323 research outputs found
Conditional Lp-spaces and the duality of modules over f-algebras
No full textMotivated by dynamic asset pricing, we extend the dual pairs’ theory of Dieudonné (1942) and Mackey (1945) to pairs of modules over a Dedekind complete f-algebra with multiplicative unit. The main tools are:
- a Hahn–Banach Theorem for modules of this kind;
- a topology on the f-algebra that has the special feature of coinciding with the norm topology when the algebra is a Banach algebra and with the strong order topology of Filipovic, Kupper, and Vogelpoth (2009), when the algebra of all random variables on a probability space (Ω, G, P) is considered.
As a leading example, we study in some detail the duality of conditional Lp-spaces
Kolmogorov-type and general extension results for nonlinear expectations
No full textDenk R, Kupper M, Nendel M. Kolmogorov-type and general extension results for nonlinear expectations. Banach Journal of Mathematical Analysis. 2018;12(3):515-540
Hopf-Lax approximation for value functions of L´evy optimal control problems
No full textKupper M, Nendel M, Sgarabottolo A. Hopf-Lax approximation for value functions of L´evy optimal control problems. Center for Mathematical Economics Working Papers. Vol 747. Bielefeld: Center for Mathematical Economics; 2025.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, depending
on the order of the composition, one obtains upper and lower bounds for the value
function of a stochastic optimal control problem associated to the drift controlled Lévy
dynamics. Dynamic consistency is restored by iterating the resulting operators. Moreover,
the value function of the control problem is approximated both from above and below
as the number of iterations tends to infinity, and we provide explicit convergence rates
and guarantees for the approximation procedure.MSC 2020 Classification: Primary 47H20; 35A35. Secondary 41A25; 93E20 41A3
A semigroup approach to nonlinear Levy processes
No full textDenk R, Kupper M, Nendel M. A semigroup approach to nonlinear Levy processes. Stochastic Processes and their Applications. 2020;130(3):1616-1642.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
A Semigroup Approach to Nonlinear Lévy Processes
No full textDenk R, Kupper M, Nendel M. A Semigroup Approach to Nonlinear Lévy Processes. Center for Mathematical Economics Working Papers. Vol 610. Bielefeld: Center for Mathematical Economics; 2019.We study the relation between Lévy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear Lévy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators () of linear Lévy processes which guarantees the existence of a nonlinear Lévy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE . The results are illustrated with several examples
Risk measures based on weak optimal transport
No full textKupper M, Nendel M, Sgarabottolo A. Risk measures based on weak optimal transport. Quantitative Finance . 2024;25(2):163-180.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 discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, divergence risk measures, uncertainty on path spaces, moment constraints, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting
Convex semigroups on Lp-like spaces
No full textDenk R, Kupper M, Nendel M. Convex semigroups on Lp-like spaces. Center for Mathematical Economics Working Papers. Vol 712. Bielefeld: Center for Mathematical Economics; 2021.In this paper, we investigate convex semigroups on Banach lattices with
order continuous norm, having -spaces in mind as a typical application. We show
that the basic results from linear -semigroup theory extend to the convex case. We
prove that the generator of a convex -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 -semigroups. The results
are discussed in several examples such as semilinear heat equations and nonlinear
integro-differential equations.AMS 2010 Subject Classification: 47H20; 35A02; 35A0
Wasserstein Perturbations of Markovian Transition Semigroups
No full textFuhrmann S, Kupper M, Nendel M. Wasserstein Perturbations of Markovian Transition Semigroups. Center for Mathematical Economics Working Papers. Vol 649. Bielefeld: Center for Mathematical Economics; 2021.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 modelled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of Lévy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.AMS 2020 Subject Classification:
Primary 60J35; 47H20; Scondary 60G65; 90C31; 62G3
Risk measures based on weak optimal transport
No full textKupper M, Nendel M, Sgarabottolo A. Risk measures based on weak optimal transport. Center for Mathematical Economics Working Papers. Vol 734. Bielefeld: Center for Mathematical Economics; 2023.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 discuss
computational aspects related to the nonlinear transform as well as approximations of
the risk measures using, for example, neural networks. Our setup comprises a variety of
examples, such as classical optimal transport penalties, parametric families of models,
uncertainty on path spaces, moment constrains, and martingale constraints. In a last
step, we show how to use the theoretical results for the numerical computation of worstcase
losses in an insurance context and no-arbitrage prices of European contingent claims
after quoted maturities in a model-free setting.AMS 2020 Subject Classification: Primary 91G70; 91B05; Secondary 68T07; 91G20;
91G6
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