 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-24/2/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

(X_{t})_{t \geq 0}

with the following properties:

$X_{0} = 0$ almost surely.
It has independent increments: increments over disjoint ordered time intervals are independent.
It has stationary increments: $X_{s + t} - X_{s}$ has the same law as $X_{t}$ for $s, t \geq 0$ .
It has stochastic continuity: $X_{s} \to X_{t}$ in probability as $s \to t$ .

One convention also includes càdlàg paths in the definition. Equivalently, under the intrinsic definition above one chooses the càdlàg modification, which exists for such a stochastic process. Thus the usual working version of a Lévy process has right-continuous paths with left limits. There is no assumption of finite moments or continuous paths.

 New to topics? Read the docs here!