Integral Stochastic Orders Research Articles

Stochastic orderings of multivariate elliptical distributions

AbstractFor two n-dimensional elliptical random vectors X and Y, we establish an identity for $\mathbb{E}[f({\bf Y})]- \mathbb{E}[f({\bf X})]$, where $f\,{:}\, \mathbb{R}^n \rightarrow \mathbb{R}$ satisfies some regularity conditions. Using this identity we provide a unified method to derive sufficient and necessary conditions for classifying multivariate elliptical random vectors according to several main integral stochastic orders. As a consequence we obtain new inequalities by applying the method to multivariate elliptical distributions. The results generalize the corresponding ones for multivariate normal random vectors in the literature.

The martingale comparison method for Markov processes

AbstractComparison results for Markov processes with respect to function-class-induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach spaces. In this paper we transfer the martingale comparison method, known for the comparison of semimartingales to Markovian semimartingales, to general Markov processes. The basic step of this martingale approach is the derivation of the supermartingale property of the linking process, giving a link between the processes to be compared. This property is achieved using the characterization of Markov processes by the associated martingale problem in an essential way. As a result, the martingale comparison method gives a comparison result for Markov processes under a general alternative but related set of regularity conditions compared to the evolution system approach.

Directional Stochastic Orders with an Application to Financial Mathematics

Relevant integral stochastic orders share a common mathematical model, they are defined by generators which are made up of increasing functions on appropriate directions. Motivated by the aim to provide a unified study of those orders, we introduce a new class of integral stochastic orders whose generators are composed of functions that are increasing on the directions of a finite number of vectors. These orders will be called directional stochastic orders. Such stochastic orders are studied in depth. In that analysis, the conical combinations of vectors in those finite subsets play a relevant role. It is proved that directional stochastic orders are generated by non-stochastic pre-orders and the class of their preserving mappings. Geometrical characterizations of directional stochastic orders are developed. Those characterizations depend on the existence of non-trivial subspaces contained in the set of conical combinations. An application of directional stochastic orders to the field of financial mathematics is developed, namely, to the comparison of investments with random cash flows.

Comparison of the multivariate skew-normal random vectors based on the integral stochastic ordering

In this paper, we derive the conditions to compare multivariate skew-normal random vectors X and Y. Then, we present an identity for where the function f fulfills some weak regularity conditions. For comparison purposes, we use six important cases of the integral stochastic orders: the usual order, the convex order, the increasing convex order, the upper orthant order, the directionally convex order, and the supermodular order.

A stochastic order for the analysis of investments affected by the time value of money

Time value of money reads that an amount of money at the present time is worth more than the same amount in future because its earning capacity and inflation. This fact is reflected in multiple financial concepts and in the final result of numerous investments by means of functions which satisfy appropriate properties. Motivated by the need to compare such investments we introduce an integral stochastic order generated by those functions. The maximal generator of the order is obtained. It is proved that the new stochastic order is generated by a non-stochastic partial order and the class of preserving mappings of such a partial order. Characterizations of the order are developed. Relevant properties, as well as connections with other stochastic orderings and examples, are studied.

Aspects of optimization with stochastic dominance

We consider stochastic optimization problems with integral stochastic order constraints. This problem class is characterized by an infinite number of constraints indexed by a function space of increasing concave utility functions. We are interested in effective numerical methods and a Lagrangian duality theory. First, we show how sample average approximation and linear programming can be combined to provide a computational scheme for this problem class. Then, we compute the Lagrangian dual problem to gain more insight into this problem class.

Optimization with a class of multivariate integral stochastic order constraints

We study convex optimization problems with a class of multivariate integral stochastic order constraints defined in terms of parametrized families of increasing concave functions. We show that utility functions act as the Lagrange multipliers of the stochastic order constraints in this general setting, and that the dual problem is a search over utility functions. Practical implementation issues are discussed.

When is an integral stochastic order generated by a poset?

Given a partial order ⪯ on a set X , one can consider the class of ⪯-preserving real functions on X characterized by x ⪯ y implies f ( x ) ≤ f ( y ) . Such a class of functions allows us the generation of a binary relation ⪯ g on the set of probabilities associated with X by means of P ⪯ g Q when ∫ f d P ≤ ∫ f d Q for all ⪯-preserving functions f. In this paper we characterize when for an integral stochastic order ⪯ ˜ on the set of probabilities associated with X , there exists a partial order ⪯ on X such that the relation ⪯ g generated by the class of ⪯-preserving functions is equal to ⪯ ˜ . The above characterization is related to the maximal generator of ⪯ ˜ , a result which can be applied for the search of maximal generators of stochastic orders generated by posets. MSC:06A06, 60E15.

Generalized Increasing Convex and Directionally Convex Orders

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

Generalized Increasing Convex and Directionally Convex Orders

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

Correlation order, merging and diversification

We investigate the influence of the dependence between random losses on the shortfall and on the diversification benefit that arises from merging these losses. We prove that increasing the dependence between losses, expressed in terms of correlation order, has an increasing effect on the shortfall, expressed in terms of an appropriate integral stochastic order. Furthermore, increasing the dependence between losses decreases the diversification benefit. We also consider merging comonotonic losses and show that even in this extreme case a strictly positive diversification benefit will often arise.

Hessian orders and multinormal distributions

Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex cone H of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are equal and the difference of their covariance matrices belongs to the dual of H . Then we show that the same conditions are also sufficient for multinormal random vectors. We study several particular cases of this general result.

Orderings and Probability Functionals Consistent with Preferences

This paper unifies the classical theory of stochastic dominance and investor preferences with the recent literature on risk measures applied to the choice problem faced by investors. First, we summarize the main stochastic dominance rules used in the finance literature. Then we discuss the connection with the theory of integral stochastic orders and we introduce orderings consistent with investors' preferences. Thus, we classify them, distinguishing several categories of orderings associated with different classes of investors. Finally, we show how we can use risk measures and orderings consistent with some preferences to determine the investors' optimal choices.

Static super-replicating strategies for a class of exotic options

In this paper, we investigate static super-replicating strategies for European-type call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the different assets are traded and hence their prices can be observed in the market. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We prove that the finite market case converges to the infinite market case when the number of observed plain vanilla option prices tends to infinity. We show how to construct a portfolio consisting of the plain vanilla options on the different assets, whose pay-off super-replicates the pay-off of the exotic option. As a consequence, the price of the super-replicating portfolio is an upper bound for the price of the exotic option. The super-hedging strategy is model-free in the sense that it is expressed in terms of the observed option prices on the individual assets, which can be e.g. dividend paying stocks with no explicit dividend process known. This paper is a generalization of the work of Simon et al. [Simon, S., Goovaerts, M., Dhaene, J., 2000. An easy computable upper bound for the price of an arithmetic Asian option. Insurance Math. Econom. 26 (2–3), 175–184] who considered this problem for Asian options in the infinite market case. Laurence and Wang [Laurence, P., Wang, T.H., 2004. What’s a basket worth? Risk Mag. 17, 73–77] and Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329–342] considered this problem for basket options, in the infinite as well as in the finite market case. As opposed to Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329–342] who use Lagrange optimization techniques, the proofs in this paper are based on the theory of integral stochastic orders and on the theory of comonotonic risks.

Level-crossing ordering of semi-Markov processes and markov chains

We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

Level-crossing ordering of semi-Markov processes and markov chains

We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

Smooth generators of integral stochastic orders

The purpose of this paper is to show that many integral stochastic orders have a generator consisting of infinitely differentiable functions. This especially holds for all stochastic orders with characterizations via difference operators. The usefulness of this result is demonstrated in two applications relating to stochastic ordering of multivariate normal distributions and ordering of random vectors in a random environment.

Stochastic Orders Generated by Integrals: a Unified Study

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

Stochastic Orders Generated by Integrals: a Unified Study

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.