Stochastic ordering

Stochastic orders are ordering relations for random variables. They generalize the concept of greater than and less on random variables and serve as the comparison of risks in the insurance industry. The theory of stochastic systems is a recent mathematical sub-region and has experienced a strong development in the last decades and found a wide application in mathematical finance, economic research and operations research. Special stochastic orders have already been explored in the post-war period, the first comprehensive monograph of the topic by Dietrich Stoyan was published in 1977. Are considered, each with different application areas many stochastic systems, the theory of integral orders, it allows the process of investigating different orders of harmonized methods.

Order in the Middle

An in practice often used ad hoc order arises from the comparison of expected values ​​. They say the real random variable is on average smaller than the random variable if and only if. This order does not consider other properties of distributions such as variance or skewness.

Ordinary stochastic order

A special role is played by the usual stochastic order ( engl.: usual stochastic order). Starting to be able to perform even under uncertainty of the need assessments and decisions, it transmits the ideological concept of order of real numbers ( formalized by the order axioms of real numbers ) on real-valued random variables. She was examined before other stochastic orders and 1947 ( Mann-Whitney ) and 1956 ( Lehmann) used in mathematical work. From Samuel Karlin it was introduced in Operations Research in 1960. In the economic literature it is known as first order stochastic dominance.

Definition: Let and be real random variables. is greater than or equal with respect to the ordinary stochastic order if and only if for all

This means that for an arbitrary barrier, the values ​​of with greater (or equal ) than the values ​​of probability over. As a symbol is often introduced. This can also be used as a criterion for the distribution functions and formulate:

An equivalent definition is

This definition can be used for random variables with values ​​in a topological space on which a partial order defined in a natural way to generalize ( where the order with the topology must be compatible and therefore the seclusion of the amount is required. )

The set of roads makes the statement that for random variables with values ​​in a Polish space with partial order is equivalent to the fact that a probability space and two random variables exist which are respectively as distributed and

. meet The proof of this existence theorem is not constructive except in simple cases.

Random variables with the same mean and different distribution can not be compared with the usual stochastic order, depending on the application, it is necessary to consider other stochastic orders.

Integral orders

Many stochastic systems of interest ( for random variables with values ​​in an ordered Polish space ) can be just like the ordinary stochastic order on classes of " test functions " (with )

Define. Such stochastic systems called integral orders, ie, the stochastic order of generated, their generator. The so -defined operation is not necessarily transitive, a way to circumvent this problem is the restriction to such functions and random variables for which there are all expected values ​​.

This is a probabilistic concept formation - ie, it only depends on the distributions of and on and integral orders can be equipped with the- algebra of Borel sets as well as for distributions on ordered Polish spaces define:

With probability measures to be.

Often an order already from a subclass of generated is (ie, it is sufficient to check the right side of for some of the functions to reason ), in the case of the ordinary stochastic order infinity, for example, for all isotonic measurable indicator functions or for all differentiable functions isotonic.

In the application order is often of interest to conclude from the left side of the validity of the right side for a specific function. The question for which functions as possible, led AW Marshall in 1991 on the concept of the maximum generator.

In determining a maximum generator, a restriction is needed on certain functions such as in securing the transitivity. It introduces a weighting function and only - limited functions and dimensions, which can be integrated with respect to considered.

The important for the actuarial stop- loss order is another example of an integral order. It is generated by the class of the real functions of the form. The stop- loss order is weaker than the ordinary stochastic order and is used, among other things, the comparison of amounts of loss and as a criterion in the choice of a premium principle. Will and interpreted as risks, means that the stop-loss reinsurance premium for the risk is greater than that for every election of the deductible.

Dependence orders

On a measure space, consider two random vectors and so is the comparison of the covariances

A comparison of the degree of mutual dependence of the components of the two vectors is possible. Dependence orders ( engl. dependence orders) generalize this concept and are also of interest to the insurance industry, where accumulation and dependency of risks such as hail or flood damage pose a financial risk to the insurer and the traditionally made ​​in the actuarial assumption of independence of risks leads to an underestimation of the ruin probability.

The dependence orders is one of the super modular order. It is generated by the class of the super- modular functions

It allows together with the stop- loss order to compare multivariate risk portfolios with dependent risks.

More precisely: Let and be two risk portfolios with the same marginal distributions. Is the vector of portfolio risks of supermodular less than the Portfolio, the random variable of losses of portfolio is smaller than that of portfolio with respect to the stop- loss order, and thus the price of a stop-loss reinsurance: