Simulate a Wiener process:

Mean and variance functions:

Covariance function:

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:

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:

GeometricBrownianMotionProcess is a transformation of a WienerProcess:

PDF of a slice distribution:

Compare to the slice distribution of the corresponding GeometricBrownianMotionProcess:

Wiener process is a transformation of BrownianBridgeProcess:

Covariance function:

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: