setPoisson: Basic constructor for Compound Poisson processes

In yuima: The YUIMA Project Package for SDEs

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

Details

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

Value

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)

yuima documentation built on Nov. 14, 2022, 3:02 p.m.