Transforming Brownian Motion

Let \(W(t)\) be standard Brownian motion on \([0,\infty)\).

  1.  Show that \(-W(t)\) is also a Brownian motion.
  2. For any \(c>0\), show that \(Y(t)= c W(t/c^2)\) is again a standard Brownian motion.
  3. Fix any \(s>0\) and define \(Z(t)=W(t+s)-W(s)\). Show that \(Z(t)\) is a standard Brownian motion. 
  4. * Define \(X(t)= t W(1/t)\) for \(t>0\) and \(X(0)=0\). Show that \(X(t)\) is a standard Brownian Motion. Do this by arguing that \(X(t)\)  is continuous almost surely, that for each \(t \geq 0\) it is a Gaussian random variable with mean zero and variance \(t\). Instead of continuity, one can rather show that \(\text{Cov}(t,s)=\mathbf E X(t) X(s) \) equals \(\min(t,s)\). To prove continuity, notice that
    \[ \lim_{t \rightarrow 0+} t W(1/t) = \lim_{s \rightarrow \infty} \frac{W(s)}{s}\]


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