# 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.

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

# Details • BrownianBridgeProcess is also known as pinned Brownian motion process.
• BrownianBridgeProcess is a continuous-time and continuous-state random process.
• The state for a Brownian bridge process satisfies and .
• The state follows NormalDistribution[a+(b-a) (t-t1)/(t2-t1), ].
• The parameters σ, t1, t2, a, and b can be any real numbers, with σ positive and t2 greater than t1.
• BrownianBridgeProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.

# 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:

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: