Stochastic Calculus for Jump Processes

The classical Itô formula applies to continuous semimartingales. When the stock price jumps, the formula acquires a jump correction term: the function value at a jump time equals the old value plus the function applied to the jump, not the infinitesimal continuous approximation. This note derives the semimartingale Itô formula for Lévy processes, establishes the quadratic variation of a jump process, and derives the stochastic exponential — the object that turns log-return SDEs into stock-price SDEs.

These tools are needed for the Fourier pricing lesson: the log-characteristic function exponent $\Psi(u)$ is computed by applying the Itô formula to $e^{iuX_t}$ and taking expectations.

Learning objectives. By the end of this note you should be able to:

state the quadratic variation of a Lévy process and distinguish the continuous and jump parts;
apply the Itô formula for semimartingales to compute $df(X_t)$ when $X$ has jumps;
derive the stochastic exponential $\mathcal{E}(X)$ for a jump process and write the stock price SDE.

In this lesson

6.1 Quadratic variation with jumps
6.2 The Itô formula for semimartingales
6.3 The stochastic exponential

Before you start

Derivation — The Lévy-Itô Decomposition and the Lévy Triple

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