Sim.DiffProc (version 4.3)

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, …)

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.

Value

dfptsde1d gives the density estimate of fpt. fptsde1d generates random of fpt.

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

# NOT RUN {
## 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')

# }
# NOT RUN {
# }