 Solution

A real Lévy process is a stochastic process $(X_t{)}_{t\ge 0}$ with $X_0=0$ almost surely, independent increments, stationary increments, and stochastic continuity. The latter means $X_t-X_s\to 0$ in probability as $t\to s$. Its usual path convention is càdlàg: paths are right-continuous and have left limits. Equivalently one starts with the increment and continuity-in-probability conditions and then chooses the càdlàg version.

The Lévy characteristic exponent is specified here bySome conventions use $e^{t\tilde{\psi }(\theta )}$ instead, with $\tilde{\psi }=-\psi$. The sign convention must be fixed before computing the exponent. The zero process satisfies this definition; excluding it requires an additional nondegeneracy condition.