Brownian Motion

Brownian Motion is the pattern for measuring the convergence of random walk through continuous timing.
discrete random walk discrete random walk is a tool used to construct Brownian Motion. It is a random walk which only takes on two discrete values at any given time: \Delta and its additive inverse -\Delta. These two cases take place at probabilities \pi and 1-\pi.
Therefore, the expected return over each time k is:

\begin{equation} \epsilon_{k} = \begin{cases} \Delta, p(\pi) \\ -\Delta, p(1-\pi) \end{cases} \end{equation}

(that, at any given time, the expectation of return is either—with probability π—\Delta, or–with probability 1-π—-\Delta.
This makes \epsilon_{k} independently and identically distributed. The price, then, is formed by:

\begin{equation} p_{k} = p_{k-1}+\epsilon_{k} \end{equation}

and therefore the price follows a random walk.
Such a discrete random walk can look like this:
We can split this time from [0,T] into n pieces; making each segment with length h=\frac{T}{n}. Then, we can parcel out:

\begin{equation} p_{n}(t) = p_{[\frac{t}{h}]} = p_{[\frac{nt}{T}]} \end{equation}

Descretized at integer intervals.
At this current, discrete moments have expected value E[p_{n}(T)] = n(\pi -(1-\pi))\Delta and variance Var[p_{n}(T)]=4n\pi (1-\pi)\Delta^{2}. #why
Now, if we want to have a continuous version of the descretized interval above, we will maintain the finiteness of p_{n}(T) but take n to \infty. To get a continuous random walk needed for Brownian Motion, we adjust \Delta, \pi, and 1-\pi such that the expected value and variance tends towards the normal (as we expect for a random walk); that is, we hope to see that:

\begin{equation} \begin{cases} n(\pi -(1-\pi))\Delta \to \mu T \\ 4n\pi (1-\pi )\Delta ^{2} \to \sigma^{2} T \end{cases} \end{equation}

To solve for these desired convergences into the normal, we have probabilities \pi, (1-\pi), \Delta such that:

\begin{equation} \begin{cases} \pi = \frac{1}{2}\left(1+\frac{\mu \sqrt{h}}{\sigma}\right)\\ (1-\pi) = \frac{1}{2}\left(1-\frac{\mu \sqrt{h}}{\sigma}\right)\\ \Delta = \sigma \sqrt{h} \end{cases} \end{equation}

where, h = \frac{1}{n}.
So looking at the expression for \Delta, we can see that as n in increases, h =\frac{1}{n} decreases and therefore \Delta decreases. In fact, we can see that the change in all three variables track the change in the rate of \sqrt{h}; namely, they vary with O(h).

\begin{equation} \pi = (1-\pi) = \frac{1}{2}+\frac{\mu \sqrt{h}}{2\sigma} = \frac{1}{2}+O\left(\sqrt{h}\right) \end{equation}

Of course:

\begin{equation} \Delta = O\left(\sqrt{h}\right) \end{equation}

So, finally, we have the conclusion that:
as n (number of subdivision pieces of the time domain T) increases, \frac{1}{n} decreases, O\left(\sqrt{h}\right) decreases with the same proportion. Therefore, as \lim_{n \to \infty} in the continuous-time case, the probability of either positive or negative delta (\pi and -\pi trends towards each to \frac{1}{2}) by the same vein, as \lim_{n \to \infty}, \Delta \to 0 Therefore, this is a cool result: in a continuous-time case of a discrete random walk, the returns (NOT! just the expect value, but literal \Delta) trend towards +0 and -0 each with \frac{1}{2} probability.
actual Brownian motion Given the final results above for the limits of discrete random walk, we can see that the price moment traced from the returns (i.e. p_{k} = p_{k-1}+\epsilon_{k}) have the properties of normality (p_{n}(T) \to \mathcal{N}(\mu T, \sigma^{2}T))
True Brownian Motion follows, therefore, three basic properties:
B_{t} is normally distributed by a mean of 0, and variance of t For some s<t, B_{t}-B_{s} is normally distributed by a mean of 0, and variance of t-s Distributions B_{j} and B_{t}-B_{s} is independent Standard Brownian Motion Brownian motion that starts at B_0=0 is called Standard Brownian Motion
quadratic variation The quadratic variation of a sequence of values is the expression that:

\begin{equation} \sum_{i=0}^{N-1} (x_{i+1}-x_i)^{2} \end{equation}

On any sequence of values x_0=0,\dots,x_{N}=1 (with defined bounds), the quadratic variation becomes bounded.