How risky is a random process?

Statistics: A Series of Textbooks and Monographs, 2005

In this paper we review and extend some key results on the stochastic ordering of risks and on bounding the influence of stochastic dependence on risk functionals. The first part of the paper is concerned with a.s. constructions of random vectors and with diffusion kernel type comparisons which are of importance for various comparison results. In the second part we consider generalizations of the classical Fréchet-bounds, in particular for the distribution of sums and maxima and for more general monotonic functionals of the risk vector. In the final part we discuss three important orderings of risks which arise from ∆-monotone, supermodular, and directionally convex functions. We give some new criteria for these orderings. For the basic results we also take care to give references to "original sources" of these results.

This paper studies and describes stochastic orderings of risk/reward positions in order to define in a natural way risk/reward measures consistent/isotonic to investors' preferences. We begin by discussing the connection among the theory of probability metrics, risk measures, distributional moments, and stochastic orderings. Then, we demonstrate how further orderings could better specify the investor's attitude toward risk. Finally, we extend these concepts in a dynamic context by defining and describing new risk measures and orderings among stochastic processes with and without considering the available information in the market.

This paper analyzes individual decision making under risk. It is assumed that an individual does not have a preference relation on the set of risky lotteries. Instead, the primitive of choice is a choice probability that captures the likelihood of one lottery being chosen over the other. Choice probabilities have a stochastic utility representation if they can be written as a non-decreasing function of the difference in expected utilities of the lotteries. Choice probabilities admit a stochastic utility representation if and only if they are complete, strongly transitive, continuous, independent of common consequences and interchangeable. Axioms of stochastic utility are consistent with systematic violations of betweenness and a common ratio effect but not with a common consequence effect. Special cases of stochastic utility include the Fechner model of random errors, Luce choice model and a tremble model of Harless and Camerer (1994).

Journal of Economic Theory, 1992

The principle that rational agents should maximize expected utility or choiceworthiness is intuitively plausible in many ordinary cases of decision-making under uncertainty. But it is less plausible in cases of extreme, low-probability risk (like Pascal's Mugging), and intolerably paradoxical in cases like the St. Petersburg and Pasadena games. In this paper I show that, under certain conditions, stochastic dominance reasoning can capture most of the plausible implications of expectational reasoning while avoiding most of its pitfalls. Specifically, given sufficient background uncertainty about the choiceworthiness of one's options, many expectation-maximizing gambles that do not stochastically dominate their alternatives "in a vacuum" become stochastically dominant in virtue of that background uncertainty. But, even under these conditions, stochastic dominance will generally not require agents to accept extreme gambles like Pascal's Mugging or the St. Petersburg game. The sort of background uncertainty on which these results depend looks unavoidable for any agent who measures the choiceworthiness of her options in part by the total amount of value in the resulting world. At least for such agents, then, stochastic dominance offers a plausible general principle of choice under uncertainty that can explain more of the apparent rational constraints on such choices than has previously been recognized.

SSRN Electronic Journal, 2000

In this paper I analyze operational measure of riskiness defined by . I give simple intuition behind their main result.

RAIRO - Operations Research, 1999

In this paper, we develop some stochastic dominance theorems for the location and scale family and linear combinations of random variables and for risk lovers as well as risk averters that extend results in Hadar and Russell (1971) and Tesfatsion (1976). The results are discussed and applied to decision-making.

Social Choice and Welfare, 2007

In this paper, we propose the infimum of the Arrow–Pratt index of absolute risk aversion as a measure of global risk aversion of a utility function. We show that, for any given arbitrary pair of distributions, there exists a threshold level of global risk aversion such that all increasing concave utility functions with at least as much global risk aversion would rank the two distributions in the same way. Furthermore, this threshold level is sharp in the sense that, for any lower level of global risk aversion, we can find two utility functions in this class yielding opposite preference relations for the two distributions.

Journal of Mathematical Economics, 2011

This paper extends to bivariate utility functions, Eeckhoudt et al.'s (2009) result for the combination of 'bad' and 'good'. The decision-maker prefers to get some of the 'good' and some of the 'bad' to taking a chance on all the 'good' or all the 'bad' where 'bad' is defined via (N, M)-increasing concave order. We generalize the concept of bivariate risk aversion introduced by Richard (1975) to higher orders.

RePEc: Research Papers in Economics, 2016

We characterise two new orders of desirability of gambles (risky assets) that are natural extensions of the stochastic dominance order to complete orders, based on choosing optimal proportions of gambles. These orders are represented by indices, which we term the S index and the G index, that are characterised axiomatically and by wealth and utility uniform dominance concepts. The S index can be viewed as a generalised Sharpe ratio, and the G index can be used for maximising the growth path of a portfolio.

Wiley-Blackwell eBooks, 2011

The goals of this chapter are the following: • To explore the relationship between preference relations and quasi-semidistances. • To introduce a universal description of probability quasisemidistances in terms of a Hausdorff structure. • To provide examples with first-, second-, and higher-order stochastic dominance and to introduce primary, simple, and compound stochastic orders. • To explore new stochastic dominance rules based on a popular risk measure. • To provide a utility-type representation of probability quasisemidistances and to describe the degree of violation utilized in almost stochastic orders in terms of quasi-semidistances.

SSRN Electronic Journal, 2000

This paper extends some well-known univariate stochastic dominance results to multivariate stochastic dominances for risk averters and risk seekers, respectively, when the attributes are assumed to be independent and the utility is assumed to be additively separable. Under these assumptions, we develop some properties of multivariate stochastic dominances for risk averters and risk seekers, respectively. For example, we prove that multivariate stochastic dominances are equivalent to the expected-utility maximization for risk averters and risk seekers, respectively. We show that the hierarchical relationship exists for multivariate stochastic dominances. We develop some properties for non-negative combinations and convex combinations of random variables of multivariate stochastic dominance.

Social Choice and Welfare, 2010

Incomplete preferences over lotteries on a finite set of alternatives satisfying, besides independence and continuity, a property called bad outcome aversion are considered. These preferences are characterized in terms of their specific multi-expected utility representations (cf. Dubra et al., 2004), and can be seen as generalized stochastic dominance preferences.

European Journal of Operational Research, 1999

Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible tradeoff analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resulting mean-variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean-risk model consistent with the second degree stochastic dominance, provided that the trade-off coefficient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome-Risk diagram, which appears to be particularly useful for comparing uncertain outcomes.