= Brownian motion under a quadratic time change
{c}
{title2=$B_{t^2}=\int_0^t\sqrt{2s}\,dW_s$}
The process $M_t=B_{t^2}$ is a martingale in the filtration $\mathcal G_t=\mathcal F_{t^2}$ and has <quadratic variation> $t^2$. The integral $W_t=\int_0^t(2s)^{-1/2}dM_s$ is well defined because its bracket is $\int_0^t(2s)^{-1}d(s^2)=t$. The <Lévy characterization of Brownian motion> makes $W$ Brownian in this filtration, and stochastic-integral associativity gives the displayed representation. The resulting integrand is continuous at zero despite the singular inverse integrand.
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