Quadratic Variation of Geometric Brownian Motion Versus Squared Increments

Summary

This question concerns a common distinction in stochastic calculus: the quadratic variation of a process is not generally the square of its total increment. For geometric Brownian motion, Itô’s rules imply that the infinitesimal squared increment has a diffusion contribution proportional to the squared volatility and the time step. Integrating that contribution describes quadratic variation over an interval.

By contrast, squaring the integral of the increments gives the square of the endpoint change, which includes cross terms and is not generally equal to the integral of squared infinitesimal increments. The document raises this identity as a point of confusion but provides no answer, derivation, or numerical evidence. Its useful lesson is therefore conceptual: distinguish quadratic variation from the squared net change, and interpret differential notation through the rules of Itô calculus.

Key ideas

• For a diffusion, the squared infinitesimal increment retains a term proportional to the time step.
• Quadratic variation accumulates these squared infinitesimal increments over time.
• The square of the total process change is generally different from its quadratic variation.
• The document poses this distinction but does not supply a resolution or supporting derivation.

Tags

# On Geometric Brownian motion and Itô's formula


# On Geometric Brownian motion and Itô's formula












Let $S_t$ be a geometric brownian motion such as $$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$ where $W$ is a standard Brownian motion.

With Itô's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=dt$, we can show that

$$ (dS_t)^2=\sigma^2 S_t^2 dt $$ Problem:

At the beginning of a demonstration, the author of an article uses the following equality: $$ (\int^T_0 dS_t)^2=\int^T_0 (dS_t)^2 $$ I can't see how he got such a result knowing that I find with Itô's lemma that : $$ (\int^T_0 dS_t)^2=\int^T_0 (dS_t)^2+2(S_0^2-S_0S_T+\int^T_0 S_tdS_t) $$ Help me see a little more clearly. Thank you in advance.

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