We extend the definition of coherent risk measures, as introduced by Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how to define such measures on the space of all random variables. We also give examples that relates the theory of coherent risk measures to game theory and to distorted probability measures. The mathematics are based on the characterisation of closed convex sets Pσ of probability measures that satisfy the property that every random variable is integrable for at least one probability measure in the set Pσ.

/pdf/coherent-risk-measures-on-general-probability-spaces-1t7s2w9ugb.pdf

Coherent Risk Measures on General Probability Spaces

The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R
3. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L
1 topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.

/pdf/the-navier-stokes-limit-of-the-boltzmann-equation-for-2znflhx7p0.pdf

The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels

We define a coherent risk measures as set-valued maps satisfying some axioms. We show that this definition is a convenient extension of the real-valued risk measures introduced by Artzner, Delbaen, Eber and Heath (1998). We then discuss the aggregation issue, i.e. the passage from valued random portofolio to valued measure of Risk. Necessary and sufficient conditions of coherent aggregation are provided

/pdf/vector-valued-coherent-risk-measures-3o5p4ujxo6.pdf

Vector-valued Coherent Risk Measures

Publisher Summary This chapter summarizes the account of the extension of the classical general equilibrium model to an infinite dimensional setting. The classical finite dimensional theory, the commodity space is the canonical finite dimensional linear space R n . By contrast, there is no canonical infinite dimensional linear space. Different economic applications require models involving different infinite dimensional linear spaces. The mathematical discipline of functional analysis has already been well developed as a tool for the abstract study of linear spaces. The chapter follows the methodology of functional analysis and attacks the existence problem. Advantage of this method is that it yields general results, capable of application in a wide variety of specific models. An important line of research in classical general equilibrium theory has been the relationship of the core to the set of competitive allocations. In the infinite dimensional setting, an extensive body of work has been developed, which centers around the infinite-dimensional version of the Debreu–Scarf core convergence theorem.

Chapter 34 Equilibrium theory in infinite dimensional spaces

Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all cadlag processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in ( −∞, ∞]. The economic interpretation of a value of ∞ is that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted cadlag processes that can be extended to coherent or convex monetary risk measures on the space of all adapted cadlag processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded cadlag processes induced by a so called m-stable set.

/pdf/coherent-and-convex-monetary-risk-measures-for-unbounded-1ist6eo7df.pdf

Coherent and convex monetary risk measures for unbounded càdlàg processes

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