Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / M / 1 / b / iv / Solution

Take $Y=U{X}_1$, where $U$ is uniform on $(0,1)$ and independent of $X_1$. Conditioning on $U$ givesFor $t$ near zero this is finite because the moment-generating function of $X_1$ is finite on a neighborhood of zero; Tonelli's theorem justifies the conditioning integral for the nonnegative exponential.