Abstract
We consider orderings between one-dimensional r.v.s X, Y (risks). An obvious example is a.s. ordering, X ≤a.s.Y . We shall, however, mainly be concerned with orderings which only involve the distributions, i.e., which are law invariant.
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Notes
- 1.
To clarify the relevance of this construction, note that the first two ways I), II) to define an ordering only involve the distributions of X, Y , and that X, Y could in principle be defined on different probability spaces \((\varOmega _X,{\mathscr {F}}_X,{\mathbb P}_X)\), \((\varOmega _Y,{\mathscr {F}}_Y,{\mathbb P}_Y)\). Even if X, Y were defined on a common probability space \((\varOmega ,{\mathscr {F}},{\mathbb P})\), the desired coupling property would not necessarily hold, cf. Exercise 1.1.
- 2.
In the definition of \({\mathcal C}_{\mathrm {incr}}\) and similar classes in the following, it is tacitly understood that the domain is some subset of \({\mathbb R}\) containing both the support of X and the support of Y . Also, it is implicit that \({\mathbb E} f(X)\le {\mathbb E} f(Y)\) is only required for functions \(f\in {\mathcal C}_{\mathrm {incr}}\) such that the expectations are well defined.
- 3.
The second condition can be restated that the stop-loss-transform of − X is dominated by the stop-loss-transform of − Y (replace b by − b).
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Asmussen, S., Steffensen, M. (2020). Chapter VIII: Orderings and Comparisons. In: Risk and Insurance. Probability Theory and Stochastic Modelling, vol 96. Springer, Cham. https://doi.org/10.1007/978-3-030-35176-2_8
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