Sim.DiffProc (version 4.9)

rsde2d: Approximate transitional densities and random generation for 2-D SDE's

Description

Transition density and random generation for the joint and marginal of (X(t-s),Y(t-s) | X(s)=x0,Y(s)=y0) of the SDE's 2-d.

Usage

rsde2d(object, ...)
dsde2d(object, ...)
# S3 method for default
rsde2d(object, at, ...)
# S3 method for default
dsde2d(object, pdf=c("Joint","Marginal"), at, ...)
# S3 method for dsde2d
plot(x,display=c("persp","rgl","image","contour"),hist=FALSE,...)

Value

dsde2d()

gives the bivariate density approximation for (X(t-s),Y(t-s) | X(s)=x0,Y(s)=y0).

rsde2d()

generates random of the couple (X(t-s),Y(t-s) | X(s)=x0,Y(s)=y0).

Arguments

object

an object inheriting from class snssde2d and bridgesde2d.

at

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

pdf

probability density function Joint or Marginal.

x

an object inheriting from class dsde2d.

display

display plots.

hist

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

...

potentially potentially arguments to be passed to methods, such as density for marginal density and kde2d fro joint density.

Author

A.C. Guidoum, K. Boukhetala.

Details

The function rsde2d returns a M random variable \(x_{t=at},y_{t=at}\) realize at time \(t=at\).

And dsde2d returns a bivariate density approximation for (X(t-s),Y(t-s) | X(s)=x0,Y(s)=y0). 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

kde2d Two-dimensional kernel density estimation in "MASS" 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.

BiGQD.density Generate the transition density of a bivariate generalized quadratic diffusion model (2D GQD) in "DiffusionRgqd" package.

Examples

## Example:1
set.seed(1234)

# SDE's 2d
fx <- expression(3*(2-y),2*x)
gx <- expression(1,y)
mod2d <- snssde2d(drift=fx,diffusion=gx,x0=c(1,2),M=1000)

# random 
r2d <- rsde2d(mod2d,at=0.5)
summary(r2d)

# Marginal density 

denM <- dsde2d(mod2d,pdf="M", at=0.5)
denM
plot(denM)

# Joint density
denJ <- dsde2d(mod2d,pdf="J",n=200, at= 0.5,lims=c(-3,4,0,6))
denJ
plot(denJ)
plot(denJ,display="contour")

## Example 2: Bivariate Transition Density of 2 Brownian motion (W1(t),W2(t)) in [0,1]

if (FALSE) {
B2d <- snssde2d(drift=rep(expression(0),2),diffusion=rep(expression(1),2),
       M=10000)
for (i in seq(B2d$Dt,B2d$T,by=B2d$Dt)){
plot(dsde2d(B2d, at = i,lims=c(-3,3,-3,3),n=100),
   display="contour",main=paste0('Transition Density \n t = ',i))
}
}

## Example 3: 

if (FALSE) {
fx <- expression(4*(-1-x)*y , 4*(1-y)*x )
gx <- expression(0.25*y,0.2*x)
mod2d1 <- snssde2d(drift=fx,diffusion=gx,x0=c(x0=1,y0=-1),
      M=5000,type="str")

# Marginal transition density
for (i in seq(mod2d1$Dt,mod2d1$T,by=mod2d1$Dt)){
plot(dsde2d(mod2d1,pdf="M", at = i),main=
      paste0('Marginal Transition Density \n t = ',i))
}

# Bivariate transition density
for (i in seq(mod2d1$Dt,mod2d1$T,by=mod2d1$Dt)){
plot(dsde2d(mod2d1, at = i,lims=c(-1,2,-1,1),n=100),
    display="contour",main=paste0('Transition Density \n t = ',i))
}
}

## Example 4: Bivariate Transition Density of 2 bridge Brownian motion (W1(t),W2(t)) in [0,1]

if (FALSE) {
B2d <- bridgesde2d(drift=rep(expression(0),2),
  diffusion=rep(expression(1),2),M=5000)
for (i in seq(0.01,0.99,by=B2d$Dt)){ 
plot(dsde2d(B2d, at = i,lims=c(-3,3,-3,3),
 n=100),display="contour",main=
 paste0('Transition Density \n t = ',i))
}
}

## Example 5: Bivariate Transition Density of bridge 
## Ornstein-Uhlenbeck process and its integral in [0,5]
## dX(t) = 4*(-1-X(t)) dt + 0.2 dW1(t)
## dY(t) = X(t) dt + 0 dW2(t)
## x01 = 0 , y01 = 0
## x02 = 0, y02 = 0 
if (FALSE) {
fx <- expression(4*(-1-x) , x)
gx <- expression(0.2 , 0)
OUI <- bridgesde2d(drift=fx,diffusion=gx,Dt=0.005,M=1000)
for (i in seq(0.01,4.99,by=OUI$Dt)){
plot(dsde2d(OUI, at = i,lims=c(-1.2,0.2,-2.5,0.2),n=100),
 display="contour",main=paste0('Transition Density \n t = ',i))
}
}