# 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:

- `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 \(X_t\):

    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 `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()`

## 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 (acguidoum@univ-tam.dz)↩︎

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