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## Monday, September 2, 2013

### Stoch Lesson 55 deltaW and dW

See Lesson 05 about stepsize_i, and h...
See Lesson 33 for a backgrounder on the canonical Wiener variable W

Note [[Hull]] uses "z" instead of w.

Now let's explain the notation deltaW in the well-known formula

S_i+1 - S_i == deltaS = driftRate * deltaT + sigma * deltaW

Here, deltaW is basically stepsize_i, generated by the noisegen at the i'th step. That's the discrete-time version. How about the dW in the continuous time SDE? Well, dW is the stepsize_i as deltaT -> 0. This dW is from a noisegen whose variance is exactly equal to deltaT. Note deltaT is the thing that we drive to 0.

In my humble opinion, the #1 key feature of a Wiener process is that the Gaussian noisegen's variance is exactly equal to deltaT.

Another name for deltaT is h. Definition is h == T/n.

Note, as Lawler said, dW/dt is meaningless for a BM, because a BM is nowhere differentiable.