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Samuel Drapeau

Publications and source records attributed to Samuel Drapeau.

27 records · Page 2Linked to original sources
arXiv2016

Stability and Markov Property of Forward Backward Minimal Supersolutions

We show stability and locality of the minimal supersolution of a forward backward stochastic differential equation with respect to the underlying forward process under weak assumptions on the generator. The forward process appears both in the generator and the terminal condition. Painlevé-Kuratowski and Convex Epi-convergence are used to establish the stability. For Markovian forward processes the minimal supersolution is shown to have the Markov property. Furthermore, it is related to a time-shifted problem and identified as the unique minimal viscosity supersolution of a corresponding PDE.

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↗
arXiv2016

Conditional Preference Orders and their Numerical Representations

We provide an axiomatic system modeling conditional preference orders which is based on conditional set theory. Conditional numerical representations are introduced, and a conditional version of the theorems of Debreu on the existence of numerical representations is proved. The conditionally continuous representations follow from a conditional version of Debreu's Gap Lemma the proof of which relies on a conditional version of the axiom of choice, free of any measurable selection argument. We give a conditional version of the von Neumann and Morgenstern representation as well as automatic conditional continuity results, and illustrate them by examples.

econ.GN↗
arXiv2014

Dynamic Assessment Indices

This paper provides a unified framework, which allows, in particular, to study the structure of dynamic monetary risk measures and dynamic acceptability indices. The main mathematical tool, which we use here, and which allows us to significantly generalize existing results is the theory of $L^0$-modules. In the first part of the paper we develop the general theory and provide a robust representation of conditional assessment indices, and in the second part we apply this theory to dynamic acceptability indices acting on stochastic processes.

math.PR↗
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↗