Building the Itô Integral: From Simple Processes to L²

In the Black-Scholes model, a trading strategy $\Delta_t$ (number of shares held at time $t$) generates a P&L of $\int_0^T \Delta_t \, dS_t$ over the trading horizon $[0, T]$. When $S_t = S_0 e^{\sigma W_t + (r-\sigma^2/2)t}$ is a GBM, understanding $\int_0^T \Delta_t \, dS_t$ requires defining an integral with respect to a BM path — and BM paths are nowhere differentiable with infinite total variation. The Stieltjes integral, which requires bounded variation, cannot handle this.

The construction proceeds in stages:

1. Elementary processes (step functions of time) — the integral is explicit.
2. Bounded continuous adapted processes — use $L^2$ approximation by elementary processes.
3. Unbounded adapted processes — localise using stopping times.

• Define a simple (elementary) process and write the Itô integral explicitly as a sum of increments.
• State the four properties of the integral of simple processes: linearity, martingale property, variance rule, continuity.
• Explain why bounded variation functions have zero quadratic variation and cannot be the integrator in a meaningful financial theory.
• Describe the $L^2$ extension: how to approximate bounded continuous adapted processes by simple processes and take the $L^2$ limit.
• State Proposition 3.2.5 (properties of the general integral) and identify when the martingale property can fail.

1. Why the Stieltjes Integral Fails
2. Step 1: Elementary (Simple) Processes
3. Four Properties of the Simple-Process Integral
4. Step 2: Extension to Bounded Continuous Adapted Processes
5. Step 3: Unbounded Adapted Processes
6. Contrast with the Riemann Integral