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Michael Kupper

Publications and source records attributed to Michael Kupper.

At least 55 records · Page 3Linked to original sources
arXiv2017

Vector duality via conditional extension of dual pairs

A Fenchel-Moreau type duality for proper convex and lower semi-continuous functions $f\colon X\to \overline{L^0}$ is established where $(X,Y,\langle \cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\overline{L^0}$ is the set of all extended real-valued measurable functions. We provide a concept of lower semi-continuity which is shown to be equivalent to the existence of a dual representation in terms of elements in the Bochner space $L^0(Y)$. To derive the duality result, several conditional completions and extensions are constructed. This is an earlier version of arXiv e-print 1708.03127, where the main results were formulated in an abstract setting of conditional completions, conditional extensions and conditional real numbers.

math.FA↗
arXiv2017

Multidimensional quadratic BSDEs with separated generators

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.

math.PR↗
arXiv2017

Solvability of multidimensional quadratic BSDEs

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.

math.PR↗
arXiv2017

Duality formulas for robust pricing and hedging in discrete time

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.

q-fin.MF↗
arXiv2017

Kolmogorov type and general extension results for nonlinear expectations

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 paper 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.

math.PR↗
arXiv2017

An equilibrium model for spot and forward prices of commodities

We consider a market model that consists of financial investors and producers of a commodity. Producers optionally store some production for future sale and go short on forward contracts to hedge the uncertainty of the future commodity price. Financial investors take positions in these contracts in order to diversify their portfolios. The spot and forward equilibrium commodity prices are endogenously derived as the outcome of the interaction between producers and investors. Assuming that both are utility maximizers, we first prove the existence of an equilibrium in an abstract setting. Then, in a framework where the consumers' demand and the exogenously priced financial market are correlated, we provide semi-explicit expressions for the equilibrium prices and analyze their dependence on the model parameters. The model can explain why increased investors' participation in forward commodity markets and higher correlation between the commodity and the stock market could result in higher spot prices and lower forward premia.

econ.GN↗
arXiv2016

Multidimensional Markov FBSDEs with superquadratic growth

We give local and global existence and uniqueness results for systems of coupled FBSDEs in the multidimensional setting and with generators allowed to grow arbitrarily fast in the control variable. Our results are based on Malliavin calculus arguments and pasting techniques.

math.PR↗
arXiv2016

Duality for increasing convex functionals with countably many marginal constraints

In this work we derive a convex dual representation for increasing convex functionals on a space of real-valued Borel measurable functions defined on a countable product of metric spaces. Our main assumption is that the functionals fulfill marginal constraints satisfying a certain 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.

math.FA↗
arXiv2016

Minimal Supersolutions of Convex BSDEs under Constraints

We study supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form $dZ = Δdt + ΓdW$. The generator may depend on the decomposition $(Δ,Γ)$ and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in $Δ$ and $Γ$. 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.

math.PR↗
arXiv2016

The algebra of conditional sets and the concepts of conditional topology and compactness

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-Šmulian theorem are shown.

math.LO↗
arXiv2015

Portfolio Optimization under Nonlinear Utility

This paper studies the utility maximization problem of an agent with non-trivial endowment, and whose preferences are modeled by the maximal subsolution of a BSDE. We prove existence of an optimal trading strategy and relate our existence result to the existence of a maximal subsolution to a controlled decoupled FBSDE. Using BSDE duality, we show that the utility maximization problem can be seen as a robust control problem admitting a saddle point if the generator of the BSDE additionally satisfies a specific growth condition. We show by convex duality that any saddle point of the robust control problem agrees with a primal and a dual optimizer of the utility maximization problem, and can be characterized in terms of a BSDE solution.

math.OC↗
arXiv2014

Conditional Analysis on R^d

This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring $L^0$ of measurable functions on a $σ$-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of $L^0$-affine sets, $L^0$-convex sets, $L^0$-convex cones, $L^0$-hyperplanes, $L^0$-half-spaces and $L^0$-convex polyhedral sets. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study $L^0$-linear, $L^0$-affine, $L^0$-convex and $L^0$-sublinear functions and introduce notions of continuity, differentiability, directional derivatives and subgradients. We use a conditional version of the Bolzano-Weierstrass theorem to show that conditional Cauchy sequences converge and give conditions under which conditional optimization problems have optimal solutions. We prove results on the separation of $L^0$-convex sets by $L^0$-hyperplanes and study $L^0$-convex conjugate functions. We provide a result on the existence of $L^0$-subgradients of $L^0$-convex functions, prove a conditional version of the Fenchel-Moreau theorem and study conditional inf-convolutions.

math.FA↗
arXiv2014

Dual Representation of Minimal Supersolutions of Convex BSDEs

We give a dual representation of minimal supersolutions of BSDEs with non-bounded, but integrable terminal conditions and under weak requirements on the generator which is allowed to depend on the value process of the equation. Conversely, we show that any dynamic risk measure satisfying such a dual representation stems from a BSDE. We also give a condition under which a supersolution of a BSDE is even a solution.

math.PR↗
arXiv2013

A Fourier Approach to the Computation of CV@R and Optimized Certainty Equivalents

We consider the class of risk measures associated with optimized certainty equivalents. This class includes several popular examples, such as CV@R and monotone mean-variance. Numerical schemes are developed for the computation of these risk measures using Fourier transform methods. This leads, in particular, to a very competitive method for the calculation of CV@R which is comparable in computational time to the calculation of V@R. We also develop methods for the efficient computation of risk contributions.

q-fin.RM↗
arXiv2013

Minimal supersolutions of convex BSDEs

We study the nonlinear operator of mapping the terminal value $ξ$ to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in $y$, convex in $z$, jointly lower semicontinuous and bounded below by an affine function of the control variable $z$. We show existence, uniqueness, monotone convergence, Fatou's lemma and lower semicontinuity of this operator. We provide a comparison principle for minimal supersolutions of BSDEs.

math.PR↗
arXiv2013

Brouwer Fixed Point Theorem in (L^0)^d

The classical Brouwer fixed point theorem states that in R^d every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let L^0 = L^0 (Ω, A,P) be the set of random variables. We consider (L^0)^d as an L^0-module and show that local, sequentially continuous functions on closed and bounded subsets have a fixed point which is measurable by construction.

math.FA↗
arXiv2012

Continuous Equilibrium in Affine and Information-Based Capital Asset Pricing Models

We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents' optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents' risk aversion on the implied volatility of simultaneously-traded European-style options.

q-fin.GN↗