Suppose that $X$ is a square integrable random-variable defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. It's characteristic function/Fourier transform is defined to be $$ \mathfrak{F}[X](\gamma)\triangleq \int_{\omega \in \Omega} e^{-\gamma X(\omega)} \mathbb{P}(d\omega) . $$

For any fixed $\gamma \in \mathbb{R}$, when is the map \begin{align} L^2(\Omega,\mathcal{F},\mathbb{P})\rightarrow \mathbb{R} \\ X \mapsto \mathfrak{F}[X](\gamma), \end{align} Gateau differentiable (for what X)? Moreover, what is its Gateau derivative?

  • $\begingroup$ Square integrability is not enough for your characteristic function to exist. This is rather called the moment generating function (with $-\gamma$ instead of $\gamma$). $\endgroup$ – Abdelmalek Abdesselam Mar 7 at 14:47
  • $\begingroup$ On the subspace of $L^2$ on which the MFG exists, what can be said about it derivative then? $\endgroup$ – AIM_BLB Mar 8 at 0:52

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