The <Lévy–Khintchine theorem> states that there is a unique triplet $(a,b,K)$, where $a\in\mathbb R$, $b\geq0$, and $K$ is a measure on $\mathbb R\setminus\{0\}$ satisfying
$$
\int_{\mathbb R}(1\wedge x^2)K(dx)<\infty,
$$
such that
$$
\boxed{
\mathbb E e^{iuX_t}
=\exp\left\{t\left(iua-\frac12bu^2
+\int_{\mathbb R\setminus\{0\}}
(e^{iux}-1-iux\mathbf1_{\{|x|\leq1\}})K(dx)
\right)\right\}.}
$$
Conversely every such triplet is the characteristic triplet of a Lévy process.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 6 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-201/6/b/solution