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Stochastic Calculus / Brownian Motion: Construction and Properties

Quadratic Variation and Non-Differentiable Paths

In classical calculus, the "infinitesimal increment" of a smooth function ff over a tiny interval [t,t+dt][t, t+dt] satisfies (df)2≈(f′(t))2(dt)2=O(dt2)(df)^2 \approx (f'(t))^2 (dt)^2 = O(dt^2) — a second-order correction that vanishes faster than dtdt. This is why ordinary calculus never needs to track squared increments.

For Brownian motion, the increment dWtdW_t is of order dt\sqrt{dt} (since Wt+dt−Wt∼N(0,dt)W_{t+dt} - W_t \sim \mathcal{N}(0, dt)). So (dWt)2≈dt(dW_t)^2 \approx dt — a first-order contribution that does not vanish. This is the origin of the Itô correction term in Itô's formula, and it is why the pricing PDE for an option contains a 12σ2S2∂2V∂S2\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} term instead of being a simple first-order PDE.
The quadratic variation [W,W]t=t[W, W]_t = t formalises this: it measures how much accumulated squared variation Brownian motion has on [0,t][0, t] and shows it equals tt in L2L^2 (and a.s. along regular partitions). The same reasoning shows BM paths have infinite total variation — hence cannot be the integral of any ordinary function.
Learning objectives. By the end of this note you should be able to:
  • Define quadratic variation of a process and compute it for SBM and BM with drift.
  • State Theorem 2.8.2 (convergence of partition sums) and understand the distinction between L2L^2 convergence and pathwise convergence.
  • Explain why infinite total variation and non-zero quadratic variation are two sides of the same coin.
  • Interpret (dWt)2=dt(dW_t)^2 = dt as the L2L^2 limit of partition sums and articulate why this forces an Itô correction in calculus.
  • Contrast BM's quadratic variation with that of a differentiable function and a bounded-variation process.

In this lesson

  1. Quadratic Variation: Definition and Motivation
  2. BM has Quadratic Variation tt
  3. Convergence: L2L^2 vs. Almost Sure
  4. Why BM Paths Have Infinite Total Variation
  5. The Itô Calculus Correction
  6. Quadratic Variation Table

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