yuima (version 1.0.81)

setPoisson: Basic constructor for Compound Poisson processes

Description

'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.

Usage

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

Value

• model
  an object of yuima.model-class.

Details

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

Examples

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")