Expected value under equivalent change of measure [closed]
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $\mathcal{X}\subset L^{\infty}$ with $\sup_{X\in\mathcal{X}}E_P[X]<\infty$. Let $Q$ be a probability measure equivalent to $P$ with $dQ/dP\in L^1$.
Question: Does it hold that $\sup_{X\in\mathcal{X}}E_Q[X]<\infty$?
Solution 1:
Counter-example: Take $\mathcal X =\{X \wedge n: n\geq 1\}$ where the non-negative r.v. $X$ is such that $E_PX<\infty$ but $E_PX^{2}=\infty$. Let $Q(E)=\int_E XdP$. Then $\frac {dQ}{dP}=X$ and $\sup_n E_QX \wedge n=\int X^{2}dP=\infty$.