We study time-consistency questions for processes of monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a process of monetary risk measures time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time, and we show how this property manifests itself in the corresponding process of acceptance sets. For processes of coherent and convex monetary risk measures admitting a robust representation with sigma-additive linear functionals, we give necessary and sufficient conditions for time-consistency in terms of the representing functionals.

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.

The goal of this thesis is the conceptual study of risk and its quantification via robust representations.In a first part, we consider risk within a context which extends the notion of "measurable uncertainty" introduced by Frank Knight [1921]. Mathematically, the risk perception of risky elements in a convex set X is expressed by a preference order having the properties of quasiconvexity and monotonicity. These properties are the appropriate translation of the two consensual statements that "diversification should not increase the risk" and "the better for sure, the less risky". Such a preference order will be called a risk order. We keep full latitude on the choice of the underlying setting and thus leave room for different interpretations of risk. Typical examples for X are the space of random variables on a given probability space, the convex set of probability distributions on the real line, or the cone of consumption streams. Risk orders can be represented by numerical representations ρ : states a one-to-one correspondence between risk orders, risk measures, and risk acceptance families. Further properties such as convexity, positive homogeneity, or cash-(sub)additivity are then characterised on these three levels.We then study risk orders on a locally convex topological vector space X . Our main theorem states that any lower semicontinuous risk measure ρ has a unique robust representation of the formwhere R :It is actually the leftinverse in the second argument of the minimal penalty functional αmin (x * , m) = sup x∈A m x * , −x . Here, K• is a polar convex cone in the dual space X * . The proof of uniqueness in this natural context of lower semicontinuity is technically involved, and it is new in the general theory of quasiconvex duality. We also prove a robust representation for risk measures on convex set as needed for risk orders on probability distributions or consumptions streams. We finally provide answers to the delicate question, under which circumstances monotonicity alone ensures lower semicontinuity of the risk order.To finish this first part, we specialize our results to various typical settings. In the case of random variables, we explicitly compute the robust representation of canonical examples such as the certainty equivalent, or the economic index of riskiness. We also show that "Value at Risk" is a risk measure on the level of probability distributions and derive its robust representation. For consumption streams, we obtain a robust representation of the intertemporal utility functional of Hindy, Huang and Kreps. For stochastic kernels, we prove a general separation theorem for risk orders which distinguishes between "model risk" and "distributional risk".In the second part of the thesis, we weaken the requirement of completeness of the preferences, that is, the necessity of deciding whether one element is preferable or not to the other. We introduce the concept of a preference order which might require additional information in order to be expressed. In a first section we ii p...

In discrete time, every time-consistent dynamic monetary risk measure can be written as a composition of one-step risk measures. We exploit this structure to give new dual representation results for time-consistent convex monetary risk measures in terms of one-step penalty functions. We first study risk measures for random variables modelling financial positions at a fixed future time. Then we consider the more general case of risk measures that depend on stochastic processes describing the evolution of financial positions or cumulated cash flows. In both cases the new representations allow for a simple composition of one-step risk measures in the dual. We discuss several explicit examples and provide connections to the recently introduced class of dynamic variational preferences.

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