BrownianBridgeProcess

BrownianBridgeProcess[σ,{t1,a},{t2,b}]

represents the Brownian bridge process from value a at time t1 to value b at time t2 with volatility σ.

BrownianBridgeProcess[{t1,a},{t2,b}]

represents the standard Brownian bridge process from value a at time t1 to value b at time t2.

BrownianBridgeProcess[t1,t2]

represents the standard Brownian bridge process pinned at 0 at times t1 and t2.

BrownianBridgeProcess[]

represents the standard Brownian bridge process pinned at 0 at time 0 and at time 1.

Details

Examples

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Basic Examples  (3)

Simulate a Brownian bridge process pinned at 0 at both ends:

Mean and variance functions:

Covariance function:

Scope  (13)

Basic Uses  (8)

Shorthand autoevaluations:

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Simulate a path of a Brownian bridge process pinned at different values:

Simulate a Brownian bridge process on an interval contained in the process support:

Process parameter estimation:

Correlation function:

Absolute correlation function:

Process Slice Properties  (5)

Univariate SliceDistribution:

First-order probability density function:

Multivariate slice distribution:

Second-order PDF:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness and kurtosis are constant:

Moment:

Generating functions:

CentralMoment and its generating function:

FactorialMoment has no closed form for symbolic order:

Cumulant and its generating function:

Generalizations & Extensions  (1)

Useful shortcuts evaluate to their full form counterparts:

Properties & Relations  (9)

A Brownian bridge is not weakly stationary:

A Brownian bridge process does not have independent increments:

Compare to the product of expectations:

Conditional cumulative probability distribution:

A Brownian bridge process is a special ItoProcess:

As well as StratonovichProcess:

A Brownian bridge process is a solution to the stochastic differential equation :

Compare with the corresponding smooth solution:

A Brownian bridge process initially follows its corresponding WienerProcess:

The distribution of the absolute supremum of a BrownianBridgeProcess follows a Kolmogorov distribution:

Simulate sample distribution:

Compare the cumulative histogram with the cumulative distribution function of a Kolmogorov distribution:

BrownianBridgeProcess can be simulated directly from WienerProcess:

Apply a transformation to the random sample:

Compare to the corresponding BrownianBridgeProcess:

A Brownian bridge is a conditioned WienerProcess:

Compare to the CDF of the slice distribution of a Brownian bridge:

Possible Issues  (1)

Simulation is supported only within the process support:

Simulate from 0 to 1:

Simulate on a part of the support:

Neat Examples  (3)

Simulate a Brownian bridge process in two dimensions:

Simulate a Brownian bridge process in three dimensions:

Simulate 500 paths from a Brownian bridge process:

Take a slice at 1/2 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 1/2:

Introduced in 2012
 (9.0)