Carr and Madan Fourier Transform

Question

I am bit confused by Carr and Madan's paper. In it the authors write that the Fourier transform $ c_T(k)$ is defined by

\begin{align} \psi_T(v) = \int_{ - \infty}^{\infty} e^{ivk} c_T(k)dk \end{align}

Yet traditionally it is know that the Fourier Transform formula is \begin{align} X( \omega) = \int_{- \infty}^{\infty} x(t) e^{ - i \omega t} dt \end{align}

The negative being the difference between the two.

Answer 1

Different fields use different conventions to define the Fourier transform. The one in Carr and Madan is often referred to as the "probabilist's Fourier transform". It coincides with the definition of the characteristic function.

See https://www.johndcook.com/blog/fourier-theorems/ for a nice overview and the link between the different definitions.