fptsde1d: Approximate densities and random generation for first passage...
In Sim.DiffProc: Simulation of Diffusion Processes

fptsde1d

R Documentation

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, ...)

## Default S3 method:
fptsde1d(object, boundary, ...)
## S3 method for class 'fptsde1d'
summary(object, digits=NULL, ...)
## S3 method for class 'fptsde1d'
mean(x, ...)
## S3 method for class 'fptsde1d'
Median(x, ...)
## S3 method for class 'fptsde1d'
Mode(x, ...)
## S3 method for class 'fptsde1d'
quantile(x, ...)
## S3 method for class 'fptsde1d'
kurtosis(x, ...)
## S3 method for class 'fptsde1d'
skewness(x, ...)
## S3 method for class 'fptsde1d'
min(x, ...)
## S3 method for class 'fptsde1d'
max(x, ...)
## S3 method for class 'fptsde1d'
moment(x, ...)
## S3 method for class 'fptsde1d'
cv(x, ...)

## Default S3 method:
dfptsde1d(object, ...)
## S3 method for class 'dfptsde1d'
plot(x, hist=FALSE, ...)

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`.

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})

fig07

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).

fig08

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

Value

`dfptsde1d()`	gives the density estimate of fpt.
`fptsde1d()`	generates random of fpt.

Author(s)

A.C. Guidoum, K. Boukhetala.

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.

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
## Not run: 
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')


## End(Not run)

Sim.DiffProc documentation built on May 29, 2024, 8:09 a.m.