Expected value of Normal random variable and Log-Normal random variable

Question

In a text book I'm currently reading but also in Bakshi, Chao, and Chen (1997), they write that a normally distributed random variable $Y = \ln(1+X)$ has mean $\ln(1 + \mu_{X}) - \frac{1}{2}\sigma_{X}^{2}$ and variance $\sigma_{X}^{2}$, where $X$ is log-normally distributed. My question is therefore, how did they derive the mean and variance. I was first thinking of using LOTUS, but for some reason I'm not getting the right results. I would appreciate some clarity on this problem.

Answer 1

They used this formula:
If pdf of $X$ is $f_X(x)$ and cdf of $X$ is $F_X(x)$ and you have $Y=\ln(1+X)$, this means that: $$F_Y(x)=F_X(e^x-1)\text{ and }f_Y(x)=e^x\cdot f_X(e^x-1)$$ If $X\sim\text{Log-Normal}(\mu,\sigma)$ this means that

$$f_X(x)={\dfrac {1}{x\sigma {\sqrt {2\pi }}}} \exp \left(-{\dfrac {\left(\ln x-\mu \right)^{2}}{2\sigma ^{2}}}\right)\text{ and }F_X(x)={\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}$$ In general if $Y=h(X)$ and $h'(x)>0$ you have that $F_Y(x)=F_X(h^{-1}(x))$