umontreal.iro.lecuyer.stochprocess

Class BrownianMotion

java.lang.Object

umontreal.iro.lecuyer.stochprocess.StochasticProcess

umontreal.iro.lecuyer.stochprocess.BrownianMotion

Direct Known Subclasses:

BrownianMotionBridge, BrownianMotionPCA, BrownianMotionPCAEqualSteps

public class BrownianMotion
extends StochasticProcess

This class represents a Brownian motion process {X(t) : t >= 0}, sampled at times 0 = t₀ < t₁ < ^... < t_d. This process obeys the stochastic differential equation

dX(t) = μdt + σdB(t),

with initial condition X(0) = x₀, where μ and σ are the drift and volatility parameters, and {B(t), t >= 0} is a standard Brownian motion (with drift 0 and volatility 1). This process has stationary and independent increments over disjoint time intervals (it is a Lévy process) and the increment over an interval of length t is normally distributed with mean μt and variance σ²t.

In this class, this process is generated using the sequential (or random walk) technique: X(0) = x₀ and

X(t_j) - X(t_j-1) = μ(t_j - t_j-1) + σ(t_j - t_j-1)^1/2Z_j

where Z_j∼N(0, 1).

Field Summary

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

path

Constructor Summary

Constructors

Constructor and Description
BrownianMotion(double x0, double mu, double sigma, NormalGen gen) Constructs a new BrownianMotion with parameters μ = mu, σ = sigma and initial value X(t0) = x0.
BrownianMotion(double x0, double mu, double sigma, RandomStream stream) Constructs a new BrownianMotion 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[]	generatePath(RandomStream stream) Same as generatePath(), but first resets the stream to stream.
NormalGen	getGen() Returns the normal random variate generator used.
double	getMu() Returns the value of μ.
double	getSigma() Returns the value of σ.
RandomStream	getStream() Returns the random stream of the normal generator.
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.
double	nextObservation(double x, double dt) Generates an observation of the process in dt time units, assuming that the process has value x at the current time.
void	setParams(double x0, double mu, double sigma) Resets the parameters X(t0) = x0, μ = mu and σ = sigma of the process.
void	setStream(RandomStream stream) Resets the random stream of the normal generator to stream.

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

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

Methods inherited from class java.lang.Object

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

Constructor Detail

BrownianMotion

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

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

BrownianMotion

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

Constructs a new BrownianMotion with parameters μ = mu, σ = sigma and initial value X(t₀) = x0. Here, the normal variate generator NormalGen is specified directly instead of specifying the stream and using inversion. The normal generator gen can use another method than inversion.

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 StochasticProcess

nextObservation

public double nextObservation(double nextTime)

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.

nextObservation

public double nextObservation(double x,
                     double dt)

Generates an observation of the process in dt time units, assuming that the process has value x at the current time. Uses the process parameters specified in the constructor. Note that this method does not affect the sample path of the process stored internally (if any).

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.

Specified by:

generatePath in class StochasticProcess

generatePath

public double[] generatePath(double[] uniform01)

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.

generatePath

public double[] generatePath(RandomStream stream)

Description copied from class: StochasticProcess

Same as generatePath(), but first resets the stream to stream.

Overrides:

generatePath in class StochasticProcess

setParams

public void setParams(double x0,
             double mu,
             double sigma)

Resets the parameters X(t₀) = x0, μ = mu and σ = sigma of the process. Warning: This method will recompute some quantities stored internally, which may be slow if called too frequently.

setStream

public void setStream(RandomStream stream)

Resets the random stream of the normal generator to stream.

Specified by:

setStream in class StochasticProcess

getStream

public RandomStream getStream()

Returns the random stream of the normal generator.

Specified by:

getStream in class StochasticProcess

getMu

public double getMu()

Returns the value of μ.

getSigma

public double getSigma()

Returns the value of σ.

getGen

public NormalGen getGen()

Returns the normal random variate generator used. The RandomStream used by that generator can be changed via getGen().setStream(stream), for example.