Sim.DiffProc (version 4.9)

fptsde1d: Approximate densities and random generation for first passage time in 1-D SDE

Description

Kernel density and random generation for first-passage-time (f.p.t) in 1-dim stochastic differential equations.

Usage

fptsde1d(object, ...)
dfptsde1d(object, ...)
# S3 method for default
fptsde1d(object, boundary, ...)
# S3 method for fptsde1d
summary(object, digits=NULL, ...)
# S3 method for fptsde1d
mean(x, ...)
# S3 method for fptsde1d
Median(x, ...)
# S3 method for fptsde1d
Mode(x, ...)
# S3 method for fptsde1d
quantile(x, ...)
# S3 method for fptsde1d
kurtosis(x, ...)
# S3 method for fptsde1d
skewness(x, ...)
# S3 method for fptsde1d
min(x, ...)
# S3 method for fptsde1d
max(x, ...)
# S3 method for fptsde1d
moment(x, ...)
# S3 method for fptsde1d
cv(x, ...)
# S3 method for default
dfptsde1d(object, ...)
# S3 method for dfptsde1d
plot(x, hist=FALSE, ...)

Value

dfptsde1d()

gives the density estimate of fpt.

fptsde1d()

generates random of fpt.

Arguments

object

an object inheriting from class snssde1d for fptsde1d, and fptsde1d for dfptsde1d.

boundary

an expression of a constant or time-dependent boundary.

x

an object inheriting from class dfptsde1d.

hist

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

digits

integer, used for number formatting.

...

potentially further arguments for (non-default) methods, such as density for dfptsde1d.

Author

A.C. Guidoum, K. Boukhetala.

Details

The function fptsde1d returns a random variable \(\tau_{(X(t),S(t))}\) "first passage time", is defined as : $$\tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \geq S(t) \},\quad if \quad X(t_{0}) < S(t_{0})$$ $$\tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \leq S(t) \},\quad if \quad X(t_{0}) > S(t_{0})$$

And dfptsde1d returns a kernel density approximation for \(\tau_{(X(t),S(t))}\) "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

fptsde2d and fptsde3d simulation fpt for 2 and 3-dim SDE.

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

## Example 1: Ito SDE
## dX(t) = -4*X(t) *dt + 0.5*dW(t)
## S(t) = 0 (constant boundary)
set.seed(1234)

# SDE 1d
f <- expression( -4*x )
g <- expression( 0.5 )
mod <- snssde1d(drift=f,diffusion=g,x0=2,M=1000)

# boundary
St <- expression(0) 

# random
out <- fptsde1d(mod, boundary=St)
out
summary(out)
# density approximate
den <- dfptsde1d(out)
den
plot(den)

## Example 2: Stratonovich SDE
## dX(t) = 0.5*X(t)*t *dt + sqrt(1+X(t)^2) o dW(t)
## S(t) = -0.5*sqrt(t) + exp(t^2) (time-dependent boundary)
set.seed(1234)

# SDE 1d
f <- expression( 0.5*x*t )
g <- expression( sqrt(1+x^2) )
mod2 <- snssde1d(drift=f,diffusion=g,x0=2,M=1000,type="srt")

# boundary
St <- expression(-0.5*sqrt(t)+exp(t^2))

# random
out2 <- fptsde1d(mod2,boundary=St)
out2
summary(out2)
# density approximate
plot(dfptsde1d(out2,bw='ucv'))


## Example 3: fptsde1d vs fptdApproximate
if (FALSE) {
f <- expression( -0.5*x+0.5*5 )
g <- expression( 1 )
St <- expression(5+0.25*sin(2*pi*t)) 
mod <- snssde1d(drift=f,diffusion=g,boundary=St,x0=3,T=10,N=10^4,M =10000)
mod

# random
out3 <- fptsde1d(mod,boundary=St)
out3
summary(out3)
# density approximate: 
library("fptdApprox")
# Under `fptdApprox':
# Define the diffusion process and give its transitional density:
OU <- diffproc(c("alpha*x + beta","sigma^2",
"dnorm((x-(y*exp(alpha*(t-s)) - beta*(1 - exp(alpha*(t-s)))/alpha))/
(sigma*sqrt((exp(2*alpha*(t-s)) - 1)/(2*alpha))),0,1)/
(sigma*sqrt((exp(2*alpha*(t-s)) - 1)/(2*alpha)))",
"pnorm(x, y*exp(alpha*(t-s)) - beta*(1 - exp(alpha*(t-s)))/alpha,
sigma*sqrt((exp(2*alpha*(t-s)) - 1)/(2*alpha)))"))
# Approximate the first passgage time density for OU, starting in X_0 = 3
# passing through 5+0.25*sin(2*pi*t) on the time interval [0,10]:
res <- Approx.fpt.density(OU, 0, 10, 3,"5+0.25*sin(2*pi*t)", list(alpha=-0.5,beta=0.5*5,sigma=1))

##
plot(dfptsde1d(out3,bw='ucv'),main = 'fptsde1d vs fptdApproximate')
lines(res$y~res$x, type = 'l',lwd=2)
legend('topright', lty = c('solid', 'dashed'), col = c(1, 2),
       legend = c('fptdApproximate', 'fptsde1d'), lwd = 2, bty = 'n')

}