A Simple Wiener Process (White Noise)

A Wiener process is a continuous process that has:

Independent increments
Gaussian increments
Continuous paths, i.e. our Wiener process W is continuous in t

The increments of a Wiener process can be described by the following equation

\[ \varepsilon_t\cdot\sqrt{dt} \]

Generating a Wiener Process

The first element in the WienerProcess(t0, W0) function is the inital value of time (when we begin the process). The second element is the inital value of our process.

We can also set $dt$ in for our process.

using DifferentialEquations, Plots
W = WienerProcess(0.0,0.0)
dt = 0.1
W.dt = dt
#

Generate more values

Now we have a Wiener Process object. However, this object doesn't have multiple points and is not plotable. We can add to our process using the code below.

setup_next_step!(W)
for i in 1:100
    accept_step!(W,dt)
end

Plotting the process

The Brownian Bridge

A Brownian bridge is a Wiener process with a predefined beginning and end point. For the function BrownianBridge(t0,tend,W0,Wend), you input the first time period the last time period and the desired intial and terminal values for the process.

BB = BrownianBridge(0.0, 100, 1.0, 1.0)
BB.dt = 0.1
#

For this Brownian Bridge the initial and terminal value of our process should be 1.0

Generate more values

Again, we need to add more values to our object

setup_next_step!(BB)
for i in 1:1000
    accept_step!(BB,dt)
end

Plotting the Brownian Bridge

Now we can look at specfic types of Brownian Motion

Geometric Brownian motion

Geometric Brownian motion can be defined as:

\[ dx_t = μ x_t dt + σ x_t dW_t \]

For the GeometricBrownianMotionProcess() function we need several inputs:

$\mu$, our "drift" parameter
$\sigma$ our "variance" parameter
Intial time period
Inital value for our process

μ = 1.0
σ = .8
GB = GeometricBrownianMotionProcess(μ, σ, 0.0, 1.0)
BB.dt = 1.0

1.0

Generate more values

setup_next_step!(GB)
for i in 1:50
    accept_step!(GB,dt)
end

Plotting Geometric Brownian Motion

An Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process is the continuous time analog of a AR(1) process. We can write this process in the following form,

\[ dx_t = θ(μ-x_t)dt + σdW_t \]

For the OrnsteinUhlenbeckProcess() function we need

$\theta$ our "drfit" parameter
$\mu$ our "mean" parameter
$\sigma$ our vairance parameter
Our intial time period t0
Our intial value for our process

θ = 1.5
μ = 2.0
σ = 1.5

OU = OrnsteinUhlenbeckProcess(θ, μ, σ, 0.0, 0.0)
OU.dt = 0.1

0.1

Generate more values

setup_next_step!(OU)
for i in 1:100
    accept_step!(OU,dt)
end

Plotting the Ornstein-Uhlenbeck Process

Backing out parameters

Another feature of the DifferentialEquations package is the ability to solve SDE problems and back out the coefficients for our Diffuction process.

OU_prob = NoiseProblem(OU, (0.0,100.0))
solve_OU = solve(OU_prob, dt = 0.12, save_noise=true)
#

Plotting this, and our graph from before we can see that this will give us back thr exact same process we had earlier