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Figure 7.3. A set of four times (f-i, t2, t3, M with non-overlapping intervals. A key assumption of the process of Brownian motion is that W(t2) - W(f1)and W(t4) - W(t3) are independent random variables, no matter how close f3 is to f2.

Time

In compromise, we will use W(t) to denote "standard Brownian motion,'' which is defined by the following four conditions:

(3) W(t) is normally distributed with mean 0 and variance t;

(4) if {t1, t2, t3, t4} represent four different, ordered times with t1 < t2 < t3 < t4 (Figure 7.3), then W(t2) — W(t1) and W(t4) — W(t3) are independent random variables, no matter how close t3 is to t2. The last property is said to be the property of independent increments (see Connections for more details) and is a key assumption.

In Figure 7.4, I show five sample trajectories, which in the business are described as ''realizations of the stochastic process.'' They all start at 0 because of property (1). The trajectories are continuous, forced by property (2). Notice, however, that although the trajectories are continuous, they are very wiggly (we will come back to that momentarily).

For much of what follows, we will work with the ''increment of Brownian motion'' (we are going to convert regular differential equations of the sort that we encountered in previous chapters into stochastic differential equations using this increment), which is defined as dW = W (t + dt) — W (t)

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