# 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:

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

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> 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 975 among 1000.
R> head(fpt1d$fpt, n = 10) [1] 0.096261 0.464695 0.509866 0.027795 0.044845 0.036563 0.100767 [8] 0.020404 0.079737 0.043660 The following statistical measures (S3 method) for class fptsde1d() can be approximated for the first-passage-time $$\tau_{S(t)}$$: R> mean(fpt1d) [1] 0.19036 R> moment(fpt1d , center = TRUE , order = 2) ## variance [1] 0.042175 R> Median(fpt1d) [1] 0.11191 R> Mode(fpt1d) [1] 0.058009 R> quantile(fpt1d) 0% 25% 50% 75% 100% 0.0094354 0.0563074 0.1119142 0.2346988 0.9950344 R> kurtosis(fpt1d) [1] 6.1617 R> skewness(fpt1d) [1] 1.8841 R> cv(fpt1d) [1] 1.0793 R> min(fpt1d) [1] 0.0094354 R> max(fpt1d) [1] 0.99503 R> moment(fpt1d , center= TRUE , order = 4) [1] 0.010982 R> moment(fpt1d , center= FALSE , order = 4) [1] 0.03391 The result summaries of the first-passage-time $$\tau_{S(t)}$$: R> summary(fpt1d) Monte-Carlo Statistics of F.P.T: |T(S(t),X(t)) = inf{t >= 0 : X(t) >= 2 * (1 - sinh(0.5 * t)) } Mean 0.19036 Variance 0.04222 Median 0.11191 Mode 0.05801 First quartile 0.05631 Third quartile 0.23470 Minimum 0.00944 Maximum 0.99503 Skewness 1.88408 Kurtosis 6.16170 Coef-variation 1.07935 3th-order moment 0.01634 4th-order moment 0.01098 5th-order moment 0.00698 6th-order moment 0.00475 Display the exact first-passage-time $$\tau_{S(t)}$$, see Figure 1: R> plot(time(mod1d),mod1d$X[,1],type="l",lty=3,ylab="X(t)",xlab="time",axes=F)
R> points(fpt1d$fpt[1],2*(1-sinh(0.5*fpt1d$fpt[1])),pch=19,col=4,cex=0.5)
R> lines(c(fpt1d$fpt[1],fpt1d$fpt[1]),c(0,2*(1-sinh(0.5*fpt1d$fpt[1]))),lty=2,col=4) R> axis(1, fpt1d$fpt[1], bquote(tau[S(t)]==.(fpt1dfpt[1])),col=4,col.ticks=4) R> legend('topleft',col=c(1,2,4),lty=c(1,1,NA),pch=c(NA,NA,19),legend=c(expression(X[t]),expression(S(t)),expression(tau[S(t)])),cex=0.8,bty = 'n') R> box() The kernel density approximation of ‘fpt1d’, using dfptsde1d() function (hist=TRUE based on truehist() function in MASS package), see e.g. Figure 2. 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> fpt1 Itô Sde 1D: | dX(t) = 0.48 * X(t) * dt + 0.07 * X(t) * dW(t) | t in [0,10]. Boundary: | S(t) = 7 + 3.2 * t + 1.4 * t * sin(1.75 * t) F.P.T: | T(S(t),X(t)) = inf{t >= 0 : X(t) >= 7 + 3.2 * t + 1.4 * t * sin(1.75 * t) } | Crossing realized 1000 among 1000. R> head(fpt1fpt, n = 10)
[1] 8.7914 5.8383 6.1322 6.6407 5.9285 8.5290 6.1353 6.1978 8.2015
[10] 5.7198
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              6.54698
Variance          0.91449
Median            6.14013
Mode              6.07867
First quartile    5.96915
Third quartile    6.43835
Minimum           5.62479
Maximum           8.98987
Skewness          1.43627
Kurtosis          3.34637
Coef-variation    0.14607
3th-order moment  1.25604
4th-order moment  2.79853
5th-order moment  5.34044
6th-order moment 10.84165

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> 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 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> fpt2
Itô Sde 1D:
| dX(t) = theta1 * 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)
| t in [1,4].
Boundary:
| S(t) = 12
F.P.T:
| T(S(t),X(t)) = inf{t >=  1 : X(t) >=  12 }
| Crossing realized 925 among 1000.
R> head(fpt2$fpt, n = 10) [1] 2.3961 1.3939 1.6272 2.6868 2.4874 1.4654 2.7769 1.3799 1.9608 [10] 2.2019 R> summary(fpt2) Monte-Carlo Statistics of F.P.T: |T(S(t),X(t)) = inf{t >= 1 : X(t) >= 12 } Mean 2.13599 Variance 0.47712 Median 2.03809 Mode 1.44323 First quartile 1.52003 Third quartile 2.57763 Minimum 1.10178 Maximum 3.98158 Skewness 0.68436 Kurtosis 2.59034 Coef-variation 0.32338 3th-order moment 0.22554 4th-order moment 0.58969 5th-order moment 0.62583 6th-order moment 1.12710 By plotting the approximations (hist=TRUE based on truehist() function in MASS package): R> plot(dfptsde1d(fpt2),hist=TRUE,nbins = "Scott",main = 'Approximation First-Passage-Time Density', ylab = 'Density', xlab = expression(tau[S(t)]), cex.main = 0.95) R> lines(res2$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') ## FPT for 2-Dim SDE’s The following $$2$$-dimensional SDE’s with a vector of drift and a diagonal matrix of diffusion coefficients: $$$\label{eq:09} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) dW_{2,t} \end{cases}$$$ $$W_{1,t}$$ and $$W_{2,t}$$ is a two independent standard Wiener process. First passage time (2D) $$(\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})$$ is defined as: $\left\{ \begin{array}{ll} \tau_{S(t),X_{t}}=\inf \left\{t: X_{t} \geq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \leq S(t_{0}) \\ \tau_{S(t),Y_{t}}=\inf \left\{t: Y_{t} \geq S(t)|Y_{t_{0}}=y_{0} \right\} & \hbox{if} \quad y_{0} \leq S(t_{0}) \end{array} \right.$ and $\left\{ \begin{array}{ll} \tau_{S(t),X_{t}}= \inf \left\{t: X_{t} \leq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \geq S(t_{0}) \\ \tau_{S(t),Y_{t}}= \inf \left\{t: Y_{t} \leq S(t)|Y_{t_{0}}=y_{0} \right\} & \hbox{if} \quad y_{0} \geq S(t_{0}) \end{array} \right.$ Assume that we want to describe the following Stratonovich SDE’s (2D): $$$\label{eq016} \begin{cases} dX_t = 5 (-1-Y_{t}) X_{t} dt + 0.5 Y_{t} \circ dW_{1,t}\\ dY_t = 5 (-1-X_{t}) Y_{t} dt + 0.5 X_{t} \circ dW_{2,t} \end{cases}$$$ and $S(t)=\sin(2\pi t)$ Set the system $$(X_t , Y_t)$$: 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> fpt2d Stratonovich Sde 2D: | dX(t) = 5 * (-1 - Y(t)) * X(t) * dt + 0.5 * Y(t) o dW1(t) | dY(t) = 5 * (-1 - X(t)) * Y(t) * dt + 0.5 * X(t) o dW2(t) | t in [0,1]. Boundary: | S(t) = sin(2 * pi * t) F.P.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) } | Crossing realized 1000 among 1000. R> head(fpt2d$fpt, n = 10)
x       y
1  0.14772 0.50296
2  0.12583 0.49905
3  0.15479 0.49246
4  0.14647 0.50153
5  0.15205 0.50438
6  0.15565 0.49350
7  0.12404 0.50258
8  0.14260 0.50512
9  0.12047 0.50759
10 0.14081 0.49838

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)
[1] 0.13391 0.50334
R> moment(fpt2d , center = TRUE , order = 2) ## variance
[1] 0.000171725 0.000028003
R> Median(fpt2d)
[1] 0.13354 0.50320
R> Mode(fpt2d)
[1] 0.13396 0.50300
R> quantile(fpt2d)
$x 0% 25% 50% 75% 100% 0.089985 0.125564 0.133543 0.142065 0.184686$y
0%     25%     50%     75%    100%
0.48606 0.49984 0.50320 0.50673 0.52257
R> kurtosis(fpt2d)
[1] 3.3818 3.2046
R> skewness(fpt2d)
[1] 0.18393 0.17609
R> cv(fpt2d)
[1] 0.097906 0.010519
R> min(fpt2d)
[1] 0.089985 0.486055
R> max(fpt2d)
[1] 0.18469 0.52257
R> moment(fpt2d , center= TRUE , order = 4)
[1] 0.000000099928 0.000000002518
R> moment(fpt2d , center= FALSE , order = 4)
[1] 0.00034038 0.06422726

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             0.13391 0.50334
Variance         0.00017 0.00003
Median           0.13354 0.50320
Mode             0.13396 0.50300
First quartile   0.12556 0.49984
Third quartile   0.14206 0.50673
Minimum          0.08999 0.48606
Maximum          0.18469 0.52257
Skewness         0.18393 0.17609
Kurtosis         3.38183 3.20459
Coef-variation   0.09791 0.01052
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

Display the exact first-passage-time $$\tau_{S(t)}$$, see Figure 5:

R> plot(ts.union(mod2d$X[,1],mod2d$Y[,1]),col=1:2,lty=3,plot.type="single",type="l",ylab= "",xlab="time",axes=F)
R> points(fpt2d$fpt$x[1],sin(2*pi*fpt2d$fpt$x[1]),pch=19,col=4,cex=0.5)
R> lines(c(fpt2d$fpt$x[1],fpt2d$fpt$x[1]),c(sin(2*pi*fpt2d$fpt$x[1]),-10),lty=2,col=4)
R> axis(1, fpt2d$fpt$x[1], bquote(tau[X[S(t)]]==.(fpt2d$fpt$x[1])),col=4,col.ticks=4)
R> points(fpt2d$fpt$y[1],sin(2*pi*fpt2d$fpt$y[1]),pch=19,col=5,cex=0.5)
R> lines(c(fpt2d$fpt$y[1],fpt2d$fpt$y[1]),c(sin(2*pi*fpt2d$fpt$y[1]),-10),lty=2,col=5)
R> axis(1, fpt2d$fpt$y[1], bquote(tau[Y[S(t)]]==.(fpt2d$fpt$y[1])),col=5,col.ticks=5)
R> legend('topright',col=1:5,lty=c(1,1,1,NA,NA),pch=c(NA,NA,NA,19,19),legend=c(expression(X[t]),expression(Y[t]),expression(S(t)),expression(tau[X[S(t)]]),expression(tau[Y[S(t)]])),cex=0.8,inset = .01)
R> box()

The marginal density of $$(\tau_{(S(t),X_{t})}$$ and $$\tau_{(S(t),Y_{t})})$$ are reported using dfptsde2d() function, see e.g. Figure 6.

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]))

## FPT for 3-Dim SDE’s

The following $$3$$-dimensional SDE’s with a vector of drift and a diagonal matrix of diffusion coefficients:

$$$\label{eq17} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) dW_{2,t}\\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) dW_{3,t} \end{cases}$$$

$$W_{1,t}$$, $$W_{2,t}$$ and $$W_{3,t}$$ is a 3 independent standard Wiener process. First passage time (3D) $$(\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})$$ is defined as:

$\left\{ \begin{array}{ll} \tau_{S(t),X_{t}}=\inf \left\{t: X_{t} \geq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \leq S(t_{0}) \\ \tau_{S(t),Y_{t}}=\inf \left\{t: Y_{t} \geq S(t)|Y_{t_{0}}=y_{0} \right\} & \hbox{if} \quad y_{0} \leq S(t_{0}) \\ \tau_{S(t),Z_{t}}=\inf \left\{t: Z_{t} \geq S(t)|Z_{t_{0}}=z_{0} \right\} & \hbox{if} \quad z_{0} \leq S(t_{0}) \end{array} \right.$ and $\left\{ \begin{array}{ll} \tau_{S(t),X_{t}}= \inf \left\{t: X_{t} \leq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \geq S(t_{0}) \\ \tau_{S(t),Y_{t}}= \inf \left\{t: Y_{t} \leq S(t)|Y_{t_{0}}=y_{0} \right\} & \hbox{if} \quad y_{0} \geq S(t_{0}) \\ \tau_{S(t),Z_{t}}= \inf \left\{t: Z_{t} \leq S(t)|Z_{t_{0}}=z_{0} \right\} & \hbox{if} \quad z_{0} \geq S(t_{0}) \\ \end{array} \right.$

Assume that we want to describe the following SDE’s (3D): $$$\label{eq0166} \begin{cases} dX_t = 4 (-1-X_{t}) Y_{t} dt + 0.2 dW_{1,t}\\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t}\\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t} \end{cases}$$$

and $S(t)=-1.5+3t$

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

R> fx <- expression(4*(-1-x)*y , 4*(1-y)*x , 4*(1-z)*y)
R> gx <- rep(expression(0.2),3)
R> mod3d <- snssde3d(drift=fx,diffusion=gx,x0=c(x=2,y=-2,z=0),M=1000)

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> fpt3d
Itô Sde 3D:
| 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)
| t in [0,1].
Boundary:
| S(t) = -1.5 + 3 * t
F.P.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 }
| Crossing realized 1000 among 1000.
R> head(fpt3d$fpt, n = 10) x y z 1 0.51515 0.022702 0.77990 2 0.53911 0.024360 0.85046 3 0.52964 0.021566 0.74697 4 0.51553 0.023404 0.77999 5 0.52309 0.020915 0.76343 6 0.49688 0.022795 0.80233 7 0.53436 0.024997 0.79127 8 0.51361 0.024625 0.82525 9 0.50504 0.022368 0.75422 10 0.53545 0.025108 0.73533 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) [1] 0.531420 0.023284 0.783810 R> moment(fpt3d , center = TRUE , order = 2) ## variance [1] 0.0001904684 0.0000016756 0.0009550880 R> Median(fpt3d) [1] 0.53100 0.02324 0.78545 R> Mode(fpt3d) [1] 0.531049 0.023309 0.787171 R> quantile(fpt3d)$x
0%     25%     50%     75%    100%
0.49521 0.52228 0.53100 0.53988 0.57769

$y 0% 25% 50% 75% 100% 0.019306 0.022376 0.023240 0.024121 0.028735$z
0%     25%     50%     75%    100%
0.68032 0.76400 0.78545 0.80508 0.86922
R> kurtosis(fpt3d)
[1] 3.0739 2.9687 3.2016
R> skewness(fpt3d)
[1]  0.22514  0.18703 -0.27444
R> cv(fpt3d)
[1] 0.025983 0.055621 0.039448
R> min(fpt3d)
[1] 0.495207 0.019306 0.680319
R> max(fpt3d)
[1] 0.577687 0.028735 0.869215
R> moment(fpt3d , center= TRUE , order = 4)
[1] 0.0000001117385477 0.0000000000083518 0.0000029263465403
R> moment(fpt3d , center= FALSE , order = 4)
[1] 0.08007775773 0.00000029944 0.38093388098

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             0.53142 0.02328  0.78381
Variance         0.00019 0.00000  0.00096
Median           0.53100 0.02324  0.78545
Mode             0.53105 0.02331  0.78717
First quartile   0.52228 0.02238  0.76400
Third quartile   0.53988 0.02412  0.80508
Minimum          0.49521 0.01931  0.68032
Maximum          0.57769 0.02874  0.86922
Skewness         0.22514 0.18703 -0.27444
Kurtosis         3.07389 2.96872  3.20162
Coef-variation   0.02598 0.05562  0.03945
3th-order moment 0.00000 0.00000 -0.00001
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

Display the exact first-passage-time $$\tau_{S(t)}$$, see Figure 9:

R> plot(ts.union(mod3d$X[,1],mod3d$Y[,1],mod3d$Z[,1]),col=1:3,lty=3,plot.type="single",type="l",ylab="",xlab="time",axes=F) R> curve(-1.5+3*x,add=TRUE,col=4) R> points(fpt3d$fpt$x[1],-1.5+3*fpt3d$fpt$x[1],pch=19,col=5,cex=0.5) R> lines(c(fpt3d$fpt$x[1],fpt3d$fpt$x[1]),c(-1.5+3*fpt3d$fpt$x[1],-10),lty=2,col=5) R> axis(1, fpt3d$fpt$x[1], bquote(tau[X[S(t)]]==.(fpt3d$fpt$x[1])),col=5,col.ticks=5) R> points(fpt3d$fpt$y[1],-1.5+3*fpt3d$fpt$y[1],pch=19,col=6,cex=0.5) R> lines(c(fpt3d$fpt$y[1],fpt3d$fpt$y[1]),c(-1.5+3*fpt3d$fpt$y[1],-10),lty=2,col=6) R> axis(1, fpt3d$fpt$y[1], bquote(tau[Y[S(t)]]==.(fpt3d$fpt$y[1])),col=6,col.ticks=6) R> points(fpt3d$fpt$z[1],-1.5+3*fpt3d$fpt$z[1],pch=19,col=7,cex=0.5) R> lines(c(fpt3d$fpt$z[1],fpt3d$fpt$z[1]),c(-1.5+3*fpt3d$fpt$z[1],-10),lty=2,col=7) R> axis(1, fpt3d$fpt$z[1], bquote(tau[Z[S(t)]]==.(fpt3d$fpt\$z[1])),col=7,col.ticks=7)
R> legend('topright',col=1:7,lty=c(1,1,1,1,NA,NA,NA),pch=c(NA,NA,NA,NA,19,19,19),legend=c(expression(X[t]),expression(Y[t]),expression(Z[t]),expression(S(t)),expression(tau[X[S(t)]]),expression(tau[Y[S(t)]]),expression(tau[Z[S(t)]])),cex=0.8,inset = .01)
R> box()

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, see e.g. Figure 10.

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

Marginal density for the F.P.T of X(t)
| T(S,X) = inf{t >= 0 : X(t) <= -1.5 + 3 * t}

Data: out[, "x"] (1000 obs.);   Bandwidth 'bw' = 0.0029707

x                f(x)
Min.   :0.48629   Min.   : 0.0016
1st Qu.:0.51137   1st Qu.: 0.9135
Median :0.53645   Median : 6.2259
Mean   :0.53645   Mean   : 9.9600
3rd Qu.:0.56152   3rd Qu.:18.3844
Max.   :0.58660   Max.   :29.8335

Marginal density for the F.P.T of Y(t)
| T(S,Y) = inf{t >= 0 : Y(t) >= -1.5 + 3 * t}

Data: out[, "y"] (1000 obs.);   Bandwidth 'bw' = 0.00029278

y                 f(y)
Min.   :0.018428   Min.   :  0.015
1st Qu.:0.021224   1st Qu.:  1.322
Median :0.024021   Median : 37.723
Mean   :0.024021   Mean   : 89.312
3rd Qu.:0.026817   3rd Qu.:167.293
Max.   :0.029614   Max.   :291.043

Marginal density for the F.P.T of Z(t)
| T(S,Z) = inf{t >= 0 : Z(t) <= -1.5 + 3 * t}

Data: out[, "z"] (1000 obs.);   Bandwidth 'bw' = 0.0069309

z                 f(z)
Min.   :0.018428   Min.   :  0.015
1st Qu.:0.021224   1st Qu.:  1.322
Median :0.024021   Median : 37.723
Mean   :0.024021   Mean   : 89.312
3rd Qu.:0.026817   3rd Qu.:167.293
Max.   :0.029614   Max.   :291.043
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")

# 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 (2018). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.3, 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 Probabilities & Statistics, Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail (acguidoum@usthb.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)