arXiv subjects
Publications and source records attributed to Freddy Delbaen.
We show that a discrete time martingale with respect to a filtration with atomless innovations is the (infinite) sum of martingales with independent increments. For the continuous time filtration coming from Brownian Motion filtration, we show that every $L^2$ martingale is the sum of a series of Gaussian martingales.
For a square integrable $m$-dimensional random variable $X$ on a probability space $(Ω,\Fc,\Pr)$ and a sub sigma algebra $\Ac$, we show that there is a constructive way to represent $X-\Er[X\mid\Ac]$ as the sum of a series of variables that are independent of $\Ac$.
We will characterise robust monetary utility functions defined on the space of real valued (bounded) continuous functions on a Polish space.
We prove that convex functions on a $C_b(X)$ space satisfying a mild continuity condition can be represented using sigma additive measures. This generalises a result of Cheridito, Kupper and Tangpi,
Given a square integrable m-dimensional random variable $X$ on a probability space $(Ω.\mathcal F,\Pr)$ and a sub sigma algebra $\mathcal A$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal A$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory.
Under appropriate integrability conditions the risk measure of the sample measures for a law invariant risk measure converge almost surely to the risk measure of the sampled random variable. The results follow from general convergence theorems based on the theory of Orlicz spaces.
We consider a group consisting of N business units. We suppose there are regulatory constraints for each unit, more precisely, the net worth of each business unit is required to belong to a set of acceptable risks, assumed to be a convex cone. Because of these requirements, there are less incentives to operate under a group structure, as creating one single business unit, or altering the liability repartition among units, may allow to reduce the required capital. We analyse the possibilities for the group to benefit from a diversification effect and economise on the cost of capital. We define and study the risk measures that allow for any group to achieve the minimal capital, as if it were a single unit, without altering the liability of business units, and despite the individual admissibility constraints. We call these risk measures cohesive risk measures.
We use the theory of cooperative games for the design of fair insurance contracts. An insurance contract needs to specify the premium to be paid and a possible participation in the benefit (or surplus) of the company. It results from the analysis that when a contract is exposed to the default risk of the insurance company, ex-ante equilibrium considerations require a certain participation in the benefit of the company to be specified in the contracts. The fair benefit participation of agents appears as an outcome of a game involving the residual risks induced by the default possibility and using fuzzy coalitions.
We give two equivalent definitions of sigma algebras that are atomless conditionally to a smaller sigma algebra.
It is proved that in suitable filtrations every pair of integrable random variables is the conditional expectation of a pair of commonotone integrable random variables.
It is proved that commonotonicity and time consistence for monetary utility functions do not go together. I also gives additional results on atomless and conditionally atomless probability spaces.
We use the theory of coherent measures to look at the problem of surplus sharing in an insurance business. The surplus share of an insured is calculated by the surplus premium in the contract. The theory of coherent risk measures and the resulting capital allocation gives a way to divide the surplus between the insured and the capital providers, i.e. the shareholders.
Lacunary trigonometric and Walsh series satisfy limiting results that are typical for i.i.d. random variables such as the central limit theorem (Salem, Zygmund 1947), the law of the iterated logarithm (Weiss 1959) and several probability related limit theorems. For Hölder continuous, periodic functions this phenomenon does not hold in general. Kac (1946, 1949) showed the validity of the central limit theorem for the sequence $\left(f(2^k x)\right)_k$ and in the case of "big gaps''. In this paper, we present an alternative approach to prove the above theorem based on martingale theory, which allows us to generalize the theorem to infinite product spaces of arbitrary probability spaces, equipped with the shift operator. In addition, we show the local limit theorems for lacunary trigonometric and Walsh series, and for Hölder continuous, periodic functions in the case of "big gaps''. We also establish Berry-Esseen bounds and moderate deviations for lacunary Walsh series. Furthermore, we identify the scale at which the validity of the Gaussian approximation for the tails breaks. To derive these limiting results, the framework of mod-Gaussian convergence has been used.
In the dual $L_{Φ^*}$ of a $Δ_2$-Orlicz space $L_Φ$, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology $τ(L_{Φ^*},L_Φ)$ if and only if on each order interval $[-ζ,ζ]=\{ξ: -ζ\leq ξ\leqζ\}$ ($ζ\in L_{Φ^*}$), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlós type result: every norm bounded sequence $(ξ_n)_n$ in $L_{Φ^*}$ admits a sequence of forward convex combinations $\barξ_n\in\mathrm{conv}(ξ_n,ξ_{n+1},...)$ such that $\sup_n|\barξ_n|\in L_{Φ^*}$ and $\barξ_n$ converges a.s.
In this paper, we relate the framework of mod-$ϕ$ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is that of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents in the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in (possibly coloured) permutations, number of prime divisors (possibly within different residue classes) of a random integer, number of irreducible factors of a random polynomial, etc. One advantage of the approach developed in this paper is that it allows us to deal with approximations in higher dimensions as well. In this setting, we can explicitly see the influence of the correlations between the components of the random vectors in our asymptotic formulas.
The paper mentioned in the title introduces the entropic value at risk. I give some extra comments and using the general theory make a relation with some commonotone risk measures.
In [3], the authors proved that uniqueness holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $γ$ ($p\textgreater{}γ$). In this paper, we consider the critical case: $p=γ$. We prove that the uniqueness holds among solutions whose exponentials are $L^γ$ under the additional assumption that the generator is strongly convex.
In the present contribution we characterize law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (2006), we show that these risk measures can be identified with a class of generalized shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (2014) and Bellini and Bignozzi (2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterization of robustness for convex risk measures in terms of a weak notion of mixture continuity.