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Figure 7.8. The time s divides the interval 0 to t into two pieces, one from 0 to just before s (s_) and one from just after s (s+) to t. The increments in Brownian motion are then independent random variables.

a billion. So go to Vegas, but go for a good time. (In spring 1981, my first year at UC Davis, I went to a regional meeting of the American Mathematical Society, held in Reno, Nevada, to speak in a session on applied stochastic processes. Many famous colleagues were they, and although our session was Friday, they had been there since Tuesday doing, you guessed it, true work in applied probability. All, of course, claimed positive gains in their holdings.)

The transition density and covariance of Brownian motion

We now return to standard Brownian motion, to learn a little bit more about it. To do this, consider the interval [0, t] and some intermediate time s (Figure 7.8). Suppose we know that W(s) = y, for s < t. What can be said about W(t)? The increment W(t) _ W(s) = W(t) _ y will be normally distributed with mean 0 and variance t _ s. Thus we conclude that

Note too that we can make this prediction knowing only W(s), and not having to know anything about the history between 0 and s. A stochastic process for which the future depends only upon the current value and not upon the past that led to the current value is called a Markov process, so that we now know that Brownian motion is a Markov process.

The integrand in Eq. (7.27) is an example of a transition density function, which tells us how the process moves from one time and value to another. It depends upon four values: s, y, t, andx, and we shall write it as q(x, t,y, s)dx = Pr{x < W(t) < x + dx| W(s) = y}

This equation should remind you of the diffusion equation encountered in Chapter 2, and the discussion that we had there about the strange properties of the right hand side as t decreases to s. In the next section all of this will be clarified. But before that, a small exercise.

W(s_)- W(0) and W(t) - W(s+) are independent random variables

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