fptsde2d: Approximate densities and random generation for first passage...
In Sim.DiffProc: Simulation of Diffusion Processes

Description Usage Arguments Details Value Author(s) References See Also Examples

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

fptsde2d(object, ...)
dfptsde2d(object, ...)

## Default S3 method:
fptsde2d(object, boundary, ...)
## S3 method for class 'fptsde2d'
summary(object, digits=NULL, ...)
## S3 method for class 'fptsde2d'
mean(x, ...)
## S3 method for class 'fptsde2d'
Median(x, ...)
## S3 method for class 'fptsde2d'
Mode(x, ...)
## S3 method for class 'fptsde2d'
quantile(x, ...)
## S3 method for class 'fptsde2d'
kurtosis(x, ...)
## S3 method for class 'fptsde2d'
skewness(x, ...)
## S3 method for class 'fptsde2d'
min(x, ...)
## S3 method for class 'fptsde2d'
max(x, ...)
## S3 method for class 'fptsde2d'
moment(x, ...)
## S3 method for class 'fptsde2d'
cv(x, ...)

## Default S3 method:
dfptsde2d(object, pdf=c("Joint","Marginal"), ...)
## S3 method for class 'dfptsde2d'
plot(x,display=c("persp","rgl","image","contour"),
                         hist=FALSE, ...)

`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.

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>=0; X(t) >= S(t)}, if X(t0) < S(t0)

tau(Y(t),S(t))={t>=0; Y(t) >= S(t)}, if Y(t0) < S(t0)

and:

tau(X(t),S(t))={t>=0; X(t) <= S(t)}, if X(t0) > S(t0)

tau(Y(t),S(t))={t>=0; Y(t) <= S(t)}, if Y(t0) > S(t0)

fig09

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).

fig10

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

dfptsde2d gives the kernel density approximation for fpt. fptsde2d generates random of fpt.

A.C. Guidoum, K. Boukhetala.

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.

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".

## 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')))