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

A.C. Guidoum1 and K. Boukhetala2

2024-03-05

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:

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

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

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 \(X_t\):

R> set.seed(1234, kind = "L'Ecuyer-CMRG")
R> f <- expression( (1-0.5*x) )
R> g <- expression( 1 )
R> 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):

R> St  <- expression(2*(1-sinh(0.5*t)) )
R> fpt1d <- fptsde1d(mod1d, boundary = St)
R> fpt1d
Itô Sde 1D:
    | dX(t) = (1 - 0.5 * X(t)) * dt + 1 * dW(t)
    | t in [0,1].
Boundary:
    | S(t) = 2 * (1 - sinh(0.5 * t))
F.P.T:
    | T(S(t),X(t)) = inf{t >=  0 : X(t) >=  2 * (1 - sinh(0.5 * t)) }
    | Crossing realized 966 among 1000.
R> head(fpt1d$fpt, n = 5)
[1] 0.038850 0.042709 0.113998 0.066493 0.244749

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

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

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

R> plot(dfptsde1d(fpt1d),hist=TRUE,nbins="FD")  ## histogramm
R> 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:

R> require(fptdApprox)
R> x <- character(4)
R> x[1] <- "m * x"
R> x[2] <- "(sigma^2) * x^2"
R> x[3] <- "dnorm((log(x) - (log(y) + (m - sigma^2/2) * (t- s)))/(sigma * sqrt(t - s)),0,1)/(sigma * sqrt(t - s) * x)"
R> x[4] <- "plnorm(x,log(y) + (m - sigma^2/2) * (t - s),sigma * sqrt(t - s))"
R> Lognormal <- diffproc(x)
R> 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:

R> ## Set the model X(t)
R> f <- expression( 0.48*x )
R> g <- expression( 0.07*x )
R> mod1 <- snssde1d(drift=f,diffusion=g,x0=1,T=10,M=1000)
R> ## Set the boundary S(t)
R> St  <- expression( 7 + 3.2 * t + 1.4 * t * sin(1.75 * t) )
R> ## Generate the fpt
R> fpt1 <- fptsde1d(mod1, boundary = St)
R> head(fpt1$fpt, n = 5)
[1] 8.4491 5.8324 6.2194 6.0628 6.0337
R> summary(fpt1)

Monte-Carlo Statistics of F.P.T:
|T(S(t),X(t)) = inf{t >=  0 : X(t) >=  7 + 3.2 * t + 1.4 * t * sin(1.75 * t) }
                         
Mean              6.45436
Variance          0.82529
Median            6.09538
Mode              6.00222
First quartile    5.93615
Third quartile    6.33197
Minimum           5.47910
Maximum           8.92931
Skewness          1.62490
Kurtosis          3.97197
Coef-variation    0.14075
3th-order moment  1.21824
4th-order moment  2.70530
5th-order moment  5.36133
6th-order moment 11.11820

By plotting the approximations:

R> 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)
R> plot(dfptsde1d(fpt1,bw="bcv"),add=TRUE)
R> legend('topright', lty = c(1, NA), col = c(1,'#BBCCEE'),pch=c(NA,15),legend = c('Approx.fpt.density()', 'fptsde1d()'), lwd = 2, bty = 'n')
`fptsde1d()` vs `Approx.fpt.density()`

fptsde1d() vs Approx.fpt.density()

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:

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

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

R> ## Set the model X(t)
R> theta1=0.5
R> f <- expression( theta1*x*(10+0.2*sin(2*pi*t)+0.3*sqrt(t)*(1+cos(3*pi*t))-x) )
R> g <- expression( sqrt(0.1)*x )
R> mod2 <- snssde1d(drift=f,diffusion=g,x0=8,t0=1,T=4,M=1000)
R> ## Set the boundary S(t)
R> St  <- expression( 12 )
R> ## Generate the fpt
R> fpt2 <- fptsde1d(mod2, boundary = St)
R> head(fpt2$fpt, n = 5)
[1] 1.9278 1.3564 2.7549 1.8239 1.5315
R> summary(fpt2)

Monte-Carlo Statistics of F.P.T:
|T(S(t),X(t)) = inf{t >=  1 : X(t) >=  12 }
                        
Mean             2.20795
Variance         0.50426
Median           2.10384
Mode             1.45117
First quartile   1.55048
Third quartile   2.68149
Minimum          1.09881
Maximum          3.99568
Skewness         0.53741
Kurtosis         2.31695
Coef-variation   0.32162
3th-order moment 0.19243
4th-order moment 0.58915
5th-order moment 0.52001
6th-order moment 0.99788

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

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

fptsde1d() vs GQD.TIpassage()

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 \((X_t , Y_t)\):

R> set.seed(1234, kind = "L'Ecuyer-CMRG")
R> fx <- expression(5*(-1-y)*x , 5*(-1-x)*y)
R> gx <- expression(0.5*y,0.5*x)
R> 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::

R> St <- expression(sin(2*pi*t))
R> fpt2d <- fptsde2d(mod2d, boundary = St)
R> head(fpt2d$fpt, n = 5)
        x       y
1 0.13622 0.50310
2 0.14040 0.50428
3 0.13260 0.50229
4 0.14323 0.50805
5 0.13881 0.51709

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

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

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

R> summary(fpt2d)

Monte-Carlo Statistics for the F.P.T of (X(t),Y(t))
    | T(S(t),X(t)) = inf{t >=  0 : X(t) <=  sin(2 * pi * t) }
    |    And
    | T(S(t),Y(t)) = inf{t >=  0 : Y(t) >=  sin(2 * pi * t) }
                  T(S,X)   T(S,Y)
Mean             0.13470  0.50335
Variance         0.00019  0.00003
Median           0.13418  0.50319
Mode             0.13300  0.50272
First quartile   0.12511  0.49986
Third quartile   0.14343  0.50675
Minimum          0.09431  0.48489
Maximum          0.22080  0.52167
Skewness         0.38774 -0.00054
Kurtosis         4.33349  3.16471
Coef-variation   0.10167  0.01046
3th-order moment 0.00000  0.00000
4th-order moment 0.00000  0.00000
5th-order moment 0.00000  0.00000
6th-order moment 0.00000  0.00000

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

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

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

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

R> 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 \((X_t , Y_t , Z_t)\):

R> set.seed(1234, kind = "L'Ecuyer-CMRG")
R> fx <- expression(4*(-1-x)*y , 4*(1-y)*x , 4*(1-z)*y) 
R> gx <- rep(expression(0.2),3)
R> Sigma <-matrix(c(1,0.3,-0.5,0.3,1,0.2,-0.5,0.2,1),nrow=3,ncol=3)
R> 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::

R> St <- expression(-1.5+3*t)
R> fpt3d <- fptsde3d(mod3d, boundary = St)
R> head(fpt3d$fpt, n = 5)
        x        y       z
1 0.51850 0.024239 0.75841
2 0.52495 0.024041 0.80606
3 0.53565 0.021115 0.75086
4 0.53324 0.023411 0.74892
5 0.54566 0.023167 0.76537

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

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

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

R> summary(fpt3d)

Monte-Carlo Statistics for the F.P.T of (X(t),Y(t),Z(t))
    | T(S(t),X(t)) = inf{t >=  0 : X(t) <=  -1.5 + 3 * t }
    |    And
    | T(S(t),Y(t)) = inf{t >=  0 : Y(t) >=  -1.5 + 3 * t }
    |    And
    | T(S(t),Z(t)) = inf{t >=  0 : Z(t) <=  -1.5 + 3 * t }
                  T(S,X)  T(S,Y)   T(S,Z)
Mean             0.53198 0.02323  0.78404
Variance         0.00015 0.00000  0.00093
Median           0.53138 0.02321  0.78402
Mode             0.53097 0.02353  0.78322
First quartile   0.52328 0.02231  0.76216
Third quartile   0.53975 0.02407  0.80498
Minimum          0.49196 0.01912  0.68248
Maximum          0.56943 0.02792  0.88831
Skewness         0.17921 0.16558 -0.01077
Kurtosis         2.96629 2.99563  2.99648
Coef-variation   0.02298 0.05735  0.03897
3th-order moment 0.00000 0.00000  0.00000
4th-order moment 0.00000 0.00000  0.00000
5th-order moment 0.00000 0.00000  0.00000
6th-order moment 0.00000 0.00000  0.00000

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.

R> denM <- dfptsde3d(fpt3d, pdf = "M")
R> 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.)

R> denJ <- dfptsde3d(fpt3d,pdf="J")
R> 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 (2024). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.9, 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.

  1. Department of Mathematics and Computer Science, Faculty of Sciences and Technology, University of Tamanghasset, Algeria, E-mail ()↩︎

  2. Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail ()↩︎