Comparison of variance of stochastic and non-stochastic integrals of the Brownian motion
Given that $B_t$ is the standard Brownian motion, I need to contrast the mean and variance of the stochastic integral $intlimits_{0}^t B_s dB_s = frac{1}{2}(B_t^2 - t)$ with the non-stochastic integral $intlimits_0^t B_s ds$. So I obtain zero means for both of the integrals. I obtain $Var(intlimits_{0}^t B_s dB_s) = frac{1}{2}t^2$ while $Var(intlimits_{0}^t B_s ds) = frac{1}{3}t^3$. 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?
stochastic-processes stochastic-calculus brownian-motion stochastic-integrals stochastic-approximation
asked Nov 29 at 22:01
user2348674
1608
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Given that $B_t$ is the standard Brownian motion, I need to contrast the mean and variance of the stochastic integral $intlimits_{0}^t B_s dB_s = frac{1}{2}(B_t^2 - t)$ with the non-stochastic integral $intlimits_0^t B_s ds$. So I obtain zero means for both of the integrals. I obtain $Var(intlimits_{0}^t B_s dB_s) = frac{1}{2}t^2$ while $Var(intlimits_{0}^t B_s ds) = frac{1}{3}t^3$. 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?
stochastic-processes stochastic-calculus brownian-motion stochastic-integrals stochastic-approximation
asked Nov 29 at 22:01
user2348674
1608
add a comment |
Given that $B_t$ is the standard Brownian motion, I need to contrast the mean and variance of the stochastic integral $intlimits_{0}^t B_s dB_s = frac{1}{2}(B_t^2 - t)$ with the non-stochastic integral $intlimits_0^t B_s ds$. So I obtain zero means for both of the integrals. I obtain $Var(intlimits_{0}^t B_s dB_s) = frac{1}{2}t^2$ while $Var(intlimits_{0}^t B_s ds) = frac{1}{3}t^3$. 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?
stochastic-processes stochastic-calculus brownian-motion stochastic-integrals stochastic-approximation
asked Nov 29 at 22:01
user2348674
1608
Given that $B_t$ is the standard Brownian motion, I need to contrast the mean and variance of the stochastic integral $intlimits_{0}^t B_s dB_s = frac{1}{2}(B_t^2 - t)$ with the non-stochastic integral $intlimits_0^t B_s ds$. So I obtain zero means for both of the integrals. I obtain $Var(intlimits_{0}^t B_s dB_s) = frac{1}{2}t^2$ while $Var(intlimits_{0}^t B_s ds) = frac{1}{3}t^3$. 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?
stochastic-processes stochastic-calculus brownian-motion stochastic-integrals stochastic-approximation
stochastic-processes stochastic-calculus brownian-motion stochastic-integrals stochastic-approximation
asked Nov 29 at 22:01
user2348674
1608
asked Nov 29 at 22:01
user2348674
1608
asked Nov 29 at 22:01
user2348674
1608
asked Nov 29 at 22:01
user2348674
1608
asked Nov 29 at 22:01
user2348674
1608
1608
add a comment |
add a comment |
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