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Stochastic Calculus / Stochastic Integration and Itô's Formula

Derivation — Itô's Formula via Taylor Expansion and Quadratic Variation

Itô's formula states f(WT)=f(W0)+∫0Tf′(Wt) dWt+12∫0Tf′′(Wt) dtf(W_T) = f(W_0) + \int_0^T f'(W_t) \, dW_t + \frac{1}{2}\int_0^T f''(W_t) \, dt for f∈C2f \in C^2. The heuristic is clear: expand Δf\Delta f in Taylor series, note (dW)2=dt(dW)^2 = dt, sum. But a heuristic is not a proof.

This note fills the gap: a clean L2L^2 (and a.s.) proof for t=1t = 1 (the general case follows by the same argument). The argument has exactly four steps:

  1. Partition — write f(W1)−f(W0)f(W_1) - f(W_0) as a telescoping sum over a partition.
  2. Taylor — expand each Δfj\Delta f_j to second order; sandwich the second-order term between m(j,n)m(j,n) and M(j,n)M(j,n) (min/max of f′′f'' on the jj-th interval).
  3. QV — show ∑(ΔWj)2→[W,W]1=1\sum (\Delta W_j)^2 \to [W,W]_1 = 1 (in L2L^2 and a.s.) to evaluate the second-order sum.
  4. Error — show the remainder ∑o((ΔWj)2)→0\sum o((\Delta W_j)^2) \to 0 in L2L^2.
The proof is structural: it works for any continuous process whose second-order increments converge, which is why the same argument extends to all Itô processes.

In this lesson

  1. Setup
  2. Evaluating Q1Q_1: Convergence to the Stochastic Integral
  3. Evaluating the Second-Order Sum: Quadratic Variation
  4. Controlling the Error: Sandwich to Itô Correction
  5. Completing the Proof
  6. Comparison: Classical Chain Rule vs Itô's Formula

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