Taylor expansion of expectation in financial modelling problem

Question

Jin and Kawczak (2003) "Birnbaum-Saunders and Lognormal Kernel Estimators for Modelling Durations in High Frequency Financial Data" expanded as $$\mathbb{E}[\lambda_x^{-1} f(\lambda_x ) ]=\mu_x^{-1} f(\mu_x )+\frac{1}{2}\color{red}{\left(2x^{-3} f(x)-2x^{-2} f'(x)+x^{-1} f'' (x)\right)} V_\lambda$$ I know that the function is expanded through Taylor series. My question is which Taylor series is used for the red part? Here $\lambda_x$ is a lognormal random variable with $(\ln x, 2 \ln (1 + b))$, $V$ is variance, $\mu_x$ is mean and $b$ is constant.

Answer 1

I can't quite figure out what your question is saying, but I'll give an answer that will hopefully be useful in figuring out what's going on underneath.

Recall the Taylor expansion of the expectation: $$ \mathbb{E}[f(x)]\approx f(\mathbb{E}[X]) + \frac{1}{2}f''(\mathbb{E}[X])\mathbb{V}[X] $$ for some RV $X$.

Let us define $$ g(\lambda_x) = \lambda_x^{-1}f(\lambda_x) $$ Further, let $\mu=\mathbb{E}[\lambda_x]$ and $\sigma^2=\mathbb{V}[\lambda_x] $. Then: $$ \mathbb{E}[g(\lambda_x)] \approx g(\mu) + \frac{g''(\mu)}{2}\sigma^2 $$ Notice that: \begin{align} g'(a) &= a^{-1}f'(a) - a^{-2}f(a)\\ g''(a) &= 2a^{-3}f(a) - 2a^{-2}f'(a) + a^{-1}f''(a) \end{align} Thus, I get: $$ \mathbb{E}[g(\lambda_x)] \approx \mu^{-1}f(\mu) + \frac{1}{2}\left( 2\mu^{-3}f(\mu) - 2\mu^{-2}f'(\mu) + \mu^{-1}f''(\mu) \right)\sigma^2 $$ which is suspiciously close to the quantity in question.