An alternative to Brownian motion the Poisson increment

This has been a long chapter, and it is nearly drawing to a close. Before closing, however, I want to introduce an alternative to Brownian motion as a model for stochastic effects. Recall that Brownian motion is a continuous process, but many processes in life are discrete. In gambling, one's holdings usually change by a discrete amount with each hand. As we discussed above, offspring come in discrete units and catastrophes may kill a large number of individuals at one time (Mangel and Tier 1993, 1994). Very often energy reserves or position changes in a more or less discrete manner.

To capture such effects, let us consider the increment of the generalized Poisson process, defined according to v with probability 2 cdt + o(dt) dn = 0 with probability 1 - cdt + o(dt) (7.96)

where v is the intensity of the process, since it describes the size of the jumps, and c is the rate of the process. It would be possible to standardize the intensity to v = 1, if we wanted to do so, but for this illustration it is better left as it is.

Exercise 7.15 (E)

Compute the mean and variance of dP.

We could imagine now a process that satisfies, for example dX = rXdt + dP (7.97)

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