BrownianMotion Class Reference

This class represents a Brownian motion process \(\{X(t) : t \geq0 \}\), sampled at times \(0 = t_0 < t_1 < \cdots< t_d\). More...

Inheritance diagram for BrownianMotion:

Collaboration diagram for BrownianMotion:

Public Member Functions
	BrownianMotion (double x0, double mu, double sigma, RandomStream stream)
	Constructs a new BrownianMotion with parameters \(\mu=\) mu, \(\sigma=\) sigma and initial value \(X(t_0) =\) x0. More...
	BrownianMotion (double x0, double mu, double sigma, NormalGen gen)
	Constructs a new BrownianMotion with parameters \(\mu=\) mu, \(\sigma=\) sigma and initial value \(X(t_0) =\) x0. More...
double	nextObservation ()
double	nextObservation (double nextTime)
	Generates and returns the next observation at time \(t_{j+1} =\) nextTime. More...
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. More...
double []	generatePath ()
double []	generatePath (double[] uniform01)
	Same as generatePath(), but a vector of uniform random numbers must be provided to the method. More...
double []	generatePath (RandomStream stream)
void	setParams (double x0, double mu, double sigma)
	Resets the parameters \(X(t_0) = \mathtt{x0}\), \(\mu= \mathtt{mu}\) and \(\sigma= \mathtt{sigma}\) of the process. More...
void	setStream (RandomStream stream)
	Resets the random stream of the normal generator to stream.
RandomStream	getStream ()
	Returns the random stream of the normal generator.
double	getMu ()
	Returns the value of \(\mu\).
double	getSigma ()
	Returns the value of \(\sigma\).
NormalGen	getGen ()
	Returns the normal random variate generator used. More...
Public Member Functions inherited from StochasticProcess
void	setObservationTimes (double[] T, int d)
	Sets the observation times of the process to a copy of T, with. More...
void	setObservationTimes (double delta, int d)
	Sets equidistant observation times at \(t_j = j\delta\), for. More...
double []	getObservationTimes ()
	Returns a reference to the array that contains the observation times. More...
int	getNumObservationTimes ()
	Returns the number \(d\) of observation times, excluding the time \(t_0\).
abstract double []	generatePath ()
	Generates, returns, and saves the sample path \(\{X(t_0), X(t_1), \dots, X(t_d)\}\). More...
double []	generatePath (RandomStream stream)
	Same as generatePath(), but first resets the stream to stream.
double []	getPath ()
	Returns a reference to the last generated sample path \(\{X(t_0), ... , X(t_d)\}\). More...
void	getSubpath (double[] subpath, int[] pathIndices)
	Returns in subpath the values of the process at a subset of the observation times, specified as the times \(t_j\) whose indices. More...
double	getObservation (int j)
	Returns \(X(t_j)\) from the current sample path. More...
void	resetStartProcess ()
	Resets the observation counter to its initial value \(j=0\), so that the current observation \(X(t_j)\) becomes \(X(t_0)\). More...
boolean	hasNextObservation ()
	Returns true if \(j<d\), where \(j\) is the number of observations of the current sample path generated since the last call to resetStartProcess. More...
double	nextObservation ()
	Generates and returns the next observation \(X(t_j)\) of the stochastic process. More...
int	getCurrentObservationIndex ()
	Returns the value of the index \(j\) corresponding to the time. More...
double	getCurrentObservation ()
	Returns the value of the last generated observation \(X(t_j)\).
double	getX0 ()
	Returns the initial value \(X(t_0)\) for this process.
void	setX0 (double s0)
	Sets the initial value \(X(t_0)\) for this process to s0, and reinitializes.
abstract void	setStream (RandomStream stream)
	Resets the random stream of the underlying generator to stream.
abstract RandomStream	getStream ()
	Returns the random stream of the underlying generator.
int []	getArrayMappingCounterToIndex ()
	Returns a reference to an array that maps an integer \(k\) to \(i_k\), the index of the observation \(S(t_{i_k})\) corresponding to the \(k\)-th observation to be generated for a sample path of this process. More...

Protected Member Functions
void	init ()
Protected Member Functions inherited from StochasticProcess
void	init ()

Protected Attributes
NormalGen	gen
double	mu
double []	mudt
Protected Attributes inherited from StochasticProcess
boolean	observationTimesSet = false
double	x0 = 0.0
int	d = -1
int	observationIndex = 0
int	observationCounter = 0
double []	t
double []	path
int []	observationIndexFromCounter

Package Attributes
double	sigma
double []	sigmasqrdt

Detailed Description

This class represents a Brownian motion process \(\{X(t) : t \geq0 \}\), sampled at times \(0 = t_0 < t_1 < \cdots< t_d\).

This process obeys the stochastic differential equation

\[ dX(t) = \mu dt + \sigma dB(t), \tag{Brownian-motion} \]

with initial condition \(X(0)= x_0\), where \(\mu\) and \(\sigma\) are the drift and volatility parameters, and \(\{B(t),  t\ge0\}\) 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 \(\mu t\) and variance \(\sigma^2 t\).

In this class, this process is generated using the sequential (or random walk) technique: \(X(0)=x_0\) and

\[ X(t_j) - X(t_{j-1}) = \mu(t_j - t_{j-1}) + \sigma\sqrt{t_j - t_{j-1}} Z_j \tag{Brownian-motion-sequential} \]

where \(Z_j \sim N(0,1)\).

Constructor & Destructor Documentation

BrownianMotion() [1/2]

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

Constructs a new BrownianMotion with parameters \(\mu=\) mu, \(\sigma=\) sigma and initial value \(X(t_0) =\) x0.

The normal variates \(Z_j\) in ( Brownian-motion-sequential ) will be generated by inversion using stream.

BrownianMotion() [2/2]

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

Constructs a new BrownianMotion with parameters \(\mu=\) mu, \(\sigma=\) sigma and initial value \(X(t_0) =\) x0.

Here, the normal variate generator umontreal.ssj.randvar.NormalGen is specified directly instead of specifying the stream and using inversion. The normal generator gen can use another method than inversion.

Member Function Documentation

generatePath()

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.

getGen()

NormalGen getGen

(

)

Returns the normal random variate generator used.

The umontreal.ssj.rng.RandomStream used by that generator can be changed via getGen().setStream(stream), for example.

nextObservation() [1/2]

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 \(\geq t_j\).

nextObservation() [2/2]

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).

setParams()

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

Resets the parameters \(X(t_0) = \mathtt{x0}\), \(\mu= \mathtt{mu}\) and \(\sigma= \mathtt{sigma}\) of the process.

Warning: This method will recompute some quantities stored internally, which may be slow if called too frequently.

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The documentation for this class was generated from the following file:

• BrownianMotion.java