Let $X$ be a continuous random variable and $Q_x$ is the associated quantile function. Show that expected shortfall $ES_X[p]$ at the confidence level $p$ which is defined as
$$ES_X[p]=\Bbb E[X|X\leq Q_x(1-p)]$$ has the representation $$ES_X[p]=\frac{1}{1-p}\int_0^{1-p} Q_x(a)da.$$
Can some one give me a hint for this?
I know the definition of quantile function $Q_X(p)=\inf\{x: F_x \geq p\}$, I can think of it on an intuitive level, but want some thoughts to get started mathematically
~length
.Dowd, K. (2007): Measuring market risk. John Wiley & Sons.
is a really good reference in my opininon. $\endgroup$ – Jan Sila Dec 8 '16 at 15:09