setPoisson function

Basic constructor for Compound Poisson processes

'setPoisson' construct a Compound Poisson model specification for a process of the form:

Mt = m0+sum_{i=0}^Nt c*Y_{tau_i}, M0=m0

where Nt is a homogeneous or time-inhomogeneous Poisson process, tau_i is the sequence of random times of Nt and Y is a sequence of i.i.d. random jumps.

setPoisson(intensity = 1, df = NULL, scale = 1, dimension=1, ...)

Arguments

• intensity: either and expression or a numerical value representing the intensity function of the Poisson process Nt.
• df: is the density of jump random variables Y.
• scale: this is the scaling factor c.
• dimension: this is the dimension of the jump component.
• ...: passed to setModel

Details

An object of yuima.model-class where the model slot is of class yuima.poisson-class.

Returns

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

Author(s)

The YUIMA Project Team

Examples

## Not run:

Terminal <- 10
samp <- setSampling(T=Terminal,n=1000)

# Ex 1. (Simple homogeneous Poisson process)
mod1 <- setPoisson(intensity="lambda", df=list("dconst(z,1)"))
set.seed(123)
y1 <- simulate(mod1, true.par=list(lambda=1),sampling=samp)
plot(y1)

# scaling the jumps
mod2 <- setPoisson(intensity="lambda", df=list("dconst(z,1)"),scale=5)
set.seed(123)
y2 <- simulate(mod2, true.par=list(lambda=1),sampling=samp)
plot(y2)

# scaling the jumps through the constant distribution
mod3 <- setPoisson(intensity="lambda", df=list("dconst(z,5)"))
set.seed(123)
y3 <- simulate(mod3, true.par=list(lambda=1),sampling=samp)
plot(y3)

# Ex 2. (Time inhomogeneous Poisson process)
mod4 <- setPoisson(intensity="beta*(1+sin(lambda*t))", df=list("dconst(z,1)"))
set.seed(123)
lambda <- 3
beta <- 5
y4 <- simulate(mod4, true.par=list(lambda=lambda,beta=beta),sampling=samp)
par(mfrow=c(2,1))
par(mar=c(3,3,1,1))
plot(y4)
f <- function(t) beta*(1+sin(lambda*t))
curve(f, 0, Terminal, col="red")

# Ex 2. (Time inhomogeneous Compound Poisson process with Gaussian Jumps)
mod5 <- setPoisson(intensity="beta*(1+sin(lambda*t))", df=list("dnorm(z,mu,sigma)"))
set.seed(123)
y5 <- simulate(mod5, true.par=list(lambda=lambda,beta=beta,mu=0, sigma=2),sampling=samp)
plot(y5)
f <- function(t)  beta*(1+sin(lambda*t))
curve(f, 0, Terminal, col="red")
## End(Not run)