Sim.DiffProc (version 4.9)

fptsde2d: Approximate densities and random generation for first passage time in 2-D SDE's

Description

Kernel density and random generation for first-passage-time (f.p.t) in 2-dim stochastic differential equations.

Usage

fptsde2d(object, ...)
dfptsde2d(object, ...)
# S3 method for default
fptsde2d(object, boundary, ...)
# S3 method for fptsde2d
summary(object, digits=NULL, ...)
# S3 method for fptsde2d
mean(x, ...)
# S3 method for fptsde2d
Median(x, ...)
# S3 method for fptsde2d
Mode(x, ...)
# S3 method for fptsde2d
quantile(x, ...)
# S3 method for fptsde2d
kurtosis(x, ...)
# S3 method for fptsde2d
skewness(x, ...)
# S3 method for fptsde2d
min(x, ...)
# S3 method for fptsde2d
max(x, ...)
# S3 method for fptsde2d
moment(x, ...)
# S3 method for fptsde2d
cv(x, ...)
# S3 method for default
dfptsde2d(object, pdf=c("Joint","Marginal"), ...)
# S3 method for dfptsde2d
plot(x,display=c("persp","rgl","image","contour"),
                         hist=FALSE, ...)

Value

dfptsde2d()

gives the kernel density approximation for fpt.

fptsde2d()

generates random of fpt.

Arguments

object

an object inheriting from class snssde2d for fptsde2d, and fptsde2d for dfptsde2d.

boundary

an expression of a constant or time-dependent boundary.

pdf

probability density function Joint or Marginal.

x

an object inheriting from class fptsde2d.

digits

integer, used for number formatting.

display

display plots.

hist

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

...

potentially further arguments for (non-default) methods. 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 fptsde1d returns a random variable \((\tau_{(X(t),S(t))},\tau_{(Y(t),S(t))})\) "first passage time", is defined as : $$\tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \geq S(t) \},\quad if \quad X(t_{0}) < S(t_{0})$$ $$\tau_{(Y(t),S(t))} = \{ t \geq 0 ; Y_{t} \geq S(t) \},\quad if \quad Y(t_{0}) < S(t_{0})$$ and: $$\tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \leq S(t) \},\quad if \quad X(t_{0}) > S(t_{0})$$ $$\tau_{(Y(t),S(t))} = \{ t \geq 0 ; Y_{t} \leq S(t) \},\quad if \quad Y(t_{0}) > S(t_{0})$$

And dfptsde2d returns a kernel density approximation for \((\tau_{(X(t),S(t))},\tau_{(Y(t),S(t))})\) "first passage time". with \(S(t)\) is through a continuous boundary (barrier).

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

References

Argyrakisa, P. and G.H. Weiss (2006). A first-passage time problem for many random walkers. Physica A. 363, 343--347.

Aytug H., G. J. Koehler (2000). New stopping criterion for genetic algorithms. European Journal of Operational Research, 126, 662--674.

Boukhetala, K. (1996) Modelling and simulation of a dispersion pollutant with attractive centre. ed by Computational Mechanics Publications, Southampton ,U.K and Computational Mechanics Inc, Boston, USA, 245--252.

Boukhetala, K. (1998a). Estimation of the first passage time distribution for a simulated diffusion process. Maghreb Math.Rev, 7(1), 1--25.

Boukhetala, K. (1998b). Kernel density of the exit time in a simulated diffusion. les Annales Maghrebines De L ingenieur, 12, 587--589.

Ding, M. and G. Rangarajan. (2004). First Passage Time Problem: A Fokker-Planck Approach. New Directions in Statistical Physics. ed by L. T. Wille. Springer. 31--46.

Roman, R.P., Serrano, J. J., Torres, F. (2008). First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Computational Statistics and Data Analysis. 52, 4132--4146.

Roman, R.P., Serrano, J. J., Torres, F. (2012). An R package for an efficient approximation of first-passage-time densities for diffusion processes based on the FPTL function. Applied Mathematics and Computation, 218, 8408--8428.

Gardiner, C. W. (1997). Handbook of Stochastic Methods. Springer-Verlag, New York.

See Also

fptsde1d for simulation fpt in sde 1-dim. fptsde3d for simulation fpt in sde 3-dim.

FPTL for computes values of the first passage time location (FPTL) function, and Approx.fpt.density for approximate first-passage-time (f.p.t.) density in package "fptdApprox".

GQD.TIpassage for compute the First Passage Time Density of a GQD With Time Inhomogeneous Coefficients in package "DiffusionRgqd".

Examples

## dX(t) = 5*(-1-Y(t))*X(t) * dt + 0.5 * dW1(t)          
## dY(t) = 5*(-1-X(t))*Y(t) * dt + 0.5 * dW2(t)
## x0 = 2, y0 = -2, and barrier -3+5*t.
## W1(t) and W2(t) two independent Brownian motion
set.seed(1234)

# SDE's 2d
fx <- expression(5*(-1-y)*x , 5*(-1-x)*y)
gx <- expression(0.5 , 0.5)
mod2d <- snssde2d(drift=fx,diffusion=gx,x0=c(2,-2),M=100)

# boundary

St <- expression(-1+5*t)

# random fpt

out <- fptsde2d(mod2d,boundary=St)
out
summary(out)

# Marginal density 

denM <- dfptsde2d(out,pdf="M")
denM
plot(denM)

# Joint density

denJ <- dfptsde2d(out,pdf="J",n=200,lims=c(0.28,0.4,0.04,0.13))
denJ
plot(denJ)
plot(denJ,display="image")
plot(denJ,display="image",drawpoints=TRUE,cex=0.5,pch=19,col.pt='green')
plot(denJ,display="contour")
plot(denJ,display="contour",color.palette=colorRampPalette(c('white','green','blue','red')))