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Stochastic Calculus / Lévy Processes and Jump Calculus

Lévy Processes: Definition, the Lévy Triple, and Key Examples

Brownian motion has three defining properties: independent increments, stationary increments, and continuous paths. Drop the continuity assumption and you get the Lévy process — the most general process with independent, stationary increments. Every Lévy process is characterised by a unique triple (b,σ2,ν)(b, \sigma^2, \nu): a drift b∈Rb \in \mathbb{R}, a diffusion coefficient σ≥0\sigma \ge 0, and a Lévy measure ν\nu on R∖{0}\mathbb{R} \setminus \{0\} that governs the frequency and size of jumps. The Lévy-Khinchin theorem says this triple completely determines the process's distribution.

In finance, the Lévy measure is the jump-intensity surface: ν(dx)\nu(dx) is roughly the expected number of jumps of size in [x,x+dx][x, x+dx] per unit time. Merton's jump-diffusion model uses ν(dx)=λ⋅fJ(x) dx\nu(dx) = \lambda \cdot f_J(x) \, dx with fJf_J log-normal; the variance-gamma model uses ν(dx)∝e−α∣x∣/∣x∣ dx\nu(dx) \propto e^{-\alpha|x|}/|x| \, dx (infinite activity, no diffusion component). Understanding the triple means understanding what jump model to reach for.

Learning objectives. By the end of this note you should be able to:
  • Define a Lévy process and explain the role of càdlàg paths.
  • State the Lévy-Khinchin theorem and interpret each component of the triple (b,σ2,ν)(b, \sigma^2, \nu).
  • Explain infinite divisibility and show BM and Poisson are infinitely divisible.
  • Classify Lévy processes as finite or infinite activity, and relate this to the integrability of ν\nu.
  • Identify the characteristic exponents of BM, Poisson, and their sum.

In this lesson

  1. Definition and Path Properties
  2. Characteristic Exponent
  3. Infinite Divisibility
  4. The Lévy-Khinchin Theorem
  5. The Lévy-Itô Decomposition
  6. Finite vs. Infinite Activity
  7. Key Examples in Finance

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