# Is first order stochastic dominance conserved under change of measure?

As the title states, my question is whether first order stochastic dominance is conserved under change of measure, for instance from the $$\mathbb{P}$$ measure to $$\mathbb{Q}$$ measure and change of numeraire to yet another measure say $$\mathbb{Q}_N$$?

I am not a probabilist, but I believe it is conserved since change of measure and/or numeraire involves multiplication by an exponential martingale which is a positive process, but if someone can give a formal proof that shows why it is or isn't invariant under change of measure that would be great.

• It is the covariance of the expotential martingale with the underlyings that matters, not the fact that it is positive. Jun 1, 2021 at 2:24

Consider a coin independently tossed 10 times. Assume under the measure $$P$$, $$Pr(H)$$ > 0.5 but not equal to 1.

Let a risk neutral person be iteratively given the gamble between getting atleast $$n$$ heads versus atleast $$n$$ tails. Clearly the person always chooses getting atleast $$n$$ heads. Let $$X$$ represent number of heads and $$Y$$ the number of tails. Thus we have

$$E(1_{X>=n})>E(1_{Y>=n})$$ for all $$n$$, and thus $$X$$ FOSD the $$Y$$ under this probability measure.

Now change to a measure where $$Pr'(H)< 0.5$$ but not equal to 0. Specify event probabilities by retaining the independence assumption. This is equivalent to the previous measure, as all event sets having 0 or 1 probability in the former retain the same probability in the latter.

Now consider the same gamble given to the same person. Now, the signs get flipped! $$E'(1_{X>=n})=n})$$ for all $$n$$, so now the $$Y$$ FOSD $$X$$. So the assertion is false.

When does it preserve FOSD? A look at some special cases below:

1. If the RV's are such that the minimum of $$X$$ exceeds the maximum of $$Y$$, then the stochastic dominance will be maintained under any change of measure, because the $$Pr(w: X(w)>X_{min})=0$$ and $$Pr(w:Y(w) under any equivalent measure, so the distributions again are non intersecting.

2. For discrete RV's $$X$$ and $$Y$$, a sufficient condition for violation of FOSD order is for the means to change order. This is because the mean can be written as the sum across all $$u$$, of $$Pr(X>=u)$$.

3. What can the Radon-Nikodym derivative (RND) look like so that FOSD is not violated?

By simple change of measure using the RND $$Z$$,

$$E'(1_{X>=u})=E(1_{X>=u})+cov(Z,1_{X>=u})$$

and

$$E'(1_{Y>=u})=E(1_{Y>=u})+cov(Z,1_{Y>=u})$$

and thus it suffices that: $$cov(Z,1_{X>=u})>cov(Z,1_{Y>=u})$$ for all $$u$$. Intuitively, $$Z$$ has more of $$X$$ in it than $$Y$$, or $$X$$ is a better predictor of $$Z$$ than $$Y$$. It also suffices of Z is independent of both X and Y.

• Great example! How to show this in more generality? Do you have a link to some source with a proof?
– user34971
Jun 1, 2021 at 5:05
• Thanks! I don't have a link unfortunately. Which assertion's proof are you looking for please? Jun 1, 2021 at 10:51
• What I'd be looking for is a theorem/proposition which states under what conditions FOSD is maintained under change of measure/numeraire and proof of this assertion. I have been googling but somehow haven't come across this (so I'm probably not searching for the right terms)
– user34971
Jun 1, 2021 at 13:34
• Please look at the added cases and see if they are of any help. Jun 1, 2021 at 13:52