Wiener Process

A Wiener Process is defined as:

A stochastic process indexed by nonnegative real numbers t with the following properties:

1. $W_0 = 0$ that is to say the starting value is 0
2. the function t -> $W_t$ is continuous in t.
3. the process $\{W_t\}_{t\ge0}$ has stationary independent increments
4. any delta increment given by ${W_{t+s} - W_{s}}$ has the Normal distribution (0,t)

Independent increments given by $0 \le s*1 < t_1 \le s_2 < t_2 ...$ produces the increment random variables $W*{t*1} - W*{s*1}, W*{t*2} - W*{s_2}$ are jointly independent. That is the increments are not related to either other. How likely is this assumption to be true?

We describe it as stationary increments because for any given increment $W_{t+s} - W_s$, the distribution of the increment will be the same distribution of $W_t$. An intitution for this is that the increments are centered at 0 and with some variance. The sum of these increments over a non-overlapping time interval will be independent. By basic variance rules, the variance of the total increment will be sum of the individual increments. Note that the constant of proportional is 1 between the increment variances and time allowing for this.

A Wiener process is an example of a L´evy process: a staochastic process with staionary, independent increments. The Wiener process is the intersection of the Gaussian process with L´evy process.

Derivation of Brownian Motion

Brownian motion can be thought of as the limit of rescaled simple random walks.

A discrete random walk is then formulated as $S_k = \sum_{j=1}^{k} X_j$ where ${X_j = \{-1,1\}}$ and both events are equally likely.

Since $X_j$ is symmetric, the mean is 0. The variance because all $X_j$ is independent is therefore the sum of all $X_j$.