# Visualization of Stochastic Processes

Stochastic processes make for an excellent source of graphical beauty.

Written on September 24, 2018

In this demo, I present the dynamics of stochastic processes along the three physical diemensions of:

• $t$: time
• $x$: the value of the stochastic process at time $t$
• $f(x)$: the probability density (or the probablity mass in the discrete case) of $x$ at time $t$

Note that the expected value (or the closest permissible value in the discrete case) at each time $t$ is also highlighted.

All plots are meant to be interactive. Drag to rotate and scroll to zoom in/out on WebGL-compatable devices. Certain feature like drift, dispersion, mean-reversion or non-negativeness might be more clearly visible from a different angle.

## Trend Stationary Model (without Drift)

$X_t = X_0 + \epsilon_t$

## Trend Stationary Model

$X_t = X_0 + \beta t + \epsilon_t$

## Brownian Motion (without Drift)

$dX_t = \sigma dW_t$

## Brownian Motion

$dX_t = \mu t + \sigma dW_t$

## Brownian Bridge

$X_t = \frac{t}{T} X_T + \frac{T-t}{T} (X_0 + \sigma W_t)$

## Geometric Brownian Motion

$dX_t = \mu X_t t + \sigma X_t dW_t$

## Vasicek Model

$dX_t = a(b-X_t)dt + \sigma dW_t$

## Cox–Ingersoll–Ross Model

$dX_t = a(b-X_t)dt + \sigma \sqrt{X_t}dW_t$

## Poisson Process

$P(X_t = k) = e^{-\lambda t} \frac{(\lambda t) ^ k}{k,!}$

## Compensated Poisson Process

$X_t = N_t - \lambda t$