Levy process

Stochastic process $X=\{X_t:t\ge 0\}$ satisfying:

• $X_0=0$ Almost surely.
• Independent increments: $X_{t_{i+1}}-X_{t_i}$ are independent for $t_i\in {\mathbb{R}}_0^{+}$, $t_{i+1}\ge t_i$.
• Stationary increments: $X_t-X_s\sim X_{t-s}$ in distribution.
• Continuity in probability: $\forall ϵ\in {\mathbb{R}}^{+},t\in {\mathbb{R}}_0^{+}$ we have $\underset{h\to 0}{\lim }\mathbb{P}[|X_{t+h}-X_t|>ϵ]=0$.