Stochastic Order¶

Definition¶

For loss variables $X,Y$ , stochastic order $X\le_{\mathrm{st}}Y$ means $F_X(t)\ge F_Y(t)$ for every real threshold $t$ . Equivalently, $\mathbb{P}(X>t)\le\mathbb{P}(Y>t)$ for every $t$ . Equal laws satisfy the weak order in both directions.

Interpretation¶

The larger ordered loss has at least as much probability above every threshold. For losses this is an adverse direction; for wealth or gains the same mathematical direction is favourable.

![Step CDFs of X ~ Bernoulli(0.2) and Y ~ Bernoulli(0.4): on 0,1) the CDF of X is 0.8 and that of Y is 0.6, and both are 0 before 0 and 1 from 1 on.

In the figure the cumulative distribution function (CDF) of $X$ is never below that of $Y$ : the two coincide below 0 and from 1 onwards, and on $[0,1)$ the shaded gap is $F_X-F_Y=0.8-0.6=0.2$ $0.2$ . Equivalently $\mathbb{P}(Y>t)-\mathbb{P}(X>t)=0.2$ there, so $Y$ has at least as much probability above every threshold, which is $X\le_{\mathrm{st}}Y$ $Y$ .

Move $\mathbb P(X=1)$ and $\mathbb P(Y=1)$ $1)$ : $X\le_{\mathrm{st}}Y$ holds exactly when the CDF of $X$ is never below that of $Y$ , here when $\mathbb P(X=1)\le\mathbb P(Y=1)$ $1)$ . Swap the order of the two probabilities to see the verdict flip.

The cumulative distribution function gives probability at or below a threshold, while the survivor function gives its complementary upper tail. Their opposite inequality directions express the same comparison of loss sizes.

Key Results¶

$X\le_{\mathrm{st}}Y$ if and only if $\mathbb{E}[h(X)]\le\mathbb{E}[h(Y)]$ for all bounded increasing $h$ .
For increasing wealth utility $u$ , $\mathbb{E}[u(w-X)]\ge\mathbb{E}[u(w-Y)]$ whenever defined.
Increasing transformations preserve stochastic order.
With finite means, the order implies $\mathbb{E}[X]\le\mathbb{E}[Y]$ $E [Y]$ ; equal means then force equal distributions.

Examples¶

If $X\sim\operatorname{Bern}(0.2)$ and $Y\sim\operatorname{Bern}(0.4)$ $Bern (0.4)$ , $X\le_{\mathrm{st}}Y$ $Y$ . The result compares laws and does not require a specified joint distribution.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 8

Aliases: first-order stochastic dominance, usual stochastic order, FSD, first-order dominance, stochastically larger, first-degree stochastic dominance Keywords: X ≤_st Y, F_X(t) ≥ F_Y(t), survivor function comparison, increasing test functions

Last updated: 17 October 2026 21:53