Let \(X\) be a integrable random variable with density \(\varphi\) such that \(\mathbb{E}(f(X))=\mathbb{E}(X f(1 / X))\) for any bounded function \(f\).

Prove that \(\varphi(x)=\frac{1}{x^{2}} \varphi\left(\frac{1}{x}\right)\). Check that the density of \(X=e^{B_{T}-T / 2}\) satisfies this equality.

Consider \(\xi(s):=\mathbb{E}\left(X^{s}\right)\) for \(s \in \mathbb{C}\), which satisfies \(\xi(s)=\xi(1-s) .