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Suppose $\{B_s,s>0\}$ is a standard brownian motion process. Is $Y_s=sB_{1\over s},\ s>0$ a brownian motion or (stardard). I have found that $Y_0=0$ and $Y_s\sim N(0,1)$ as $B_s\sim N(0,s)$, so it remains to show that it is stationary increment and independent increment. But i am not sure how to do it.

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    It is, but $\lambda N(0,\sigma^2) \sim N(0,\lambda^2\sigma^2)$.2012-12-05

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Have you heard of Gaussian processes ? If you have, you only have to check that $(Y_s)$ has the same covariance function as the Brownian motion.

If you haven't, don't worry, it's very simple here: you are interested in the law of the couple $(sB_{1/s},tB_{1/t}-sB_{1/s})$ when $0 < s . This is a 2 dimensional centered Gaussian vector, so its law is entirely determined by its covariance matrix. In the end, you have to compute $E(sB_{1/s} tB_{1/t})$.

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    First, the two methods are in fact the same. Second, I'm not trying to show anything, you are. Anyway trying to show that $cov(Y_t,Y_s)=\min(s,t)$ is a good intuition.2012-12-05