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Stochastic Calculus / Lévy Processes and Jump Calculus

Compound Poisson Processes and Jump SDEs

The Poisson process jumps by exactly 1 at each event time. The compound Poisson process (CPP) generalises this: jumps arrive at rate λ\lambda but their sizes Y1,Y2,…Y_1, Y_2, \ldots are i.i.d. with distribution μ#\mu^\#. The Lévy measure μ=λμ#\mu = \lambda\mu^\# encodes both. This is the core of Merton's (1976) jump-diffusion model for equity prices: stock returns are a GBM plus a compound Poisson sum of log-normal shocks.

The CPP also provides the prototype for jump SDEs — stochastic differential equations driven by a combination of a diffusion BtB_t and a CPP. The Itô formula extends to these processes with an extra term that captures the effect of discrete jumps.

Learning objectives. By the end of this note you should be able to:
  • Construct a compound Poisson process from a Poisson process and i.i.d. jump sizes.
  • Compute the characteristic exponent and generator of a CPP.
  • Derive the mean and variance of a CPP from the Lévy measure.
  • Define the compensated CPP and prove it is a martingale.
  • State the Itô-type formula for a function of a CPP and apply it to moments.
  • Write down the SDE for the Merton jump-diffusion model.

In this lesson

  1. Construction
  2. Characteristic Exponent and Generator
  3. Moments and Compensated CPP
  4. Quadratic Variation
  5. Itô's Formula for Compound Poisson Processes
  6. Jump SDEs and the Merton Model

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