umontreal.iro.lecuyer.stochprocess

Class BrownianMotionBridge

java.lang.Object

umontreal.iro.lecuyer.stochprocess.StochasticProcess

umontreal.iro.lecuyer.stochprocess.BrownianMotion

umontreal.iro.lecuyer.stochprocess.BrownianMotionBridge

public class BrownianMotionBridge
extends BrownianMotion

Represents a Brownian motion process {X(t) : t >= 0} sampled using the bridge sampling technique (see for example). This technique generates first the value X(t_d) at the last observation time, then the value at time t_d/2 (or the nearest integer), then the values at time t_d/4 and at time t_3d/4 (or the nearest integers), and so on. If the process has already been sampled at times t_i < t_k but not in between, the next sampling point in that interval will be t_j where j = floor((i + k)/2). For example, if the sampling times used are { t₀, t₁, t₂, t₃, t₄, t₅}, then the observations are generated in the following order: X(t₅), X(t₂), X(t₁), X(t₃), X(t₄).

Warning: Both the generatePath and the nextObservation methods from BrownianMotion are modified to use the bridge method. In the case of nextObservation, the user should understand that the observations returned are not ordered chronologically. However they will be once an entire path is generated and the observations are read from the internal array (referenced by the getPath method) that contains them.

The method nextObservation(double nextTime) differs from that of the class BrownianMotion in that nextTime represents the next observation time of the Brownian bridge. However, the t_i supplied must still be non-decreasing with i.

Note also that, if the path is not entirely generated before being read from this array, there will be ``pollution'' from the previous path generated, and the observations will not represent a sample path of this process.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.stochprocess.StochasticProcess

path

Constructor Summary

Constructors

Constructor and Description
BrownianMotionBridge(double x0, double mu, double sigma, NormalGen gen) Constructs a new BrownianMotionBridge with parameters μ = mu, σ = sigma and initial value X(t0) = x0.
BrownianMotionBridge(double x0, double mu, double sigma, RandomStream stream) Constructs a new BrownianMotionBridge with parameters μ = mu, σ = sigma and initial value X(t0) = x0.

Method Summary

Methods

Modifier and Type	Method and Description
double[]	generatePath() Generates, returns, and saves the sample path {X(t0), X(t1),…, X(td)}.
double[]	generatePath(double[] uniform01) Same as generatePath(), but a vector of uniform random numbers must be provided to the method.
double	nextObservation() Generates and returns the next observation X(tj) of the stochastic process.
double	nextObservation(double nextTime) Generates and returns the next observation at time tj+1 = nextTime.
void	resetStartProcess() Resets the observation counter to its initial value j = 0, so that the current observation X(tj) becomes X(t0).

Methods inherited from class umontreal.iro.lecuyer.stochprocess.BrownianMotion

generatePath, getGen, getMu, getSigma, getStream, nextObservation, setParams, setStream

Methods inherited from class umontreal.iro.lecuyer.stochprocess.StochasticProcess

getArrayMappingCounterToIndex, getCurrentObservation, getCurrentObservationIndex, getNbObservationTimes, getObservation, getObservationTimes, getPath, getSubpath, getX0, hasNextObservation, setObservationTimes, setObservationTimes, setX0

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

BrownianMotionBridge

public BrownianMotionBridge(double x0,
                    double mu,
                    double sigma,
                    RandomStream stream)

Constructs a new BrownianMotionBridge with parameters μ = mu, σ = sigma and initial value X(t₀) = x0. The normal variates will be generated by inversion using the RandomStream stream.

BrownianMotionBridge

public BrownianMotionBridge(double x0,
                    double mu,
                    double sigma,
                    NormalGen gen)

Constructs a new BrownianMotionBridge with parameters μ = mu, σ = sigma and initial value X(t₀) = x0. The normal variates will be generated by the NormalGen gen.

Method Detail

nextObservation

public double nextObservation()

Description copied from class: StochasticProcess

Generates and returns the next observation X(t_j) of the stochastic process. The processes are usually sampled sequentially, i.e. if the last observation generated was for time t_j-1, the next observation returned will be for time t_j. In some cases, subclasses extending this abstract class may use non-sequential sampling algorithms (such as bridge sampling). The order of generation of the t_j's is then specified by the subclass. All the processes generated using principal components analysis (PCA) do not have this method.

Overrides:

nextObservation in class BrownianMotion

nextObservation

public double nextObservation(double nextTime)

Description copied from class: BrownianMotion

Generates and returns the next observation at time t_j+1 = nextTime. It uses the previous observation time t_j defined earlier (either by this method or by setObservationTimes), as well as the value of the previous observation X(t_j). Warning: This method will reset the observations time t_j+1 for this process to nextTime. The user must make sure that the t_j+1 supplied is >= t_j.

Overrides:

nextObservation in class BrownianMotion

generatePath

public double[] generatePath()

Description copied from class: StochasticProcess

Generates, returns, and saves the sample path {X(t₀), X(t₁),…, X(t_d)}. It can then be accessed via getPath, getSubpath, or getObservation. The generation method depends on the process type.

Overrides:

generatePath in class BrownianMotion

generatePath

public double[] generatePath(double[] uniform01)

Description copied from class: BrownianMotion

Same as generatePath(), but a vector of uniform random numbers must be provided to the method. These uniform random numbers are used to generate the path.

Overrides:

generatePath in class BrownianMotion

resetStartProcess

public void resetStartProcess()

Description copied from class: StochasticProcess

Resets the observation counter to its initial value j = 0, so that the current observation X(t_j) becomes X(t₀). This method should be invoked before generating observations sequentially one by one via nextObservation, for a new sample path.

Overrides:

resetStartProcess in class StochasticProcess