Sim.DiffProc (version 4.9)

rsde1d: Approximate transitional densities and random generation for 1-D SDE

Description

Transition density and random generation for X(t-s) | X(s)=x0 of the 1-dim SDE.

Usage

rsde1d(object, ...)
dsde1d(object, ...)
# S3 method for default
rsde1d(object, at, ...)
# S3 method for default
dsde1d(object, at, ...)
# S3 method for dsde1d
plot(x,hist=FALSE, ...)

Value

dsde1d()

gives the transition density estimate of X(t-s) | X(s)=x0.

rsde1d()

generates random of X(t-s) | X(s)=x0.

Arguments

object

an object inheriting from class snssde1d and bridgesde1d.

at

time between s=t0 and t=T. The default at = T.

x

an object inheriting from class dsde1d.

hist

if hist=TRUE plot histogram. Based on truehist function.

...

potentially arguments to be passed to methods, such as density for kernel density.

Author

A.C. Guidoum, K. Boukhetala.

Details

The function rsde1d returns a M random variable \(x_{t=at}\) realize at time \(t=at\) defined by : $$x_{ t=at } = \{ t \geq 0 ; x = X_{ t=at } \}$$

And dsde1d returns a transition density approximation for X(t-s) | X(s)=x0. with \(t= at \) is a fixed time between t0 and T.

An overview of this package, see browseVignettes('Sim.DiffProc') for more informations.

See Also

density Kernel density estimation in "stats" package.

kde Kernel density estimate for 1- to 6-dimensional data in "ks" package.

sm.density Nonparametric density estimation in one, two or three dimensions in "sm" package.

rng random number generators in "yuima" package.

dcSim Pedersen's simulated transition density in "sde" package.

rcBS, rcCIR, rcOU and rsOU in package "sde".

dcBS, dcCIR, dcOU and dsOU in package "sde".

GQD.density Generate the transition density of a scalar generalized quadratic diffusion in "DiffusionRgqd" package.

Examples

## Example 1:  
## dX(t) = (-2*(X(t)<=0)+2*(X(t)>=0)) *dt + 0.5 * dW(t)
set.seed(1234)

f <- expression(-2*(x<=0)+2*(x>=0))
g <- expression(0.5)
res1 <- snssde1d(drift=f,diffusion=g,M=5000)
x <- rsde1d(res1, at = 1)
summary(x)
dens1 <-  dsde1d(res1, at = 1)
dens1
plot(dens1,main="Transition density of X(t=1)|X(s=0)=0") # kernel estimated
plot(dens1,hist=TRUE) # histogramme

## Example 2:
## Transition density of standard Brownian motion W(t) at time = 0.5
set.seed(1234)

f <- expression(0)
g <- expression(1)
res2 <- snssde1d(drift=f,diffusion=g,M=5000)
plot(dsde1d(res2, at = 0.5),dens=function(x) dnorm(x,0,sqrt(0.5)))
plot(dsde1d(res2, at = 0.5),dens=function(x) dnorm(x,0,sqrt(0.5)),hist=TRUE)

## Example 3: Transition density of Brownian motion W(t) in [0,1]

if (FALSE) {
for (i in seq(res2$t0,res2$T,by=res2$Dt)){
plot(dsde1d(res2, at = i),main=paste0('Transition Density \n t = ',i))
}
}

## Example 4:
## Transition density of bridge Brownian motion W(t) at time = 0.25 and 0.75 
set.seed(1234)
if (FALSE) {
f <- expression(0)
g <- expression(1)
Bd <- bridgesde1d(drift=f,diffusion=g,M=5000)
Bd
plot(dsde1d(Bd, at = 0.25))         ## Transition Density at time=0.25
plot(dsde1d(Bd, at = 0.75),add=TRUE)## Transition Density at time=0.75
}