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

The Goodman-Nguyen Relation within Imprecise Probability Theory

The Goodman-Nguyen relation is a partial order generalising the implication (inclusion) relation to conditional events. As such, with precise probabilities it both induces an agreeing probability ordering and is a key tool in a certain common extension problem. Most previous work involving this relation is concerned with either conditional event algebras or precise probabilities. We investigate here its role within imprecise probability theory, first in the framework of conditional events and then proposing a generalisation of the Goodman-Nguyen relation to conditional gambles. It turns out that this relation induces an agreeing ordering on coherent or C-convex conditional imprecise previsions. In a standard inferential problem with conditional events, it lets us determine the natural extension, as well as an upper extension. With conditional gambles, it is useful in deriving a number of inferential inequalities.Comment: Published version: http://www.sciencedirect.com/science/article/pii/S0888613X1400101

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

Coherent Risk Measures and Upper Previsions

In this paper coherent risk measures and other currently used risk measures, notably Value-at-Risk (VaR), are studied from the perspective of the theory of coherent imprecise previsions. 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. We also show that Value-at-Risk does not necessarily satisfy a weaker notion of coherence called ‘avoiding sure loss’ (ASL), and discuss both sufficient conditions for VaR to avoid sure loss and ways of modifying VaR into a coherent risk measure.Coherent risk measure, imprecise prevision, Value-at-Risk, avoiding sure loss condition

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

Weak consistency for imprecise conditional 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, centered 2-convex previsions satisfy the Generalized Bayes Rule and always have a 2-convex natural extension. We discuss then the rationality requirements of 2-convexity and 2-coherence from a desirability perspective. Among the uncertainty concepts that can be modelled by 2-convexity, we mention generalizations of capacities and niveloids to a conditional framework

A Sandwich Theorem for Natural Extensions

The recently introduced weak consistency notions of 2-coherence and 2-convexity are endowed with a concept of 2-coherent, respectively, 2-convex natural extension, whose properties parallel those of the natural extension for coherent lower previsions. We show that some of these extensions coincide in various common instances, thus producing the same inferences

Jensen's and Cantelli's Inequalities with Imprecise Previsions

We investigate how basic probability inequalities can be extended to an imprecise framework, where (precise) probabilities and expectations are replaced by imprecise probabilities and lower/upper previsions. We focus on inequalities giving information on a single bounded random variable $X$, considering either convex/concave functions of $X$ (Jensen's inequalities) or one-sided bounds such as $(X\geq c)$ or $(X\leq c)$ (Markov's and Cantelli's inequalities). As for the consistency of the relevant imprecise uncertainty measures, our analysis considers coherence as well as weaker requirements, notably $2$-coherence, which proves to be often sufficient. Jensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. After discussing upper and lower Markov's inequalities, Cantelli-like inequalities are proven with different degrees of consistency for the related lower/upper previsions. In the case of coherent imprecise previsions, the corresponding Cantelli's inequalities make use of Walley's lower and upper variances, generally ensuring better bounds.Comment: Published in Fuzzy Sets and Systems - https://dx.doi.org/10.1016/j.fss.2022.06.02

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