In a Wiener process w(t):

- If the time intervals
`[s, t]`

and`[s', t']`

do not intersect, then the increments`w(t) - w(s)`

and`w(t') - w(s')`

are independent random variables. - As the increments
`w(t) - w(s)`

are normal random variables with`E{w(t) - w(s)} = 0`

and`Var{w(t) - w(s)} = t - s`

This way we have that w(t_{i+1}) - w(t_{i}) ~ N(0, t_{i+1} - t_{i}) for the simulation.

## Simulation

Given a normal random variable w(t_{i+1}) = w(t_{i}) + √(t_{i+1} - t_{i}) due time instants t_{i} is posible to pick them as we see fit, as the moment in which two instants are -sufficiently short- to match to other process simulated on the system.

## Brownian Movement

Is given by the following equation:

For the simulation we will use

## Geometric Brownian Movement

Is given by the following stochastic differential equation

Its solution can be written in the following way

## Result

### Implementation

```
Random random = new Random();
const int miu = 1;
const double k = 0.1; //0.02
int days = (int) inputDays.Value;
double price = (double) inputPrice.Value;
chart1.Series[0].Points.AddXY(0, price);
for (int i = 1; i <= days; i++)
{
//price = price * (1 + k * (random.NextDouble() - 0.5));
// e^((miu - (price*price/2)) * k + price * random)
double w = miu + Math.Sqrt(k) * (random.NextDouble() - 0.5);
price = Math.Exp((miu - (price*price/2)) * k + (price * w));
chart1.Series[0].Points.AddXY(i, price);
}
```