# CHAPTER 3

# Stochastic Calculus

## 3.1. WIENER PROCESS

We have already looked at the concept of a random walk, starting at zero,

The result of *N* steps of size 1 in a randomly selected direction gets the walker to position *X*; the mean of *X* is zero and the standard deviation of *X* is in the large *N* limit. The same is true for any shock of the same standard deviation and mean, for example, if each *ε _{i}* is selected from a normal distribution of standard deviation 1 and mean 0.

In this chapter, we modify the analysis slightly. We will make the steps occur every increment of time Δ*t*, and the step is going to be selected from a random distribution of mean zero and standard deviation ,

Then define

This formula is mathematically well defined for any finite *N*; because the behavior is smooth over large values of *N*, the limit for *N* → ∞, is well-defined as well. Visually we have a wiggly path. But it is impossible to correctly draw! This is because it has a fractal nature to it. No matter how small you ...

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