yuima (version 1.0.81)

setCogarch: Continuous-time GARCH (p,q) process

Description

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}

Usage

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

Value

• model
  an object of yuima.cogarch-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: [object Object] and the same slots in an object of yuima.model-class .

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., & 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.

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")