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...

We study the nonlinear operator of mapping the terminal value $\xi$ 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.Comment: Published in at http://dx.doi.org/10.1214/13-AOP834 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The ongoing concern about systemic risk since the outburst of the global financial crisis has highlighted the need for risk measures at the level of sets of interconnected financial components, such as portfolios, institutions or members of clearing houses. The two main issues in systemic risk measurement are the computation of an overall reserve level and its allocation to the different components according to their systemic relevance. We develop here a pragmatic approach to systemic risk measurement and allocation based on multivariate shortfall risk measures, where acceptable allocations are first computed and then aggregated so as to minimize costs. We analyze the sensitivity of the risk allocations to various factors and highlight its relevance as an indicator of systemic risk. In particular, we study the interplay between the loss function and the dependence structure of the components. Moreover, we address the computational aspects of risk allocation. Finally, we apply this methodology to the allocation of the default fund of a CCP on real data.

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.framework of Drapeau et al. [6]. The H 1 -L ∞ duality turns out to be the right candidate to constitute the basis of our representation. As a starting point, we consider the set of essentially bounded terminal conditions. In this case, we obtain a dual representation of the minimal supersolution at time 0 and a pointwise robust representation in the dynamic case. We show that when the generator of the equation is decreasing in the value process, the minimal supersolution defines a time consistent cash-subadditive risk measure. It allows for a dual representation on the space of essentially bounded random variables, which agrees with the representation of El Karoui and Ravanelli [8] obtained for BSDE solutions. Our dual representation is obtained by showing that the representation of El Karoui and Ravanelli [8] can be restricted on a smaller set. Then we can use truncation and approximation arguments to obtain the representation in the general case, due to monotone stability of minimal supersolutions. A direct consequence of our representation is the identification of BSDEs solution and minimal supersolution in the case of linear growth generators. Note that our truncation technique appears already in the work of Delbaen et al. [4] where it is used to construct a sequence of µ-dominated risk measures. Furthermore, prior to us Barrieu and El Karoui [1] and Bion-Nadal [2] already used the BM O-martingale theory in the study of financial risk measures, but in different settings from ours. Using standard convex duality arguments such as the Fenchel-Moreau theorem and the properties of the Fenchel-Legendre transform of a convex functional, we extend our dual representation to the set of random variables that can be identified to H 1 -martingales. Notice that this representation is obtained in the static case. Our representation results can be seen as extensions of the dual representation of the minimal superreplicating cost of El Karoui and Quenez [7] to the case where we allow for a nonlinear cost function in the dynamics of the wealth process. The second theme of this work is to give conditions based on convex duality under which a dynamic cash-subadditive risk measure with a given representation can be seen as the solution, or the minimal supersolution of a BSDE. The cash-additive case has been studied by Delbaen et al. [5]. Their results are based on m-stability of the dual space, some supermartingale property and Dood-Meyer decomposition of the risk measure. We shall show that in the cash-subadditive case, discounting the risk measure yields similar results, hence showing an equivalent relationship betwee...