# WienerProcess

WienerProcess[μ,σ]

represents a Wiener process with a drift μ and volatility σ.

represents a standard Wiener process with drift 0 and volatility 1.

# Details • WienerProcess is also known as Brownian motion, a continuous-time random walk, or integrated white Gaussian noise.
• WienerProcess is a continuous-time and continuous-state random process.
• The state at time t follows NormalDistribution[μ t,σ ].
• The parameter μ can be any real number and the parameter σ can be any positive real number.
• WienerProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.

# Examples

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## Basic Examples(3)

Simulate a Wiener process:

Mean and variance functions:

Covariance function:

## Scope(12)

### Basic Uses(7)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Compare paths for different values of the drift parameter:

Compare paths for different values of the volatility parameter:

Process parameter estimation:

Correlation function:

Absolute correlation function:

### Process Slice Properties(5)

Univariate SliceDistribution:

First-order probability density function:

Compare with the density function of a normal distribution:

Multivariate slice distributions:

Second-order PDF:

Higher-order PDF:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness and kurtosis are constant:

Moment of order r:

Generating functions:

CentralMoment and its generating function:

FactorialMoment has no closed form for symbolic order:

Cumulant and its generating function:

## Generalizations & Extensions(1)

Useful shortcut evaluates to its full form counterpart:

## Applications(7)

Define a two-dimensional Bessel process:

Mean and variance functions:

Define a martingale process using a quadratic WienerProcess:

Define the stochastic exponential function:

Slice properties:

The corresponding differential equation is u[t] u[t] w[t]:

Slice properties:

Use WienerProcess directly to simulate GeometricBrownianMotionProcess:

Apply a transformation to the random sample:

Compare to the corresponding GeometricBrownianMotionProcess:

Use WienerProcess directly to simulate BrownianBridgeProcess:

Apply a transformation to the random sample:

Compare to the corresponding BrownianBridgeProcess:

Use Wiener process to simulate a solution to the stochastic differential equation :

Use the simulation to plot the solution:

Find the mean function of the simulated paths:

Compare with the corresponding smooth solution:

Find the distribution of the time a WienerProcess with positive drift takes to reach 2:

Remove empty lists and extract times:

Select positive values:

Fit InverseGaussianDistribution to the data:

Compare its histogram to the PDF:

## Properties & Relations(11)

A Wiener process is not weakly stationary:

Wiener process has independent increments:

Compare to the product of expectations:

Conditional cumulative distribution function:

The correlation function of the Wiener process is the same as that of RandomWalkProcess:

A Wiener process is a special ItoProcess:

As well as StratonovichProcess:

Simulate the proportion of time spent on the positive side by a standard WienerProcess:

In the limit, the ratio follows ArcSinDistribution:

Find the distribution of the last time WienerProcess changed sign between times 0 and 1:

Calculate the differences of signs to find sign changes:

Extract paths and find times of the last sign change for each path:

In the limit, the times follow ArcSinDistribution:

Find the distribution of the time corresponding to the maximum value of WienerProcess until time 1:

From each path, extract the times corresponding to the maximum for that path:

In the limit, the times follow ArcSinDistribution:

Wiener process is scaling invariant:

Covariance function:

Compare to the WienerProcess:

Wiener process is invariant under a difference transformation:

Covariance function:

Compare to the WienerProcess:

Wiener process is a transformation of BrownianBridgeProcess:

Covariance function:

## Neat Examples(3)

Simulate a Wiener process in two dimensions:

Simulate a Wiener process in three dimensions:

Simulate 500 paths from a Wiener process:

Take a slice at 1 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 1: