setCogarch function

Continuous-time GARCH (p,q) Process

setCogarch describes the Cogarch(p,q) model introduced in Brockwell et al. (2006):

dGt = sqrt(Vt)dZt

Vt = a0 + (a1 Yt(1) + ... + a(p) Yt(p))

dYt(1) = Yt(2) dt

...

dYt(q-1) = Yt(q) dt

dYt(q) = (-b(q) Yt(1) - ... - b(1) Yt(q))dt + (a0 + (a1 Yt(1) + ... +a(p) Yt(p))d[ZtZt]^{q}

setCogarch(p, q, ar.par = "b", ma.par = "a", loc.par = "a0", Cogarch.var = "g",
   V.var = "v", Latent.var = "y", jump.variable = "z",  time.variable = "t",
   measure = NULL, measure.type = NULL, XinExpr = FALSE, startCogarch = 0,
   work = FALSE, ...)

Arguments

• p: a non negative integer that is the number of the moving average coefficients of the Variance process.
• q: a non-negative integer that indicates the number of the autoregressive coefficients of the Variance process.
• ar.par: a character-string that is the label of the autoregressive coefficients.
• ma.par: a character-string that is the label of the autoregressive coefficients.
• loc.par: the location coefficient.
• Cogarch.var: a character-string that is the label of the observed cogarch process.
• V.var: a character-string that is the label of the latent variance process.
• Latent.var: a character-string that is the label of the latent process in the state space representation for the variance process.
• jump.variable: the jump variable.
• time.variable: the time variable.
• measure: Levy measure of jump variables.
• measure.type: type specification for Levy measure.
• XinExpr: a vector of expressions identyfying the starting conditions for Cogarch model.
• startCogarch: Start condition for the Cogarch process
• work: Internal Variable. In the final release this input will be removed.
• ...: Arguments to be passed to setCogarch such as the slots of the yuima.model-class

Details

We remark that yuima describes a Cogarch(p,q) model using the formulation proposed in Brockwell et al. (2006). This representation has the Cogarch(1,1) model introduced in Kluppelberg et al. (2004) as a special case. Indeed, by choosing beta = a0 b1, eta = b1 and phi = a1, we obtain the Cogarch(1,1) model proposed in Kluppelberg et al. (2004) defined as the solution of the SDEs:

dGt = sqrt(Vt)dZt

dVt = (beta - eta Vt) dt + phi Vt d[ZtZt]^{q}

Please refer to the vignettes and the examples.

An object of yuima.cogarch-class contains:

• info:: It is an object of cogarch.info-class which is a list of arguments that identifies the Cogarch(p,q) model

and the same slots in an object of yuima.model-class .

Returns

• model: an object of yuima.cogarch-class.

Author(s)

The YUIMA Project Team

Note

There may be missing information in the model description. Please contribute with suggestions and fixings.

References

Brockwell, P., Chadraa, E. and Lindner, A. (2006) Continuous-time GARCH processes, The Annals of Applied Probability, 16 , 790-826.

Kluppelberg, C., Lindner, A., and Maller, R. (2004) A continuous-time GARCH process driven by a Levy process: Stationarity and second-order behaviour, Journal of Applied Probability, 41 , 601-622.

Stefano M. Iacus, Lorenzo Mercuri, Edit Rroji (2017) COGARCH(p,q): Simulation and Inference with the yuima Package, Journal of Statistical Software, 80 (4), 1-49.

Examples

# Ex 1. (Continuous time GARCH process driven by a compound poisson process)
prova<-setCogarch(p=1,q=3,work=FALSE,
                  measure=list(intensity="1", df=list("dnorm(z, 0, 1)")),
                  measure.type="CP",
                  Cogarch.var="y",
                  V.var="v",
                  Latent.var="x")