Sim.DiffProc (version 4.9)

bridgesde2d: Simulation of 2-D Bridge SDE's

Description

The (S3) generic function bridgesde2d for simulation of 2-dim bridge stochastic differential equations,Itô or Stratonovich type, with different methods.

Usage

bridgesde2d(N, ...)
# S3 method for default
bridgesde2d(N = 1000, M = 1, x0 = c(0, 0), 
   y = c(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 bridgesde2d
summary(object, at,
     digits=NULL, ...)								  
# S3 method for bridgesde2d
time(x, ...)
# S3 method for bridgesde2d
mean(x, at, ...)
# S3 method for bridgesde2d
Median(x, at, ...)
# S3 method for bridgesde2d
Mode(x, at, ...)
# S3 method for bridgesde2d
quantile(x, at, ...)
# S3 method for bridgesde2d
kurtosis(x, at, ...)
# S3 method for bridgesde2d
skewness(x, at, ...)
# S3 method for bridgesde2d
min(x, at, ...)
# S3 method for bridgesde2d
max(x, at, ...)
# S3 method for bridgesde2d
moment(x, at, ...)
# S3 method for bridgesde2d
cv(x, at, ...)
# S3 method for bridgesde2d
bconfint(x, at, ...)
# S3 method for bridgesde2d
plot(x, ...)
# S3 method for bridgesde2d
lines(x, ...)
# S3 method for bridgesde2d
points(x, ...)	
# S3 method for bridgesde2d
plot2d(x, ...)
# S3 method for bridgesde2d
lines2d(x, ...)
# S3 method for bridgesde2d
points2d(x, ...)

Value

bridgesde2d returns an object inheriting from class

"bridgesde2d".

X, Y

an invisible mts (2-dim) object (X(t),Y(t)).

driftx, drifty

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

diffx, diffy

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

Cx, Cy

indices of crossing realized of X(t) and Y(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 2) of the process \(X_t\) and \(Y_t\) at time \(t_0\).

y

terminal value (numeric vector of length 2) of the process \(X_t\) and \(Y_t\) 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 three variables t, x and y for process \(X_t\) and \(Y_t\).

diffusion

diffusion coefficient: an expression of three variables t, x and y for process \(X_t\) and \(Y_t\).

corr

the correlation structure of two Brownian motions W1(t) and W2(t); must be a real symmetric positive-definite square matrix of dimension 2.

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 snssde2d.

x, object

an object inheriting from class "bridgesde2d".

at

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

digits

integer, used for number formatting.

...

potentially further arguments for (non-default) methods.

Author

A.C. Guidoum, K. Boukhetala.

Details

The function bridgesde2d returns a mts of the diffusion bridge starting at x at time t0 and ending at y at time T. W1(t) and W2(t) are two 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

See Also

bridgesde1d for simulation of 1-dim SDE.

DBridge in package "sde".

Examples

## 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 
## W1(t) and W2(t) two correlated Brownian motion with matrix Sigma=matrix(c(1,0.7,0.7,1),nrow=2)
set.seed(1234)

fx <- expression(4*(-1-x) , x)
gx <- expression(0.2 , 0)
Sigma= matrix(c(1,0.7,0.7,1),nrow=2)
res <- bridgesde2d(drift=fx,diffusion=gx,Dt=0.005,M=500,corr=Sigma)
res
summary(res) ## Monte-Carlo statistics at time T/2=2.5
summary(res,at=1) ## Monte-Carlo statistics at time 1
summary(res,at=4) ## Monte-Carlo statistics at time 4
##
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)
legend("topright",c(expression(E(X[t])),expression(E(Y[t]))),lty=1,inset = .7,col=c(3,4))
##
plot2d(res)