# Association between a random variable and Radon-Nikodym derivative

Suppose that $$X$$ is a random variable and $$\frac{d\mathbb{Q}}{d\mathbb{P}}$$ is the Radon-Nikodym derivative. The quantity under consideration is as follows:

$$\begin{equation} Cov(X, \frac{d\mathbb{Q}}{d\mathbb{P}}) \end{equation}$$

My question is here: Under which conditions we can say that the above quantity is positive or negative?

• This is too generic. Unless there's something specific to those variable, you're basically asking when the integral of a product of 2 functions, one non-negative is positive. Jun 16, 2021 at 12:33
• You can suppose that X is a positive random variable. The reason I am asking this question is here. I want to show when the following inequality holds: \begin{equation} \mathbb{E}^{\mathbb{Q}}[X] < \mathbb{E}^{\mathbb{P}}[X] \end{equation} where $\mathbb{Q}$ and $\mathbb{P}$ are two equivalent measures. We know that there is a relation between $\mathbb{Q}$ and $\mathbb{P}$ via Radon-Nikodym derivative. In other words, using the change of measures technique, we have that Jun 16, 2021 at 12:42
• I guess the question is: is $X$ traded (or attainable)? Any traded asset should yield the risk-free rate under the risk-neutral measure. On the other hand, the risk-free rate is the smallest possible return in an economy (otherwise there would be arbitrage, this is trivial). If the asset is risky, then its return under the physical measure $\mathbb{P}$ should be greater than the risk-free one. Hence, if $X$ is a traded asset, we always have $\mathbb{E}^\mathbb{Q}(X)<\mathbb{E}^\mathbb{P}(X)$. Jun 16, 2021 at 13:08
• If the random variable $X$ represents the loss linked to mortality risks or catastrophe risks (which have nothing to do with the asset price), then is it possible to say something similar? Jun 16, 2021 at 21:50
• @AntoineConze This is sort of an intuitive assumption I think and boils down to the concept of risk-aversion. In the physical world, investors are risk averse, and thus by definition will not purchase an asset with higher (non-zero in this case) volatility than the risk free asset unless $\mathbb{E}^\mathbb{P}[X] > r$. I.e. you will only take on risk if you have some feeling that it is worth it to take on risk. Jun 17, 2021 at 14:21