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Stochastic Calculus / Lévy Processes and Jump Calculus

The Lévy-Khinchin Representation and Infinite Divisibility

Every compound Poisson process has finitely many jumps in any bounded time interval — it is a finite-activity process. But the Lévy-Khinchin theorem covers much more: processes with infinitely many jumps in any interval (infinite activity). The variance-gamma model has this property; so does the α\alpha-stable process used to model heavy-tailed returns. To accommodate infinite activity, the Lévy measure ν\nu can satisfy ν(R)=∞\nu(\mathbb{R}) = \infty, as long as small jumps are integrable: ∫∣x∣≤1x2 ν(dx)<∞\int_{|x| \le 1} x^2 \, \nu(dx) < \infty.

The Lévy-Khinchin theorem is the analogue of the BM-is-the-only-continuous-Lévy-process theorem: it says every process with i.i.d. increments (i.e., every Lévy process) is completely characterised by a triple (b,σ2,ν)(b, \sigma^2, \nu), and every valid triple corresponds to a Lévy process. This is the master classification of jump processes.

Learning objectives. By the end of this note you should be able to:
  • Construct an infinite-activity generalised Poisson process via limits of truncated CPPs.
  • State the Lévy-Khinchin formula and explain the role of the compensation term −iξx1∣x∣≤1-i\xi x \mathbf{1}_{|x|\le 1}.
  • Interpret (b,σ2,ν)(b, \sigma^2, \nu) for the BM, Poisson, and variance-gamma processes.
  • Explain finite vs. infinite activity and finite vs. infinite variation in terms of ν\nu.
  • State the Lévy-Itô decomposition (Theorem 6.8.1) and sketch why it holds.

In this lesson

  1. From Finite to Infinite Activity
  2. The Compensation for Small Jumps
  3. The Lévy-Khinchin Theorem
  4. The Lévy-Itô Decomposition
  5. Examples of the Triple (b,σ2,ν)(b, \sigma^2, \nu)
  6. The α\alpha-Stable Processes
  7. Characteristic Exponent Examples

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