Max Nendel
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Upper Comonotonicity and Risk Aggregation Under Dependence Uncertainty Open
In this paper, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multimarginal coup…
Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups Open
In this work, we provide conditions for nonlinear monotone semigroups on locally convex vector lattices to give rise to a generalized notion of viscosity solutions to a related nonlinear partial differential equation. The semigroup needs t…
Hopf-Lax approximation for value functions of Lévy optimal control problems Open
In this paper, we investigate stochastic versions of the Hopf-Lax formula which are based on compositions of the Hopf-Lax operator with the transition kernel of a Lévy process taking values in a separable Banach space. We show that, depend…
Pasting of Equilibria and Donsker-type Results for Mean Field Games Open
This paper studies the relation between equilibria in single-period, discrete-time and continuous-time mean field game models. First, for single-period mean field games, we establish the existence of equilibria and then prove the propagati…
A comparison principle based on couplings of partial integro-differential operators Open
This paper is concerned with a comparison principle for viscosity solutions to Hamilton-Jacobi (HJ), -Bellman (HJB), and -Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach innovates in three…
A Hypothesis Test for the Long-Term Calibration in Rating Systems with Overlapping Time Windows Open
We present a statistical test for the long-term calibration in rating systems that can deal with overlapping time windows as required by the guidelines of the European Banking Authority (EBA), which apply to major financial institutions in…
Operator semigroups in the mixed topology and the infinitesimal description of Markov processes Open
We define a class of not necessarily linear C 0 -semigroups (P t ) t≥0 on C b (E) (more generally, on C κ (E) := 1 κ C b (E), for some bounded function κ, which is the pointwise limit of a decreasing sequence of continuous functions) equip…
Upper Comonotonicity and Risk Aggregation under Dependence Uncertainty Open
In this paper, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multi-marginal cou…
A hypothesis test for the long-term calibration in rating systems with overlapping time windows Open
We present a statistical test that can be used to verify supervisory requirements concerning overlapping time windows for the long-term calibration in rating systems. In a first step, we show that the long-run default rate is approximately…
Risk measures based on weak optimal transport Open
In this paper, we study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we…
Convex Monotone Semigroups on Lattices of Continuous Functions Open
We consider convex monotone C_0 -semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a \sigma -Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and th…
Convergence of infinitesimal generators and stability of convex monotone semigroups Open
Based on the convergence of their infinitesimal generators in the mixed topology, we provide a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions. In contrast to previous results, we do no…
An axiomatic approach to default risk and model uncertainty in rating systems Open
In this paper, we deal with an axiomatic approach to default risk. We introduce the notion of a default risk measure, which generalizes the classical probability of default (PD), and allows to incorporate model risk in various forms. We di…
A parametric approach to the estimation of convex risk functionals based on Wasserstein distance Open
In this paper, we explore a static setting for the assessment of risk in the context of mathematical finance and actuarial science that takes into account model uncertainty in the distribution of a possibly infinite-dimensional risk factor…
Lower semicontinuity of monotone functionals in the mixed topology on $C_b$ Open
The main result of this paper characterizes the continuity from below of monotone functionals on the space $C_b$ of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular…
A Unifying Framework for Submodular Mean Field Games Open
We provide an abstract framework for submodular mean field games and identify verifiable sufficient conditions that allow us to prove the existence and approximation of strong mean field equilibria in models where data may not be continuou…
Operator semigroups in the mixed topology and the infinitesimal description of Markov processes Open
We define a class of not necessarily linear $C_0$-semigroups $(P_t)_{t\geq0}$ on $C_b(E)$ (more generally, on $C_κ(E):=\frac1κC_b(E)$, for some bounded function $κ$, which is the pointwise limit of a decreasing sequence of continuous funct…
Convex monotone semigroups and their generators with respect to $Γ$-convergence Open
We study semigroups of convex monotone operators on spaces of continuous functions and their behaviour with respect to $Γ$-convergence. In contrast to the linear theory, the domain of the generator is, in general, not invariant under the s…
A unifying framework for submodular mean field games Open
We provide an abstract framework for submodular mean field games and identify verifiable sufficient conditions that allow to prove existence and approximation of strong mean field equilibria in models where data may not be continuous with …
Submodular mean field games: Existence and approximation of solutions Open
We study mean field games with scalar Itô-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly,…
Wasserstein perturbations of Markovian transition semigroups Open
In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition probabili…
Upper Envelopes of Families of Feller Semigroups and Viscosity Solutions to a Class of Nonlinear Cauchy Problems Open
In this paper, we consider the (upper) semigroup envelope, i.e. the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields visco…
Convex semigroups on L<sup>p</sup>-like spaces Open
In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having Lp-spaces in mind as a typical application. We show that the basic results from linear C0-semigroup theory extend to the convex case. We …
Markov chains under nonlinear expectation Open
In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operator s as an extension of Q ‐matrices or rate matrices to a nonlinear setup, where the nonlinearity …
Convex monotone semigroups on lattices of continuous functions Open
We consider convex monotone $C_0$-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a $σ$-Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the s…
Separability vs. robustness of Orlicz spaces: financial and economic perspectives Open
We investigate robust Orlicz spaces as a generalisation of robust $L^p$-spaces. Two constructions of such spaces are distinguished, a top-down approach and a bottom-up approach. We show that separability of robust Orlicz spaces or their su…