Given that $B_t$ is the standard Brownian motion, I need to contrast the mean and variance of the stochastic integral $\int\limits_{0}^t B_s dB_s = \frac{1}{2}(B_t^2 - t)$ with the non-stochastic integral $\int\limits_0^t B_s ds$. So I obtain zero means for both of the integrals. I obtain $Var(\int\limits_{0}^t B_s dB_s) = \frac{1}{2}t^2$ while $Var(\int\limits_{0}^t B_s ds) = \frac{1}{3}t^3$. I am wondering what is the implication of this result? How should I interpret the observation that the stochastic integral of the Brownian motion has lower variance compared with the non-stochastic integral?
2026-03-25 11:16:22.1774437382
Comparison of variance of stochastic and non-stochastic integrals of the Brownian motion
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