Lévy Processes

The family built from independent identical pieces of time, with Brownian motion and the Poisson process as its two poles.

The idea

A Lévy process is the general model of motion by independent, stationary increments; Brownian motion and the Poisson process are its two fundamental examples. The two look nothing alike — one moves continuously and never jumps, the other is constant between jumps of $1$ — but they are assembled in the same way: cut $[0, t]$ into equal blocks, and in either process the increments across the blocks are independent and share one law that depends only on the block's length. The definition isolates exactly this structure.

A process $X = (X_{t})_{t \ge 0}$ is a Lévy process when it starts at $X_{0} = 0$, has independent increments (increments over disjoint time intervals are independent), has stationary increments (the law of $X_{t} - X_{s}$ depends on $s$ and $t$ only through the elapsed time $t - s$), and is continuous in probability ($\mathbb{P}(|X_{t+h} - X_{t}| > \varepsilon) \to 0$ as $h \to 0$, for every $\varepsilon > 0$).

The continuity condition is milder than it sounds: it forbids a jump at a fixed, predetermined time and says nothing against jumps at random times, so the Poisson process satisfies it. The conditions admit both examples and still force a great deal. The value at time $t$ is a sum of $n$ independent identical increments for every $n$, a severe restriction on which laws can appear; and the structural theorem for the family says the two examples are its building materials — every Lévy process is a steady drift, plus a Brownian motion, plus jumps, running independently of one another.

Ways to work on it

• Walkthrough. The definition and a non-example, why the characteristic function is exponential in time, and the Lévy-Itô decomposition.
• Practice. Decide which candidate processes satisfy the definition, and compute covariances and interval probabilities.
• Hardest. Laws at fractional times, and processes that rescale exactly under a speeded-up clock.

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