Package umontreal.ssj.stochprocess

Stochastic Processes. More...

Classes
class	BrownianMotion
	This class represents a Brownian motion process \(\{X(t) : t \geq0 \}\), sampled at times \(0 = t_0 < t_1 < \cdots< t_d\). More...
class	BrownianMotionBridge
	Represents a Brownian motion process \(\{X(t) : t \geq0 \}\) sampled using the bridge sampling technique (see for example [69] ). More...
class	BrownianMotionPCA
	A Brownian motion process \(\{X(t) : t \geq0 \}\) sampled using the principal component decomposition (PCA) [69], [95], [153] . More...
class	BrownianMotionPCAEqualSteps
	Same as BrownianMotionPCA, but uses a trick to speed up the calculation when the time steps are equidistant. More...
class	CIRProcess
	This class represents a CIR (Cox, Ingersoll, Ross) process [36]  \(\{X(t) : t \geq0 \}\), sampled at times \(0 = t_0 < t_1 < \cdots< t_d\). More...
class	CIRProcessEuler
	This class represents a CIR process as in CIRProcess, but the process is generated using the simple Euler scheme. More...
class	GammaProcess
	This class represents a gamma process [171]  (page 82) \(\{ S(t) = G(t; \mu, \nu) : t \geq0 \}\) with mean parameter \(\mu\) and variance parameter \(\nu\). More...
class	GammaProcessBridge
	This class represents a gamma process \(\{ S(t) = G(t; \mu, \nu) : t \geq0 \}\) with mean parameter \(\mu\) and variance parameter \(\nu\), sampled using the gamma bridge method (see for example [208], [11] ). More...
class	GammaProcessPCA
	Represents a gamma process sampled using the principal component analysis (PCA). More...
class	GammaProcessPCABridge
	Same as GammaProcessPCA, but the generated uniforms correspond to a bridge transformation of the BrownianMotionPCA instead of a sequential transformation. More...
class	GammaProcessPCASymmetricalBridge
	Same as GammaProcessPCABridge, but uses the fast inversion method for the symmetrical beta distribution, proposed by L’Ecuyer and Simard [134] , to accelerate the generation of the beta random variables. More...
class	GammaProcessSymmetricalBridge
	This class differs from GammaProcessBridge only in that it requires the number of interval of the path to be a power of 2 and of equal size. More...
class	GeometricBrownianMotion
	Represents a geometric Brownian motion (GBM) process \(\{S(t), t\ge0\}\), which evolves according to the stochastic differential equation. More...
class	GeometricLevyProcess
	Abstract class used as a parent class for the exponentiation of a Lévy process \(X(t)\): \[ S(t) = S(0) \exp\left(X(t) + (r - \omega_{RN}) t\right). \] The interest rate is denoted \(r\) and is referred to as muGeom in the class below. More...
class	GeometricNormalInverseGaussianProcess
	The geometric normal inverse gaussian (GNIG) process is the exponentiation of a NormalInverseGaussianProcess : \[ S(t) = S_0 \exp\left[ (r-\omega_{RN})t + \mbox{NIG}(t;\alpha,\beta,\mu,\delta) \right], \] where \(r\) is the interest rate. More...
class	GeometricVarianceGammaProcess
	This class represents a geometric variance gamma process \(S(t)\) (see [171]  (page 86)). More...
class	InverseGaussianProcess
	The inverse Gaussian process is a non-decreasing process where the increments are additive and are given by the inverse gaussian distribution, umontreal.ssj.probdist.InverseGaussianDist. More...
class	InverseGaussianProcessBridge
	Samples the path by bridge sampling: first finding the process value at the final time and then the middle time, etc. More...
class	InverseGaussianProcessMSH
	Uses a faster generating method (MSH) [180]  than the simple inversion of the distribution function used by InverseGaussianProcess. More...
class	InverseGaussianProcessPCA
	Approximates a principal component analysis (PCA) decomposition of the InverseGaussianProcess. More...
class	MultivariateBrownianMotion
	This class represents a multivariate Brownian motion process \(\{\mathbf{X}(t) = (X_1(t),…, X_c(t)),  t \geq0 \}\), sampled at times \(0 = t_0 < t_1 < \cdots< t_d\). More...
class	MultivariateBrownianMotionBridge
	A multivariate Brownian motion process \(\{\mathbf{X}(t) : t \geq0 \}\) sampled via bridge sampling. More...
class	MultivariateBrownianMotionPCA
	A multivariate Brownian motion process \(\{\mathbf{X}(t) : t \geq0 \}\) sampled entirely using the principal component decomposition (PCA), as explained in [69] , page 92. More...
class	MultivariateBrownianMotionPCABigSigma
	A multivariate Brownian motion process \(\{\mathbf{X}(t) : t \geq0 \}\) sampled entirely using the principal component decomposition (PCA). More...
class	MultivariateGeometricBrownianMotion
	This class is a multivariate version of GeometricBrownianMotion. More...
class	MultivariateStochasticProcess
	This class is a multivariate version of StochasticProcess where the process evolves in the \(c\)-dimensional real space. More...
class	NormalInverseGaussianProcess
	This class represents a normal inverse gaussian process (NIG). More...
class	OrnsteinUhlenbeckProcess
	This class represents an Ornstein-Uhlenbeck process \(\{X(t) : t \geq0 \}\), sampled at times \(0 = t_0 < t_1 < \cdots< t_d\). More...
class	OrnsteinUhlenbeckProcessEuler
	This class represents an Ornstein-Uhlenbeck process as in OrnsteinUhlenbeckProcess, but the process is generated using the simple Euler scheme. More...
class	StochasticProcess
	Abstract base class for a stochastic process \(\{X(t) : t \geq 0 \}\) sampled (or observed) at a finite number of time points, \(0 = t_0 < t_1 < \cdots< t_d\). More...
class	VarianceGammaProcess
	This class represents a variance gamma (VG) process \(\{S(t) = X(t; \theta, \sigma, \nu) : t \geq0\}\). More...
class	VarianceGammaProcessAlternate
	This is a VarianceGammaProcess for which the successive random numbers are used in a different order to generate the sample path. More...
class	VarianceGammaProcessDiff
	This class represents a variance gamma (VG) process \(\{S(t) = X(t; \theta, \sigma, \nu) : t \geq0\}\). More...
class	VarianceGammaProcessDiffPCA
	Same as VarianceGammaProcessDiff, but the two inner GammaProcess ’es are of PCA type. More...
class	VarianceGammaProcessDiffPCABridge
	Same as VarianceGammaProcessDiff, but the two inner GammaProcess ’es are of the type PCABridge. More...
class	VarianceGammaProcessDiffPCASymmetricalBridge
	Same as VarianceGammaProcessDiff, but the two inner GammaProcess ’es are of the PCASymmetricalBridge type. More...

Detailed Description

Stochastic Processes.

This package provides classes to define stochastic processes \(\{X(t), t\ge0\}\) in the real space, and to simulate their sample paths at a finite number of observation times \(t_0 \le t_1 \le\cdots\le t_d\), i.e., skeletons of their sample paths. The generated path skeleton is a vector \((X(t_0),X(t_1),\dots,X(t_d)) \in \mathbb{R}\).

The observation times \(t_0, \dots, t_d\) can be specified (and can be changed) after defining the process, via the method setObservationTimes. In some cases, the observation times can also be specified one by one when generating the value at the next observation time. This may be convenient or even necessary if the observation times are random, for example. The random stream used to generate the sample path can also be set or changed, using setStream, and it can also be passed each time to the method that generates the paths.

The available processes include the Brownian motion (or Gaussian process), Gamma Process, Inverse Gaussian, versions with a random clock (or subordinate process) such as the variance-gamma and normal inverse Gaussian processes,
exponential (or geometric) versions of all of these, and more.
Many of those processes can be simulated in different ways, such as standard sequential generation of the increments (which are independent in the caes of L\'evy processes), or using a bridge sampling strategy as in BrownianMotionBridge, or by using a principal component decomposition as in BrownianMotionPCA. The choice of simulation strategy can have a significant impact on the variance when combined with randomized quasi-Monte Carlo (RQMC).

There are also multivariate versions in which the state is a \(c\)-dimensional vector, \(\mathbb{X}(t) \in \mathbb{R}\). At each time step, a new state vector is generated.

Remarks

: In the future, it might be useful to have a common interface implemented by StochasticProcess and umontreal.ssj.markovchainrqmc.MarkovChainDouble.