Show that a Brownian path has infinite total variation on [0,T] almost surely, and explain what that forces on the construction of the integral of a process against dW.

Show that a Brownian path has infinite total variation on [0,T] almost surely, and explain what that forces on the construction of the integral of a process against dW.

Approach: Bound the sum of squared increments by the largest increment times the sum of absolute increments, then use continuity of the path and the known quadratic variation limit.

A continuous path of finite total variation has zero quadratic variation, so a Brownian path, whose quadratic variation is T > 0, has infinite total variation, and no pathwise Riemann-Stieltjes integral against dW exists. The bound is sum_i (dW_i)^2 ≤ max_i |dW_i| * sum_i |dW_i|. If the total variation sum_i |dW_i| stayed bounded by V, then uniform continuity of the path on a compact interval sends max_i |dW_i| to 0 as the mesh shrinks, so the left side would tend to 0, contradicting the quadratic variation limit T. Consequently the Ito integral is defined as an L^2 limit of integrals of simple adapted integrands rather than path by path, and the Ito isometry E[(∫_0^T H dW)^2] = E[∫_0^T H^2 ds] supplies the norm that makes that limit well defined.

Follow-up: Where in this argument does adaptedness of the integrand become necessary, and what breaks if the integrand looks one step ahead?

Key concepts: total variation, quadratic variation, Riemann-Stieltjes integral, Ito isometry.