In Example 14.45, we constructed a (canonical) process (X t )t∈[0,∞) with independent stationary normally distributed increments. For example, such a process can be used to describe the motion of a particle immersed in water or the change of prices in the stock market. We are now interested in properties of this process X that cannot be described in terms of finite-dimensional distributions but reflect the whole path t → X t . For example, we want to compute the distribution of the functional F(X) := supt∈[0,1] X t . The first problem that has to be resolved is to show that F is a random variable.
In this chapter, we investigate continuity properties of paths of stochastic processes and show how they ensure measurability of some path functionals. Then we construct a version of X that has continuous paths, the so-called Wiener process or Brownian motion. Without exaggeration, it can be stated that Brownian motion is the central object of probability theory.
KeywordsBrownian Motion Continuous Path Local Martingale Brownian Bridge Path Property
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