Monte-Carlo Simulation and Kernel Density Estimation of First passage time"

In Sim.DiffProc: Simulation of Diffusion Processes

library(Sim.DiffProc)
library(knitr)
knitr::opts_chunk$set(comment="", prompt=TRUE, fig.show='hold',warning=FALSE, message=FALSE)
options(prompt="R> ",scipen=16,digits=5,warning=FALSE, message=FALSE,
        width = 70)

The fptsdekd() functions

A new algorithm based on the Monte Carlo technique to generate the random variable FPT of a time homogeneous diffusion process (1, 2 and 3D) through a time-dependent boundary, order to estimate her probability density function.

Let (X_t) be a diffusion process which is the unique solution of the following stochastic differential equation:

\begin{equation}\label{eds01} dX_t = \mu(t,X_t) dt + \sigma(t,X_t) dW_t,\quad X_{t_{0}}=x_{0} \end{equation}

if (S(t)) is a time-dependent boundary, we are interested in generating the first passage time (FPT) of the diffusion process through this boundary that is we will study the following random variable:

[ \tau_{S(t)}= \left{ \begin{array}{ll} inf \left{t: X_{t} \geq S(t)|X_{t_{0}}=x_{0} \right} & \hbox{if} \quad x_{0} \leq S(t_{0}) \ inf \left{t: X_{t} \leq S(t)|X_{t_{0}}=x_{0} \right} & \hbox{if} \quad x_{0} \geq S(t_{0}) \end{array} \right. ]

The main arguments to 'random' fptsdekd() (where k=1,2,3) consist:

• object an object inheriting from class snssde1d, snssde2d and snssde3d.
• boundary an expression of a constant or time-dependent boundary $S(t)$.

The following statistical measures (S3 method) for class fptsdekd() can be approximated for F.P.T $\tau_{S(t)}$:

• The expected value $\text{E}(\tau_{S(t)})$, using the command mean.
• The variance $\text{Var}(\tau_{S(t)})$, using the command moment with order=2 and center=TRUE.
• The median $\text{Med}(\tau_{S(t)})$, using the command Median.
• The mode $\text{Mod}(\tau_{S(t)})$, using the command Mode.
• The quartile of $\tau_{S(t)}$, using the command quantile.
• The maximum and minimum of $\tau_{S(t)}$, using the command min and max.
• The skewness and the kurtosis of $\tau_{S(t)}$, using the command skewness and kurtosis.
• The coefficient of variation (relative variability) of $\tau_{S(t)}$, using the command cv.
• The central moments up to order $p$ of $\tau_{S(t)}$, using the command moment.
• The result summaries of the results of Monte-Carlo simulation, using the command summary.

The main arguments to 'density' dfptsdekd() (where k=1,2,3) consist:

• object an object inheriting from class fptsdekd() (where k=1,2,3).
• pdf probability density function Joint or Marginal.

Examples

FPT for 1-Dim SDE

Consider the following SDE and linear boundary:

\begin{align} dX_{t}= & (1-0.5 X_{t}) dt + dW_{t},~x_{0} =1.7.\ S(t)= & 2(1-sinh(0.5t)) \end{align}

Generating the first passage time (FPT) of this model through this boundary: [ \tau_{S(t)}= \inf \left{t: X_{t} \geq S(t) |X_{t_{0}}=x_{0} \right} ~~ \text{if} \quad x_{0} \leq S(t_{0}) ]

Set the model $X_t$:

set.seed(1234, kind = "L'Ecuyer-CMRG")
f <- expression( (1-0.5*x) )
g <- expression( 1 )
mod1d <- snssde1d(drift=f,diffusion=g,x0=1.7,M=1000,method="taylor")

Generate the first-passage-time $\tau_{S(t)}$, with fptsde1d() function ( based on density() function in [base] package):

St  <- expression(2*(1-sinh(0.5*t)) )
fpt1d <- fptsde1d(mod1d, boundary = St)
fpt1d
head(fpt1d$fpt, n = 5)

The following statistical measures (S3 method) for class fptsde1d() can be approximated for the first-passage-time $\tau_{S(t)}$:

mean(fpt1d)
moment(fpt1d , center = TRUE , order = 2) ## variance
Median(fpt1d)
Mode(fpt1d)
quantile(fpt1d)
kurtosis(fpt1d)
skewness(fpt1d)
cv(fpt1d)
min(fpt1d)
max(fpt1d)
moment(fpt1d , center= TRUE , order = 4)
moment(fpt1d , center= FALSE , order = 4)

The kernel density approximation of 'fpt1d', using dfptsde1d() function (hist=TRUE based on truehist() function in MASS package)

plot(dfptsde1d(fpt1d),hist=TRUE,nbins="FD")  ## histogramm
plot(dfptsde1d(fpt1d))              ## kernel density

Since fptdApprox and DiffusionRgqd packages can very effectively handle first passage time problems for diffusions with analytically tractable transitional densities we use it to compare some of the results from the Sim.DiffProc package.

fptsde1d() vs Approx.fpt.density()

Consider for example a diffusion process with SDE:

\begin{align} dX_{t}= & 0.48 X_{t} dt + 0.07 X_{t} dW_{t},~x_{0} =1.\ S(t)= & 7 + 3.2 t + 1.4 t \sin(1.75 t) \end{align} The resulting object is then used by the Approx.fpt.density() function in package fptdApprox to approximate the first passage time density:

require(fptdApprox)
x <- character(4)
x[1] <- "m * x"
x[2] <- "(sigma^2) * x^2"
x[3] <- "dnorm((log(x) - (log(y) + (m - sigma^2/2) * (t- s)))/(sigma * sqrt(t - s)),0,1)/(sigma * sqrt(t - s) * x)"
x[4] <- "plnorm(x,log(y) + (m - sigma^2/2) * (t - s),sigma * sqrt(t - s))"
Lognormal <- diffproc(x)
res1 <- Approx.fpt.density(Lognormal, 0, 10, 1, "7 + 3.2 * t + 1.4 * t * sin(1.75 * t)",list(m = 0.48,sigma = 0.07))

Using fptsde1d() and dfptsde1d() functions in the Sim.DiffProc package:

## Set the model X(t)
f <- expression( 0.48*x )
g <- expression( 0.07*x )
mod1 <- snssde1d(drift=f,diffusion=g,x0=1,T=10,M=1000)
## Set the boundary S(t)
St  <- expression( 7 + 3.2 * t + 1.4 * t * sin(1.75 * t) )
## Generate the fpt
fpt1 <- fptsde1d(mod1, boundary = St)
head(fpt1$fpt, n = 5)
summary(fpt1)

By plotting the approximations:

plot(res1$y ~ res1$x, type = 'l',main = 'Approximation First-Passage-Time Density', ylab = 'Density', xlab = expression(tau[S(t)]),cex.main = 0.95,lwd=2)
plot(dfptsde1d(fpt1,bw="bcv"),add=TRUE)
legend('topright', lty = c(1, NA), col = c(1,'#BBCCEE'),pch=c(NA,15),legend = c('Approx.fpt.density()', 'fptsde1d()'), lwd = 2, bty = 'n')

knitr::include_graphics("Figures/fig01.png")

fptsde1d() vs GQD.TIpassage()

Consider for example a diffusion process with SDE:

\begin{align} dX_{t}= & \theta_{1}X_{t}(10+0.2\sin(2\pi t)+0.3\sqrt(t)(1+\cos(3\pi t))-X_{t}) ) dt + \sqrt(0.1) X_{t} dW_{t},~x_{0} =8.\ S(t)= & 12 \end{align} The resulting object is then used by the GQD.TIpassage() function in package DiffusionRgqd to approximate the first passage time density:

require(DiffusionRgqd)
G1 <- function(t)
     {
 theta[1] * (10+0.2 * sin(2 * pi * t) + 0.3 * prod(sqrt(t),
 1+cos(3 * pi * t)))
 }
G2 <- function(t){-theta[1]}
Q2 <- function(t){0.1}
res2 = GQD.TIpassage(8, 12, 1, 4, 1 / 100, theta = c(0.5))

Using fptsde1d() and dfptsde1d() functions in the Sim.DiffProc package:

## Set the model X(t)
theta1=0.5
f <- expression( theta1*x*(10+0.2*sin(2*pi*t)+0.3*sqrt(t)*(1+cos(3*pi*t))-x) )
g <- expression( sqrt(0.1)*x )
mod2 <- snssde1d(drift=f,diffusion=g,x0=8,t0=1,T=4,M=1000)
## Set the boundary S(t)
St  <- expression( 12 )
## Generate the fpt
fpt2 <- fptsde1d(mod2, boundary = St)
head(fpt2$fpt, n = 5)
summary(fpt2)

By plotting the approximations (hist=TRUE based on truehist() function in MASS package):

plot(dfptsde1d(fpt2),hist=TRUE,nbins = "Scott",main = 'Approximation First-Passage-Time Density', ylab = 'Density', xlab = expression(tau[S(t)]), cex.main = 0.95)
lines(res2$density ~ res2$time, type = 'l',lwd=2)
legend('topright', lty = c(1, NA), col = c(1,'#FF00004B'),pch=c(NA,15),legend = c('GQD.TIpassage()', 'fptsde1d()'), lwd = 2, bty = 'n')

knitr::include_graphics("Figures/fig02.png")

FPT for 2-Dim SDE's

Assume that we want to describe the following Stratonovich SDE's (2D):

\begin{equation}\label{eq016} \begin{cases} dX_t = 5 (-1-Y_{t}) X_{t} dt + 0.5 Y_{t} \circ dW_{1,t}\ dY_t = 5 (-1-X_{t}) Y_{t} dt + 0.5 X_{t} \circ dW_{2,t} \end{cases} \end{equation}

and [ S(t)=\sin(2\pi t) ]

Set the system $(X_t , Y_t)$:

set.seed(1234, kind = "L'Ecuyer-CMRG")
fx <- expression(5*(-1-y)*x , 5*(-1-x)*y)
gx <- expression(0.5*y,0.5*x)
mod2d <- snssde2d(drift=fx,diffusion=gx,x0=c(x=1,y=-1),M=1000,type="str")

Generate the couple ((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})), with fptsde2d() function::

St <- expression(sin(2*pi*t))
fpt2d <- fptsde2d(mod2d, boundary = St)
head(fpt2d$fpt, n = 5)

The following statistical measures (S3 method) for class fptsde2d() can be approximated for the couple ((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})):

mean(fpt2d)
moment(fpt2d , center = TRUE , order = 2) ## variance
Median(fpt2d)
Mode(fpt2d)
quantile(fpt2d)
kurtosis(fpt2d)
skewness(fpt2d)
cv(fpt2d)
min(fpt2d)
max(fpt2d)
moment(fpt2d , center= TRUE , order = 4)
moment(fpt2d , center= FALSE , order = 4)

The result summaries of the couple ((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})):

summary(fpt2d)

The marginal density of ((\tau_{(S(t),X_{t})}) and (\tau_{(S(t),Y_{t})})) are reported using dfptsde2d() function.

denM <- dfptsde2d(fpt2d, pdf = 'M')
plot(denM)

A contour and image plot of density obtained from a realization of system ((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})).

denJ <- dfptsde2d(fpt2d, pdf = 'J',n=100)
plot(denJ,display="contour",main="Bivariate Density of F.P.T",xlab=expression(tau[x]),ylab=expression(tau[y]))
plot(denJ,display="image",main="Bivariate Density of F.P.T",xlab=expression(tau[x]),ylab=expression(tau[y]))

A $3$D plot of the Joint density with:

plot(denJ,display="persp",main="Bivariate Density of F.P.T",xlab=expression(tau[x]),ylab=expression(tau[y]))

Return to fptsde2d()

FPT for 3-Dim SDE's

Assume that we want to describe the following SDE's (3D): \begin{equation}\label{eq0166} \begin{cases} dX_t = 4 (-1-X_{t}) Y_{t} dt + 0.2 dB_{1,t}\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dB_{2,t}\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dB_{3,t} \end{cases} \end{equation} with $(B_{1,t},B_{2,t},B_{3,t})$ are three correlated standard Wiener process: $$ \Sigma= \begin{pmatrix} 1 & 0.3 &-0.5\ 0.3 & 1 & 0.2 \ -0.5 &0.2&1 \end{pmatrix} $$ and $$ S(t)=-1.5+3t $$

Set the system $(X_t , Y_t , Z_t)$:

set.seed(1234, kind = "L'Ecuyer-CMRG")
fx <- expression(4*(-1-x)*y , 4*(1-y)*x , 4*(1-z)*y) 
gx <- rep(expression(0.2),3)
Sigma <-matrix(c(1,0.3,-0.5,0.3,1,0.2,-0.5,0.2,1),nrow=3,ncol=3)
mod3d <- snssde3d(drift=fx,diffusion=gx,x0=c(x=2,y=-2,z=0),M=1000,corr=Sigma)

Generate the triplet $(\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})$, with fptsde3d() function::

St <- expression(-1.5+3*t)
fpt3d <- fptsde3d(mod3d, boundary = St)
head(fpt3d$fpt, n = 5)

The following statistical measures (S3 method) for class fptsde3d() can be approximated for the triplet $(\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})$:

mean(fpt3d)
moment(fpt3d , center = TRUE , order = 2) ## variance
Median(fpt3d)
Mode(fpt3d)
quantile(fpt3d)
kurtosis(fpt3d)
skewness(fpt3d)
cv(fpt3d)
min(fpt3d)
max(fpt3d)
moment(fpt3d , center= TRUE , order = 4)
moment(fpt3d , center= FALSE , order = 4)

The result summaries of the triplet $(\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})$:

summary(fpt3d)

The marginal density of $\tau_{(S(t),X_{t})}$ ,$\tau_{(S(t),Y_{t})}$ and $\tau_{(S(t),Z_{t})})$ are reported using dfptsde3d() function.

denM <- dfptsde3d(fpt3d, pdf = "M")
plot(denM)

For an approximate joint density for $(\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})$ (for more details, see package sm or ks.)

denJ <- dfptsde3d(fpt3d,pdf="J")
plot(denJ,display="rgl")

Return to fptsde3d()

Further reading

1. snssdekd() & dsdekd() & rsdekd()- Monte-Carlo Simulation and Analysis of Stochastic Differential Equations.
2. bridgesdekd() & dsdekd() & rsdekd() - Constructs and Analysis of Bridges Stochastic Differential Equations.
3. fptsdekd() & dfptsdekd() - Monte-Carlo Simulation and Kernel Density Estimation of First passage time.
4. MCM.sde() & MEM.sde() - Parallel Monte-Carlo and Moment Equations for SDEs.
5. TEX.sde() - Converting Sim.DiffProc Objects to LaTeX.
6. fitsde() - Parametric Estimation of 1-D Stochastic Differential Equation.

References

1. Boukhetala K (1996). Modelling and Simulation of a Dispersion Pollutant with Attractive Centre, volume 3, pp. 245-252. Computer Methods and Water Resources, Computational Mechanics Publications, Boston, USA.

2. Boukhetala K (1998). Estimation of the first passage time distribution for a simulated diffusion process. Maghreb Mathematical Review, 7, pp. 1-25.

3. Boukhetala K (1998). Kernel density of the exit time in a simulated diffusion. The Annals of The Engineer Maghrebian, 12, pp. 587-589.

4. Guidoum AC, Boukhetala K (2020). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.8, URL https://cran.r-project.org/package=Sim.DiffProc.

5. Pienaar EAD, Varughese MM (2016). DiffusionRgqd: An R Package for Performing Inference and Analysis on Time-Inhomogeneous Quadratic Diffusion Processes. R package version 0.1.3, URL https://CRAN.R-project.org/package=DiffusionRgqd.

6. 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.

7. 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.

Sim.DiffProc documentation built on Nov. 8, 2020, 4:27 p.m.