Sim.DiffProc (version 2.8)

fptsde3d: First Passage Time in 3-Dim SDE

Description

The (S3) generic function fptsde3d for simulate first-passage-time (f.p.t) in 3-dim stochastic differential equations.

Usage

fptsde3d(N, ...)
## S3 method for class 'default':
fptsde3d(N = 1000, M = 100, x0 = 0, y0 = 0, z0 = 0, t0 = 0, T = 1, Dt, 
   boundary, driftx, diffx, drifty, diffy, driftz, diffz, alpha = 0.5, mu = 0.5, 
   type = c("ito", "str"), method = c("euler", "milstein", "predcorr", 
   "smilstein", "taylor", "heun", "rk1", "rk2", "rk3"), ...)
						   
						   
## S3 method for class 'fptsde3d':
summary(object, \dots)	
## S3 method for class 'fptsde3d':
mean(x, \dots)
## S3 method for class 'fptsde3d':
median(x, \dots)
## S3 method for class 'fptsde3d':
quantile(x, \dots)
## S3 method for class 'fptsde3d':
kurtosis(x, \dots)
## S3 method for class 'fptsde3d':
skewness(x, \dots)
## S3 method for class 'fptsde3d':
moment(x, order = 2, \dots)
## S3 method for class 'fptsde3d':
bconfint(x, level=0.95, \dots)	
## S3 method for class 'fptsde3d':
plot(x, \dots)

Arguments

N

size of sde.

M

size of fpt.

x0, y0, z0

initial value of the process $X_{t}$, $Y_{t}$ and $Z_{t}$ at time t0.

t0

initial time.

T

final time.

Dt

time step of the simulation (discretization). If it is missing a default $\Delta t = \frac{T-t_{0}}{N}$.

boundary

an expression of a constant or time-dependent boundary.

driftx, drifty, driftz

drift coefficient: an expression of four variables t, x, y and z for process $X_t$, $Y_t$ and $Z_t$.

diffx, diffy, diffz

diffusion coefficient: an expression of four variables t, x, y and z for process $X_t$, $Y_t$ and $Z_t$.

alpha, mu

weight of the predictor-corrector scheme; the default alpha = 0.5 and mu = 0.5.

type

sde of the type Ito or Stratonovich.

method

numerical methods of simulation, the default method = "euler"; see snssde3d.

x, object

an object inheriting from class "fptsde3d".

order

order of moment.

level

the confidence level required.

...

further arguments for (non-default) methods.

Value

• fptsde3d returns an object inheriting from class "fptsde3d".
• fptx, fpty, fptz
  a vector of triplet 'fpt' $(\tau_{(X(t),S(t))},\tau_{(Y(t),S(t))},\tau_{(Z(t),S(t))})$.

newcommand

\CRANpkg

href

http://CRAN.R-project.org/package=#1

pkg

#1

Details

The function fptsde3d returns a random variable $(\tau_{(X(t),S(t))},\tau_{(Y(t),S(t))},\tau_{(Z(t),S(t))})$ "first passage time", is defined as : $$\tau_{(X(t),S(t))} = { t \geq 0 ; X_{t} \geq S(t) },\quad if \quad X(t_{0}) < S(t_{0})$$ $$\tau_{(Y(t),S(t))} = { t \geq 0 ; Y_{t} \geq S(t) },\quad if \quad Y(t_{0}) < S(t_{0})$$ $$\tau_{(Z(t),S(t))} = { t \geq 0 ; Z_{t} \geq S(t) },\quad if \quad Z(t_{0}) < S(t_{0})$$ and: $$\tau_{(X(t),S(t))} = { t \geq 0 ; X_{t} \leq S(t) },\quad if \quad X(t_{0}) > S(t_{0})$$ $$\tau_{(Y(t),S(t))} = { t \geq 0 ; Y_{t} \leq S(t) },\quad if \quad Y(t_{0}) > S(t_{0})$$ $$\tau_{(Z(t),S(t))} = { t \geq 0 ; Z_{t} \leq S(t) },\quad if \quad Z(t_{0}) > S(t_{0})$$ with $S(t)$ is through a continuous boundary (barrier).

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. 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.

Examples

## dX(t) = 4*(-1-X(t))*Y(t) dt + 0.2 * dW1(t) 
## dY(t) = 4*(1-Y(t)) *X(t) dt + 0.2 * dW2(t) 
## dZ(t) = 4*(1-Z(t)) *Y(t) dt + 0.2 * dW3(t) 
## x0 = 0, y0 = -2, z0 = 0, and barrier -3+5*t.       
## W1(t), W2(t) and W3(t) three independent Brownian motion      

fx <- expression(4*(-1-x)*y)
gx <- expression(0.2)
fy <- expression(4*(1-y)*x)
gy <- expression(0.2)
fz <- expression(4*(1-z)*y)
gz <- expression(0.2)

St <- expression(-3+5*t)

res <- fptsde3d(driftx=fx,diffx=gx,drifty=fy,diffy=gy,driftz=fz,diffz=gz,boundary=St,
                x0=2,y0=-2,z0=0,M=50)
res
summary(res)
plot(res,union=TRUE)
dev.new()
plot(res,union=FALSE)
##

fptx <- res$fptx
fpty <- res$fpty
fptz <- res$fptz
X1 <- cbind(fptx,fpty,fptz)
## library(sm)
## sm.density(X1,display="rgl")