Convex Imprecise Previsions for Risk Measurement

In this paper we introduce convex imprecise previsions as a special class of imprecise previsions, showing that they retain or generalise most of the relevant properties of coherent imprecise previsions but are not necessarily positively homogeneous. The broader class of weakly convex imprecise previsions is also studied and its fundamental properties are demonstrated. The notions of weak convexity and convexity are then applied to risk measurement, leading to a more general definition of convex risk measure than the one already known in risk measurement literature.imprecise previsions, risk measures, weakly convex imprecise previsions, convex imprecise previsions

Convex Imprecise Previsions: Basic Issues and Applications

In this paper we study two classes of imprecise previsions, which we termed convex and centered convex previsions, in the framework of Walley's theory of imprecise previsions. We show that convex previsions are related with a concept of convex natural estension, which is useful in correcting a large class of inconsistent imprecise probability assessments. This class is characterised by a condition of avoiding unbounded sure loss. Convexity further provides a conceptual framework for some uncertainty models and devices, like unnormalised supremum preserving functions. Centered convex previsions are intermediate between coherent previsions and previsions avoiding sure loss, and their not requiring positive homogeneity is a relevant feature for potential applications. Finally, we show how these concepts can be applied in (financial) risk measurement.Comment: Proceedings of ISIPTA'0

Le misure di rischio nell’ambito della teoria delle probabilità imprecise

2noNell’ambito della finanza matematica hanno di recente riscosso un interesse crescente la ricerca di metodi e lo sviluppo di modelli teorici per la valutazione dei rischi connessi a posizioni finanziarie. Ha così assunto notevole rilievo la nozione di misura di rischio coerente, introdotta da P. Artzner, F. Delbaen, S. Eber e D. Heath in alcuni articoli [1, 2, 5] nei quali tali autori hanno individuato alcuni requisiti ritenuti, a loro giudizio, fondamentali e che ogni misura di rischio dovrebbe ragionevolmente soddisfare. In questo lavoro, dopo aver ricordato tale nozione ed averne illustrato le principali caratteristiche nella Sezione 2, ne viene evidenziata, nella Sezione 3, la stretta connessione con la teoria delle previsioni imprecise, seguendo la linea introdotta in [14]. Vengono successivamente illustrati alcuni problemi rilevanti per la teoria delle misure di rischio coerenti, tra i quali la generalizzazione della nozione di coerenza a spazi di numeri aleatori limitati privi di struttura. Inoltre, qualora una misura non sia coerente, si pu`o porre la necessit`a di determinarne una “correzione”, cio`e di individuare una misura di ¢ONVEGNO eCONOMIA E iNCERTEZZA 191 rischio coerente che le sia in qualche modo “vicina”. Analogamente, vi pu`o essere la necessit`a di determinare un’estensione di una misura di rischio coerente che sia definita su un insieme di numeri aleatori non sufficientemente ampio. Questi problemi, e la corrispondente nozione di estensione naturale, vengono affrontati nella Sezione 4. Nella Sezione 5 viene invece illustrata la nozione di misura di rischio convessa, una generalizzazione del concetto di misura di rischio coerente che consente di prendere in considerazione anche il cosiddetto liquidity risk e per la quale si provano, con riferimento alla teoria delle previsioni imprecise, risultati simili a quelli ottenuti per le misure coerenti. Nella Sezione 6 vengono infine fornite alcune indicazioni su ulteriori sviluppi e su alcuni modelli specifici nei quali la teoria della previsioni imprecise viene impiegata nella misurazione del rischio.nonemixedPelessoni R.; Vicig P.Pelessoni, Renato; Vicig, Paol

2-coherent and 2-convex Conditional Lower Previsions

In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of $n$-coherent and $n$-convex conditional previsions, at the varying of $n$. We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) $2$-convex or, if positive homogeneity and conjugacy is needed, $2$-coherent lower previsions. Basic properties of these previsions are studied. In particular, we prove that they satisfy the Generalized Bayes Rule and always have a $2$-convex or, respectively, $2$-coherent natural extension. The role of these extensions is analogous to that of the natural extension for coherent lower previsions. On the contrary, $n$-convex and $n$-coherent previsions with $n\geq 3$ either are convex or coherent themselves or have no extension of the same type on large enough sets. Among the uncertainty concepts that can be modelled by $2$-convexity, we discuss generalizations of capacities and niveloids to a conditional framework and show that the well-known risk measure Value-at-Risk only guarantees to be centered $2$-convex. In the final part, we determine the rationality requirements of $2$-convexity and $2$-coherence from a desirability perspective, emphasising how they weaken those of (Williams) coherence.Comment: This is the authors' version of a work that was accepted for publication in the International Journal of Approximate Reasoning, vol. 77, October 2016, pages 66-86, doi:10.1016/j.ijar.2016.06.003, http://www.sciencedirect.com/science/article/pii/S0888613X1630079

Characterizing coherence, correcting incoherence

Lower previsions defined on a finite set of gambles can be looked at as points in a finite-dimensional real vector space. Within that vector space, the sets of sure loss avoiding and coherent lower previsions form convex polyhedra. We present procedures for obtaining characterizations of these polyhedra in terms of a minimal, finite number of linear constraints. As compared to the previously known procedure, these procedures are more efficient and much more straightforward. Next, we take a look at a procedure for correcting incoherent lower previsions based on pointwise dominance. This procedure can be formulated as a multi-objective linear program, and the availability of the finite characterizations provide an avenue for making these programs computationally feasible

Addressing ambiguity in randomized reinsurance stop-loss treaties using belief functions

The aim of the paper is to model ambiguity in a randomized reinsurance stop-loss treaty. For this, we consider the lower envelope of the set of bivariate joint probability distributions having a precise discrete marginal and an ambiguous Bernoulli marginal. Under an independence assumption, since the lower envelope fails 2-monotonicity, inner/outer Dempster-Shafer approximations are considered, so as to select the optimal retention level by maximizing the lower expected insurer's annual profit under reinsurance. We show that the inner approximation is not suitable in the reinsurance problem, while the outer approximation preserves the given marginal information, weakens the independence assumption, and does not introduce spurious information in the retention level selection problem. Finally, we provide a characterization of the optimal retention level

Characterizing Coherence, Correcting Incoherence

Abstract Lower previsions defined on a finite set of gambles can be looked at as points in a finite-dimensional real vector space. Within that vector space, the sets of sure loss avoiding and coherent lower previsions form convex polyhedra. We present procedures for obtaining characterizations of these polyhedra in terms of a minimal, finite number of linear constraints. As compared to the previously known procedure, these procedures are more efficient and much more straightforward. Next, we take a look at a procedure for correcting incoherent lower previsions based on pointwise dominance. This procedure can be formulated as a multi-objective linear program, and the availability of the finite characterizations provide an avenue for making these programs computationally feasible

Computational Determination of Coherence of Financial Risk Measure as a Lower Prevision of Imprecise Probability

This study is about developing some further ideas in imprecise probability models of financial risk measures. A financial risk measure has been interpreted as an upper prevision of imprecise probability, which through the conjugacy relationship can be seen as a lower prevision. The risk measures selected in the study are value-at-risk (VaR) and conditional value-at-risk (CVaR). The notion of coherence of risk measures is explained. Stocks that are traded in the financial markets (the risky assets) are seen as the gambles. The study makes a determination through computation from actual assets data whether the risk measure assessments of gambles (assets) are coherent as an imprecise probability. It is observed that coherence of assessments depends on the asset's returns distribution characteristic

Risk Measures and Upper Probabilities: Coherence and Stratification

Machine learning typically presupposes classical probability theory which implies that aggregation is built upon expectation. There are now multiple reasons to motivate looking at richer alternatives to classical probability theory as a mathematical foundation for machine learning. We systematically examine a powerful and rich class of alternative aggregation functionals, known variously as spectral risk measures, Choquet integrals or Lorentz norms. We present a range of characterization results, and demonstrate what makes this spectral family so special. In doing so we arrive at a natural stratification of all coherent risk measures in terms of the upper probabilities that they induce by exploiting results from the theory of rearrangement invariant Banach spaces. We empirically demonstrate how this new approach to uncertainty helps tackling practical machine learning problems

Imprecise Previsions for Risk Measurement

In this paper the theory of coherent imprecise previsions is applied to risk measurement. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. Consistency properties of Value-at-Risk (V aR), currently one of the most used risk measures, are investigated too, showing that it does not necessarily satisfy a weaker notion of consistency called \u2018avoiding sure loss\u2019. We introduce sufficient conditions for VaR to avoid sure loss and to be coherent. Finally we discuss ways of modifying incoherent risk measures into coherent ones