Levy過程 (Levy process)

• Levy process為一般化的隨機過程，Brownian process與Poisson (Jump) process均為Levy process的特例。

　Cadlag function

• Cadlag: : right continuous function with left limits.

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• A function $f: E \rightarrow \mathbb{R}$ is called a cadlag function if $\forall t \in E$
  • The left limit of function $f(t-) = \lim_{s \rightarrow t-}f(s)$ exists.
  • The right limit of function $f(t+) = \lim_{s \rightarrow t+}f(s)$ exists.
  • $f(t+) = f(t)$.
• Then $f$ is right continuous with left limits

Levy process

• Stochastic process $X(0) = 0$ a.s.
• (Independent increment) $0 \leq t_1 < t_2 < \cdots <t_n$, $X(t_1)$, $X(t_2) - X(t_1)$, $\cdots$, $X(t_n)-X(t_{n-1})$ are all mutual independent.
• (Stationary distribution) $0 \leq S < t < \infty$, the probability distribution of $X(t)-X(s)$ coincides with that of $X(t-s)$.