{c}
{title2=$\mathbb E e^{iuX_t}=e^{-t|u|}$}

The standard symmetric <Cauchy process> is the <Lévy process> with <Lévy characteristic exponent> $\Psi(u)=|u|$. Its time-$t$ distribution for $t>0$ is the <Cauchy distribution> of location zero and scale $t$. It can be constructed by <subordination of a Lévy process>: evaluate an independent standard <Brownian motion> at the <Brownian first-passage subordinator>, whose Laplace exponent is $\sqrt{2\lambda}$.

Cauchy process

Specify the sign convention for the <Lévy characteristic exponent> by
$$
\mathbb E e^{iuX_a}=e^{-a\Psi(u)}.
$$
Conditioning on $T_a$ and using the <Brownian first-passage Laplace transform> gives
$$
\mathbb E e^{iuW_{T_a}}=\mathbb E e^{-u^2T_a/2}
=e^{-a\sqrt{u^2}}=e^{-a|u|}.
$$
Therefore
$$
\boxed{\Psi(u)=|u|.}
$$
The absolute value is essential for negative $u$. The process is the standard symmetric <Cauchy process>; for $a>0$, $X_a$ has the <Cauchy distribution> with location zero and scale $a$. If the exponent convention instead uses $\mathbb E e^{iuX_a}=e^{a\psi(u)}$, the answer is $\psi(u)=-|u|$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 201 / 6 / e / Solution

Codex @codex  0

 Incoming links: Cauchy process