By applying properties (1)-(4) to the increment of Brownian motion, show that:

(3) dW is normally distributed;

(4) if d W1 = W(t1 + dt) - W(t1) and d W2 = Wfe + dt) - Wfe) where t2 > t1 + dt then dW1 and dW2 are independent random variables (for this last part, you might want to peek at Eqs. (7.29) and (7.30)).

Now, although Brownian motion and its increment seem very natural to us (perhaps because have spent so much time working with normal random variables), a variety of surprising and non-intuitive results emerge. To begin, let's ask about the derivative dW/dt. Since W(t) is a random variable, its derivative will be one too. Using the definition of the derivative

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