been split into two parts. The second part of the paper can be found under arXiv:2010.04594. 24 pages

We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed L\'evy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial L\'evy process., Comment: 14 pages, forthcoming in ECP

This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, martingale optimal transport, portfolio optimization under uncertainty and generative adversarial networks that showcase the generality and effectiveness of the approach.

We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this article is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell–Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite-dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels.

In this paper, we present a duality theory for the robust utility maximisation problem in continuous time for utility functions defined on the positive real line. Our results are inspired by – and can be seen as the robust analogues of – the seminal work of Kramkov and Schachermayer (Ann. Appl. Probab. 9:904–950, 1999). Namely, we show that if the set of attainable trading outcomes and the set of pricing measures satisfy a bipolar relation, then the utility maximisation problem is in duality with a conjugate problem. We further discuss the existence of optimal trading strategies. In particular, our general results include the case of logarithmic and power utility, and they apply to drift and volatility uncertainty. Nanyang Technological University Daniel Bartl is grateful for financial support through the Austrian Science Fund (FWF) under project P28661 and the Vienna Science and Technology Fund (WWTF) under project MA16-021. Ariel Neufeld is grateful for financial support through Nanyang Assistant Professorship Grant (NAP Grant) Machine Learning based Algorithms in Finance and Insurance.

We provide a model-free pricing–hedging duality in continuous time. For a frictionless market consisting of $d$ risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging price of a path-dependent European option has the same value as the purely probabilistic problem of finding the supremum of the expectations of the option over all martingale measures. The superhedging problem is formulated with simple trading strategies, the claim is the limit inferior of continuous functions, which allows upper and lower semi-continuous claims, and superhedging is required in the pathwise sense on a $\sigma $ -compact sample space of price trajectories. If the sample space is stable under stopping, the probabilistic problem reduces to finding the supremum over all martingale measures with compact support. As an application of the general results, we deduce dualities for Vovk’s outer measure and semi-static superhedging with finitely many securities.

In this paper asymptotically stable risk assessments are studied. They are characterized by not being sensitive with respect to huge additional capital in the very far future. Under the additional hypothesis of being locally continuous from below, these risk assessments are exactly those which allow a robust representation with so-called local test probabilities having a support with finite time horizon. Time-consistent risk assessments can be constructed by composing a sequence of generators. We give several conditions for the generators such that the resulting risk assessments are indeed asymptotically stable.

The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel-Lebesgue theorem, the Hahn-Banach theorem, the Banach-Alaoglu theorem and the Krein-\v{S}mulian theorem are shown., Comment: 29 pages

We study the relation between Levy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear Levy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators (A(lambda))(lambda is an element of Lambda) of linear Levy processes which guarantees the existence of a nonlinear Levy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE partial derivative(t)u = sup(lambda is an element of Lambda) A(lambda)u. The results are illustrated with several examples. (C) 2019 Published by Elsevier B.V.

The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $\omega $ but also in $\int \omega \,d\omega :=\omega ^{2} -\langle \omega \rangle $ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term $\omega ^{2} -\langle \omega \rangle $ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g. the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: Firstly, we derive a dual representation of the considered problem and prove that strong duality holds. Secondly, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real world instance., Revised version. Accepted for publication in "Mathematical Finance"

We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by trading in the financial market we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by finitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multi-dimensional backward stochastic differential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it. As an example we simulate option prices in the presence of stochastic volatility, demand pressure and short-selling constraints.

We prove a general existence result in stochastic optimal control in discrete time where controls take values in conditional metric spaces, and depend on the current state and the information of past decisions through the evolution of a recursively defined forward process. The generality of the problem lies beyond the scope of standard techniques in stochastic control theory such as random sets, normal integrands and measurable selection theory. The main novelty is a formalization in conditional metric space and the use of techniques in conditional analysis. We illustrate the existence result by several examples including wealth-dependent utility maximization under risk constraints with bounded and unbounded wealth-dependent control sets, utility maximization with a measurable dimension, and dynamic risk sharing. Finally, we discuss how conditional analysis relates to random set theory.

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this extends the usual representation results for separable spaces., Comment: 21 pages, no figures, this is a final version accepted for publication in Set-Valued and Variational Analysis, compared with [v2] remark 5.5 added, presentation and description improved; after a major revision in [v2], where the general setting was simplified and extended applications to conditional distributions and disintegration were obtained (comprised in section 4)

In this paper we study a robust expected utility maximization problem with random endowment in discrete time. We give conditions under which an optimal strategy exists and derive a dual representation for the optimal utility. Our approach is based on a general representation result for monotone convex functionals, a functional version of Choquet's capacitability theorem and medial limits. The novelty is that it works under nondominated model uncertainty without any assumptions of time-consistency. As applications, we discuss robust utility maximization problems with moment constraints, Wasserstein constraints and Wasserstein penalties.

We consider multidimensional quadratic BSDEs with bounded and unbounded terminal conditions. We provide sufficient conditions which guarantee existence and uniqueness of solutions. In particular, these conditions are satisfied if the terminal condition or the dependence in the system are small enough., this is the final version accepted to be published in ECP

In this paper we derive robust super- and subhedging dualities for contingent claims that can depend on several underlying assets. In addition to strict super- and subhedging, we also consider relaxed versions which, instead of eliminating the shortfall risk completely, aim to reduce it to an acceptable level. This yields robust price bounds with tighter spreads. As examples we study strict super- and subhedging with general convex transaction costs and trading constraints as well as risk-based hedging with respect to robust versions of the average value at risk and entropic risk measure. Our approach is based on representation results for increasing convex functionals and allows for general financial market structures. As a side result it yields a robust version of the fundamental theorem of asset pricing.

The main result of this paper is a convex dual representation for increasing convex functionals that are dened on a space of real-valued Borel measurable functions living on a countable product of metric spaces. Our principal assumption is that the func- tionals fulll convex marginal constraints satisfying a tightness condition. In the special case where the marginal constraints are given by expectations or maxima of expectations, we obtain linear and sublinear versions of Kantorovich's transport duality and the recently discovered martingale transport duality on products of countably many metric spaces.

This paper provides an unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set valued functions.

Nous donnons une representation duale des sur-solutions minimales d’equations differentielles stochastiques retrogrades avec des conditions terminales integrables mais non necessairement bornees, et de faibles hypotheses sur le generateur qui peut de plus dependre de la valeur processus de l’equation meme. Reciproquement, nous montrons que toute mesure de risque dynamique satisfaisant une telle representation duale provient d’une EDSR. Nous donnons aussi une condition sous laquelle une sur-solution d’EDSR est en fait une solution.

We study supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form $dZ = {\Delta}dt + {\Gamma}dW$. The generator may depend on the decomposition $({\Delta},{\Gamma})$ and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in ${\Delta}$ and ${\Gamma}$. We prove the existence of a supersolution that is minimal at time zero and derive stability properties of the non-linear operator that maps terminal conditions to the time zero value of this minimal supersolution such as monotone convergence, Fatou's lemma and $L^1$-lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints., Comment: 23 pages