This class represents a variance gamma (VG) process \(\{S(t) = X(t; \theta, \sigma, \nu) : t \geq0\}\). More...
Public Member Functions | |
| VarianceGammaProcess (double s0, double theta, double sigma, double nu, RandomStream stream) | |
Constructs a new VarianceGammaProcess with parameters \(\theta= \mathtt{theta}\), \(\sigma= \mathtt{sigma}\), \(\nu= \mathtt{nu}\) and initial value \(S(t_0) = \mathtt{s0}\). More... | |
| VarianceGammaProcess (double s0, BrownianMotion BM, GammaProcess Gamma) | |
Constructs a new VarianceGammaProcess. More... | |
| double | nextObservation () |
| Generates the observation for the next time. More... | |
| double [] | generatePath () |
| Generates and returns the path. More... | |
| double [] | generatePath (double[] uniform01) |
Similar to the usual generatePath(), but here the uniform random numbers used for the simulation must be provided to the method. More... | |
| void | resetStartProcess () |
Resets the observation index and counter to 0 and applies the resetStartProcess method to the BrownianMotion and the GammaProcess objects used to generate this process. | |
| void | setParams (double s0, double theta, double sigma, double nu) |
Sets the parameters \(S(t_0) =\) s0, \(\theta=\) theta, \(\sigma=\) sigma and \(\nu=\) nu of the process. More... | |
| double | getTheta () |
| Returns the value of the parameter \(\theta\). | |
| double | getSigma () |
| Returns the value of the parameter \(\sigma\). | |
| double | getNu () |
| Returns the value of the parameter \(\nu\). | |
| void | setObservationTimes (double t[], int d) |
Sets the observation times on the VarianceGammaProcess as usual, but also sets the observation times of the underlying GammaProcess. More... | |
| void | setStream (RandomStream stream) |
| Resets the umontreal.ssj.rng.RandomStream ’s. More... | |
| RandomStream | getStream () |
| Returns the random stream of the BrownianMotion process, which should be the same as for the GammaProcess. | |
| BrownianMotion | getBrownianMotion () |
| Returns a reference to the inner BrownianMotion. | |
| GammaProcess | getGammaProcess () |
| Returns a reference to the inner GammaProcess. | |
 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 | |
| GammaProcess | randomTime |
| BrownianMotion | BM |
| double | theta |
| 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 | nu |
Detailed Description
This class represents a variance gamma (VG) process \(\{S(t) = X(t; \theta, \sigma, \nu) : t \geq0\}\).
This process is obtained as a subordinate of the Brownian motion process \(B(t;\theta,\sigma)\) using the operational time \(G(t;1,\nu)\) (see [61], [10] ):
\[ X(t; \theta, \sigma, \nu) := B(G(t;1,\nu),\theta, \sigma). \tag{VGeqn} \]
See also [171], [169], [170] for applications to modelling asset returns and option pricing.
The process is sampled as follows: when generatePath() is called, the method generatePath() of the inner GammaProcess is called; its path is then used to set the observation times of the BrownianMotion. Finally, the method generatePath() of the BrownianMotion is called. Warning: If one wants to reduced the variance as much as possible in a QMC simulation, this way of proceeding is not optimal. Use the method generatePath(uniform01) instead.
If one calls the nextObservation method, the operational time is generated first, followed by the corresponding brownian motion increment, which is then returned.
Note that if one wishes to use bridge sampling with the nextObservation method, both the gamma process \(G\) and the Brownian motion process \(B\) should use bridge sampling so that their observations are synchronized.
Constructor & Destructor Documentation
◆ VarianceGammaProcess() [1/2]
| VarianceGammaProcess | ( | double | s0, |
| double | theta, | ||
| double | sigma, | ||
| double | nu, | ||
| RandomStream | stream | ||
| ) |
Constructs a new VarianceGammaProcess with parameters \(\theta= \mathtt{theta}\), \(\sigma= \mathtt{sigma}\), \(\nu= \mathtt{nu}\) and initial value \(S(t_0) = \mathtt{s0}\).
stream is used to generate both the BrownianMotion \(B\) and the GammaProcess \(G\) in ( VGeqn ).
◆ VarianceGammaProcess() [2/2]
| VarianceGammaProcess | ( | double | s0, |
| BrownianMotion | BM, | ||
| GammaProcess | Gamma | ||
| ) |
Constructs a new VarianceGammaProcess.
The parameters \(\theta\) and \(\sigma\) are set to the parameters \(\mu\) and \(\sigma\), respectively, of the BrownianMotion BM and the parameter \(\nu\) is set to the parameter \(\nu\) of the GammaProcess Gamma. The parameters \(\mu\) and \(x0\) of the GammaProcess are overwritten to equal 1 and 0 respectively. The initial value of the process is \(S(t_0) = {\mathtt{s0}}\).
Member Function Documentation
◆ generatePath() [1/2]
| double [] generatePath | ( | ) |
Generates and returns the path.
To do so, it first generates the complete path of the inner GammaProcess and sets the observation times of the inner BrownianMotion to this path. This method is not optimal to reduce the variance in QMC simulations; use generatePath(double[] uniform01) for that.
◆ generatePath() [2/2]
| double [] generatePath | ( | double [] | uniform01 | ) |
Similar to the usual generatePath(), but here the uniform random numbers used for the simulation must be provided to the method.
This allows to properly use the uniform random variates in QMC simulations. This method divides the table of uniform random numbers uniform01 in two smaller tables, the first one, containing the odd indices of uniform01 which are used to generate the path of the inner GammaProcess, and the even indices (in the second table) are used to generate the path of the inner BrownianMotion. This way of proceeding reduces the variance as much as possible for QMC simulations.
◆ nextObservation()
| double nextObservation | ( | ) |
Generates the observation for the next time.
It also works with bridge sampling; however both BrownianMotionBridge and GammaProcessBridge must be used in the constructor in that case. Furthermore, for bridge sampling, the order of the observations is that of the bridge, not sequential order.
◆ setObservationTimes()
| void setObservationTimes | ( | double | t[], |
| int | d | ||
| ) |
Sets the observation times on the VarianceGammaProcess as usual, but also sets the observation times of the underlying GammaProcess.
It furthermore sets the starting value of the GammaProcess to t[0].
◆ setParams()
| void setParams | ( | double | s0, |
| double | theta, | ||
| double | sigma, | ||
| double | nu | ||
| ) |
Sets the parameters \(S(t_0) =\) s0, \(\theta=\) theta, \(\sigma=\) sigma and \(\nu=\) nu of the process.
Warning: This method will recompute some quantities stored internally, which may be slow if called repeatedly.
◆ setStream()
| void setStream | ( | RandomStream | stream | ) |
Resets the umontreal.ssj.rng.RandomStream ’s.
Warning: this method sets both the umontreal.ssj.rng.RandomStream of the BrownianMotion and of the GammaProcess to the same umontreal.ssj.rng.RandomStream.
The documentation for this class was generated from the following file:
- VarianceGammaProcess.java
