bridgesde3d {Sim.DiffProc}

Simulation of 3-D Bridge SDE's

Description

The (S3) generic function bridgesde3d for simulation of 3-dim bridge stochastic differential equations,Itô or Stratonovich type, with different methods.

Usage

bridgesde3d(N, ...)
## Default S3 method:
bridgesde3d(N=1000,M=1, x0=c(0,0,0), 
   y=c(0,0,0), t0 = 0, T = 1, Dt, drift, diffusion, corr = NULL, 
   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 'bridgesde3d'
summary(object, at,
     digits=NULL, ...)								  
## S3 method for class 'bridgesde3d'
time(x, ...)
## S3 method for class 'bridgesde3d'
mean(x, at, ...)
## S3 method for class 'bridgesde3d'
Median(x, at, ...)
## S3 method for class 'bridgesde3d'
Mode(x, at, ...)
## S3 method for class 'bridgesde3d'
quantile(x, at, ...)
## S3 method for class 'bridgesde3d'
kurtosis(x, at, ...)
## S3 method for class 'bridgesde3d'
skewness(x, at, ...)
## S3 method for class 'bridgesde3d'
min(x, at, ...)
## S3 method for class 'bridgesde3d'
max(x, at, ...)
## S3 method for class 'bridgesde3d'
moment(x, at, ...)
## S3 method for class 'bridgesde3d'
cv(x, at, ...)
## S3 method for class 'bridgesde3d'
bconfint(x, at, ...)

## S3 method for class 'bridgesde3d'
plot(x, ...)
## S3 method for class 'bridgesde3d'
lines(x, ...)
## S3 method for class 'bridgesde3d'
points(x, ...)	
## S3 method for class 'bridgesde3d'
plot3D(x, display = c("persp","rgl"), ...)						  

Arguments

N

number of simulation steps.

M

number of trajectories.

x0

initial value (numeric vector of length 3) of the process X_t, Y_t and Z_t at time t_0.

y

terminal value (numeric vector of length 3) of the process X_t, Y_t and Z_t 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 four variables t, x, y and z for process X_t, Y_t and Z_t.

diffusion

diffusion coefficient: an expression of four variables t, x, y and z for process X_t, Y_t and Z_t.

corr

the correlation structure of three Brownian motions W1(t), W2(t) and W3(t); must be a real symmetric positive-definite square matrix of dimension 3.

alpha

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

mu

weight mu of the predictor-corrector scheme; the default 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 snssde3d.

x, object

an object inheriting from class "bridgesde3d".

at

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

digits

integer, used for number formatting.

display

"persp" perspective and "rgl" plots.

...

potentially further arguments for (non-default) methods.

Details

The function bridgesde3d returns a mts of the diffusion bridge starting at x at time t0 and ending at y at time T. W1(t), W2(t) and W3(t) three standard Brownian motion independent if corr=NULL.

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

bridgesde3d returns an object inheriting from class "bridgesde3d".

X, Y, Z

an invisible mts (3-dim) object (X(t),Y(t),Z(t)).

driftx, drifty, driftz

drift coefficient of X(t), Y(t) and Z(t).

diffx, diffy, diffz

diffusion coefficient of X(t), Y(t) and Z(t).

Cx, Cy, Cz

indices of crossing realized of X(t), Y(t)) and Z(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

bridgesde1d for simulation of 1-dim SDE. DBridge in package "sde".

bridgesde2d for simulation of 2-dim SDE.

Examples

## dX(t) = 4*(-1-X(t))*Y(t) dt + 0.2 * dW1(t) ; x01 = 0 and y01 = 0
## dY(t) = 4*(1-Y(t)) *X(t) dt + 0.2 * dW2(t) ; x02 = -1 and y02 = -2
## dZ(t) = 4*(1-Z(t)) *Y(t) dt + 0.2 * dW3(t) ; x03 = 0.5 and y03 = 0.5       
## W1(t), W2(t) and W3(t) are three correlated Brownian motions with Sigma
set.seed(1234)

fx <- expression(4*(-1-x)*y, 4*(1-y)*x, 4*(1-z)*y)
gx <- rep(expression(0.2),3)
# correlation matrix
Sigma <-matrix(c(1,0.3,0.5,0.3,1,0.2,0.5,0.2,1),nrow=3,ncol=3) 

res <- bridgesde3d(x0=c(0,-1,0.5),y=c(0,-2,0.5),drift=fx,diffusion=gx,corr=Sigma,M=200)
res
summary(res) ## Monte-Carlo statistics at time T/2=0.5
summary(res,at=0.25) ## Monte-Carlo statistics at time 0.25
summary(res,at=0.75) ## Monte-Carlo statistics at time 0.75
##
plot(res,type="n")
lines(time(res),apply(res$X,1,mean),col=3,lwd=2)
lines(time(res),apply(res$Y,1,mean),col=4,lwd=2)
lines(time(res),apply(res$Z,1,mean),col=5,lwd=2)
legend("topleft",c(expression(E(X[t])),expression(E(Y[t])),
       expression(E(Z[t]))),lty=1,inset = .01,col=c(3,4,5))
##
plot3D(res,display = "persp",main="3-dim bridge sde")