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Stochastic Calculus / Lévy Processes and Jump Calculus

Stochastic Integration for Lévy Processes and Itô Formula with Jumps

The Itô integral ∫0tHs dBs\int_0^t H_s \, dB_s was built for continuous integrators. When the integrator XtX_t is a Lévy process with jumps, the integral requires two modifications. First, the integrand HsH_s must be left-continuous (predictable): a bettor cannot react to a jump the instant it occurs. Second, the Itô formula acquires an extra term capturing the discrete jump in ff:
df(Xt)=f′(Xt−) dXtc+σ22f′′(Xt−) dt+[f(Xt−+ΔXt)−f(Xt−)],df(X_t) = f'(X_{t-}) \, dX_t^c + \frac{\sigma^2}{2}f''(X_{t-}) \, dt + [f(X_{t-} + \Delta X_t) - f(X_{t-})],

where dXtc=m dt+σ dBtdX^c_t = m \, dt + \sigma \, dB_t is the continuous part and the last term activates only at jump times. Recognising this extra term is essential for pricing in jump-diffusion models (Merton), where the BS PDE gains a jump-integral term.

Learning objectives. By the end of this note you should be able to:
  • Define the stochastic integral ∫H dY\int H \, dY with respect to a square-integrable jump martingale, state its linearity, martingale property, and variance rule.
  • Explain why the integrand must be left-continuous (predictable), and why right-continuous integrands fail for CPPs.
  • State and apply the Itô formula for a general Lévy process.
  • Use the Itô formula with jumps to derive the BS-like PDE for the Merton model.
  • Compute quadratic variation for a jump process.

In this lesson

  1. Integration with Respect to a CPP
  2. Integration with Respect to a General Lévy Process
  3. The Itô Formula with Jumps
  4. The Merton Model PDE
  5. Quadratic Variation for Jump Processes

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