bridgesde1d {Sim.DiffProc}

Simulation of 1-D Bridge SDE

Description

The (S3) generic function bridgesde1d for simulation of 1-dim bridge stochastic differential equations,Itô or Stratonovich type, with different methods.

Usage

bridgesde1d(N, ...)
## Default S3 method:
bridgesde1d(N = 1000, M=1, x0 = 0, y = 0, t0 = 0, T = 1, Dt, 
   drift, diffusion, alpha = 0.5, mu = 0.5, type = c("ito", "str"), 
   method = c("euler", "milstein", "predcorr", "smilstein", "taylor", 
   "heun", "rk1", "rk2", "rk3"), ...)
   
## S3 method for class 'bridgesde1d'
summary(object, at ,digits=NULL, ...)							  
## S3 method for class 'bridgesde1d'
time(x, ...)
## S3 method for class 'bridgesde1d'
mean(x, at, ...)
## S3 method for class 'bridgesde1d'
Median(x, at, ...)
## S3 method for class 'bridgesde1d'
Mode(x, at, ...)
## S3 method for class 'bridgesde1d'
quantile(x, at, ...)
## S3 method for class 'bridgesde1d'
kurtosis(x, at, ...)
## S3 method for class 'bridgesde1d'
skewness(x, at, ...)
## S3 method for class 'bridgesde1d'
min(x, at, ...)
## S3 method for class 'bridgesde1d'
max(x, at, ...)
## S3 method for class 'bridgesde1d'
moment(x, at, ...)
## S3 method for class 'bridgesde1d'
cv(x, at, ...)
## S3 method for class 'bridgesde1d'
bconfint(x, at,  ...)

## S3 method for class 'bridgesde1d'
plot(x, ...)
## S3 method for class 'bridgesde1d'
lines(x, ...)
## S3 method for class 'bridgesde1d'
points(x, ...)							  

Arguments

N

number of simulation steps.

M

number of trajectories.

x0

initial value of the process at time t0.

y

terminal value of the process at time T.

t0

initial time.

T

final time.

Dt

time step of the simulation (discretization). If it is missing a default \Delta t = \frac{T-t_{0}}{N}.

drift

drift coefficient: an expression of two variables t and x.

diffusion

diffusion coefficient: an expression of two variables t and x.

alpha, mu

weight of the predictor-corrector scheme; the default alpha = 0.5 and mu = 0.5.

type

if type="ito" simulation diffusion bridge of Itô type, else type="str" simulation diffusion bridge of Stratonovich type; the default type="ito".

method

numerical methods of simulation, the default method = "euler"; see snssde1d.

x, object

an object inheriting from class "bridgesde1d".

at

time between t0 and T. Monte-Carlo statistics of the solution X_{t} at time at. The default at = T/2.

digits

integer, used for number formatting.

...

potentially further arguments for (non-default) methods.

Details

The function bridgesde1d returns a trajectory of the diffusion bridge starting at x at time t0 and ending at y at time T.

The methods of approximation are classified according to their different properties. Mainly two criteria of optimality are used in the literature: the strong and the weak (orders of) convergence. The method of simulation can be one among: Euler-Maruyama Order 0.5, Milstein Order 1, Milstein Second-Order, Predictor-Corrector method, Itô-Taylor Order 1.5, Heun Order 2 and Runge-Kutta Order 1, 2 and 3.

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

Value

bridgesde1d returns an object inheriting from class "bridgesde1d".

X

an invisible ts object.

drift

drift coefficient.

diffusion

diffusion coefficient.

C

indices of crossing realized of X(t).

type

type of sde.

method

the numerical method used.

Author(s)

A.C. Guidoum, K. Boukhetala.

References

Bladt, M. and Sorensen, M. (2007). Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Working Paper, University of Copenhagen.

Iacus, S.M. (2008). Simulation and inference for stochastic differential equations: with R examples. Springer-Verlag, New York

See Also

bridgesde2d and bridgesde3d for 2 and 3-dim.

DBridge in package "sde".

Examples

## Example 1: Ito bridge sde
## Ito Bridge sde
## dX(t) = 2*(1-X(t)) *dt + dW(t)
## x0 = 2 at time t0=0 , and y = 1 at time T=1
set.seed(1234)

f <- expression( 2*(1-x) )
g <- expression( 1 )
mod1 <- bridgesde1d(drift=f,diffusion=g,x0=2,y=1,M=1000)
mod1
summary(mod1) ## Monte-Carlo statistics at T/2=0.5
summary(mod1,at=0.75) ## Monte-Carlo statistics at 0.75
## Not run: 
plot(mod1)
lines(time(mod1),apply(mod1$X,1,mean),col=2,lwd=2)
lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[1,],col=4,lwd=2)
lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[2,],col=4,lwd=2)
legend("topleft",c("mean path",paste("bound of", 95," percent confidence")),
       inset = .01,col=c(2,4),lwd=2,cex=0.8)

## End(Not run)

## Example 2: Stratonovich sde
## dX(t) = ((2-X(t))/(2-t)) dt + X(t) o dW(t)
## x0 = 2 at time t0=0 , and y = 2 at time T=1
set.seed(1234)

f <- expression((2-x)/(2-t))
g <- expression(x)
mod2 <- bridgesde1d(type="str",drift=f,diffusion=g,M=1000,x0=2,y=2)
mod2
summary(mod2,at = 0.25) ## Monte-Carlo statistics at 0.25
summary(mod2,at = 0.5)  ## Monte-Carlo statistics at 0.5
summary(mod2,at = 0.75 )## Monte-Carlo statistics at 0.75
## Not run: 
plot(mod2)
lines(time(mod2),apply(mod2$X,1,mean),col=2,lwd=2)
lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[1,],col=4,lwd=2)
lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[2,],col=4,lwd=2)
legend("topright",c("mean path",paste("bound of", 95," percent confidence")),
       inset = .01,col=c(2,4),lwd=2,cex=0.8)

## End(Not run)