bridgesde3d: 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, ...)
# S3 method for default
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 bridgesde3d
summary(object, at,
     digits=NULL, ...)								  
# S3 method for bridgesde3d
time(x, ...)
# S3 method for bridgesde3d
mean(x, at, ...)
# S3 method for bridgesde3d
Median(x, at, ...)
# S3 method for bridgesde3d
Mode(x, at, ...)
# S3 method for bridgesde3d
quantile(x, at, ...)
# S3 method for bridgesde3d
kurtosis(x, at, ...)
# S3 method for bridgesde3d
skewness(x, at, ...)
# S3 method for bridgesde3d
min(x, at, ...)
# S3 method for bridgesde3d
max(x, at, ...)
# S3 method for bridgesde3d
moment(x, at, ...)
# S3 method for bridgesde3d
cv(x, at, ...)
# S3 method for bridgesde3d
bconfint(x, at, ...)
# S3 method for bridgesde3d
plot(x, ...)
# S3 method for bridgesde3d
lines(x, ...)
# S3 method for bridgesde3d
points(x, ...)	
# S3 method for bridgesde3d
plot3D(x, display = c("persp","rgl"), ...)

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.

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.

Author

A.C. Guidoum, K. Boukhetala.

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.

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

Examples

Run this code

## 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")

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