# Bridges SDE’s

Consider now a $$d$$-dimensional stochastic process $$X_{t}$$ defined on a probability space $$(\Omega, \mathfrak{F},\mathbb{P})$$. We say that the bridge associated to $$X_{t}$$ conditioned to the event $$\{X_{T}= a\}$$ is the process: $\{\tilde{X}_{t}, t_{0} \leq t \leq T \}=\{X_{t}, t_{0} \leq t \leq T: X_{T}= a \}$ where $$T$$ is a deterministic fixed time and $$a \in \mathbb{R}^d$$ is fixed too.

# The bridgesdekd() functions

The (S3) generic function bridgesdekd() (where k=1,2,3) for simulation of 1,2 and 3-dim bridge stochastic differential equations,Itô or Stratonovich type, with different methods. The main arguments consist:

• The drift and diffusion coefficients as R expressions that depend on the state variable x (y and z) and time variable t.
• The number of simulation steps N.
• The number of the solution trajectories to be simulated by M (default: M=1).
• Initial value x0 at initial time t0.
• Terminal value y final time T
• The integration step size Dt (default: Dt=(T-t0)/N).
• The choice of process types by the argument type="ito" for Ito or type="str" for Stratonovich (by default type="ito").
• The numerical method to be used by method (default method="euler").

By Monte-Carlo simulations, the following statistical measures (S3 method) for class bridgesdekd() (where k=1,2,3) can be approximated for the process at any time $$t \in [t_{0},T]$$ (default: at=(T-t0)/2):

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

We can just make use of the rsdekd() function (where k=1,2,3) to build our random number for class bridgesdekd() (where k=1,2,3) at any time $$t \in [t_{0},T]$$. the main arguments consist:

• object an object inheriting from class bridgesdekd() (where k=1,2,3).
• at time between $$s=t0$$ and $$t=T$$.

The function dsde() (where k=1,2,3) approximate transition density for class bridgesdekd() (where k=1,2,3), the main arguments consist:

• object an object inheriting from class bridgesdekd() (where k=1,2,3).
• at time between $$s=t0$$ and $$t=T$$.
• pdf probability density function Joint or Marginal.

The following we explain how to use this functions.

# bridgesde1d()

Assume that we want to describe the following bridge sde in Itô form: $$$\label{eq0166} dX_t = \frac{1-X_t}{1-t} dt + X_t dW_{t},\quad X_{t_{0}}=3 \quad\text{and}\quad X_{T}=1$$$

We simulate a flow of $$1000$$ trajectories, with integration step size $$\Delta t = 0.001$$, and $$x_0 = 3$$ at time $$t_0 = 0$$, $$y = 1$$ at terminal time $$T=1$$.

R> f <- expression((1-x)/(1-t))
R> g <- expression(x)
R> mod <- bridgesde1d(drift=f,diffusion=g,x0=3,y=1,M=1000,method="milstein")
R> mod
Itô Bridge Sde 1D:
| dX(t) = (1 - X(t))/(1 - t) * dt + X(t) * dW(t)
Method:
| First-order Milstein scheme
Summary:
| Size of process   | N = 1001.
| Crossing realized | C = 969 among 1000.
| Initial value     | x0 = 3.
| Ending value      | y = 1.
| Time of process   | t in [0,1].
| Discretization    | Dt = 0.001.
R> summary(mod) ## default: summary at time = (T-t0)/2

Monte-Carlo Statistics for X(t) at time t = 0.5
| Crossing realized 969 among 1000

Mean                1.99633
Variance            1.68234
Median              1.63854
Mode                1.20091
First quartile      1.12992
Third quartile      2.47647
Minimum             0.28446
Maximum            11.31831
Skewness            2.16858
Kurtosis           10.50985
Coef-variation      0.64972
3th-order moment    4.73203
4th-order moment   29.74570
5th-order moment  195.18963
6th-order moment 1466.57231

In Figure 1, we present the flow of trajectories, the mean path (red lines) of solution of $$X_{t}|X_{0}=3,X_{T}=1$$:

R> plot(mod,ylab=expression(X[t]))
R> lines(time(mod),apply(mod$X,1,mean),col=2,lwd=2) R> legend("topleft","mean path",inset = .01,col=2,lwd=2,cex=0.8,bty="n") Figure 2, show approximation results for $$m(t)=\text{E}(X_{t}|X_{0}=3,X_{T}=1)$$ and $$S(t)=\text{V}(X_{t}|X_{0}=3,X_{T}=1)$$: R> m <- apply(mod$X,1,mean)
R> S  <- apply(mod$X,1,var) R> out <- data.frame(m,S) R> matplot(time(mod), out, type = "l", xlab = "time", ylab = "", col=2:3,lwd=2,lty=2:3,las=1) R> legend("topright",c(expression(m(t),S(t))),col=2:3,lty=2:3,lwd=2,bty="n") The following statistical measures (S3 method) for class bridgesde1d() can be approximated for the $$X_{t}|X_{0}=3,X_{T}=1$$ process at any time $$t$$, for example at=0.55: R> s = 0.55 R> mean(mod, at = s) [1] 1.9133 R> moment(mod, at = s , center = TRUE , order = 2) ## variance [1] 1.8066 R> Median(mod, at = s) [1] 1.574 R> Mode(mod, at = s) [1] 1.0493 R> quantile(mod , at = s)  0% 25% 50% 75% 100% 0.38514 1.03459 1.57398 2.32453 17.82485  R> kurtosis(mod , at = s) [1] 27.906 R> skewness(mod , at = s) [1] 3.398 R> cv(mod , at = s ) [1] 0.70287 R> min(mod , at = s) [1] 0.38514 R> max(mod , at = s) [1] 17.825 R> moment(mod, at = s , center= TRUE , order = 4) [1] 91.267 R> moment(mod, at = s , center= FALSE , order = 4) [1] 207.59 The result summaries of the $$X_{t}|X_{0}=3,X_{T}=1$$ process at time $$t=0.55$$: R> summary(mod, at = 0.55)  Monte-Carlo Statistics for X(t) at time t = 0.55 | Crossing realized 969 among 1000 Mean 1.91328 Variance 1.80844 Median 1.57398 Mode 1.04934 First quartile 1.03459 Third quartile 2.32453 Minimum 0.38514 Maximum 17.82485 Skewness 3.39796 Kurtosis 27.90642 Coef-variation 0.70287 3th-order moment 8.26372 4th-order moment 91.26714 5th-order moment 1212.20003 6th-order moment 17899.61922 Hence we can just make use of the rsde1d() function to build our random number generator for $$X_{t}|X_{0}=3,X_{T}=1$$ at time $$t=0.55$$: R> x <- rsde1d(object = mod, at = s) R> head(x, n = 10)  [1] 1.68801 0.76792 1.77977 0.94930 1.04856 1.65776 3.65794 3.48042 [9] 1.11518 1.46935 R> summary(x)  Min. 1st Qu. Median Mean 3rd Qu. Max. 0.385 1.035 1.574 1.913 2.325 17.825  Display the random number generator for $$X_{t}|X_{0}=3,X_{T}=1$$, see Figure 3: R> plot(time(mod),mod$X[,1],type="l",ylab="X(t)",xlab="time",axes=F,lty=3)
R> points(s,x[1],pch=19,col=2,cex=0.5)
R> lines(c(s,s),c(0,x[1]),lty=2,col=2)
R> lines(c(0,s),c(x[1],x[1]),lty=2,col=2)
R> axis(1, s, bquote(at==.(s)),col=2,col.ticks=2)
R> axis(2, x[1], bquote(X[t==.(s)]),col=2,col.ticks=2)
R> legend('topright',col=2,pch=19,legend=bquote(X[t==.(s)]==.(x[1])),bty = 'n')
R> box()

The function dsde1d() can be used to show the kernel density estimation for $$X_{t}|X_{0}=3,X_{T}=1$$ at time $$t=0.55$$ (hist=TRUE based on truehist() function in MASS package):

R> dens <- dsde1d(mod, at = s)
R> dens

Density of X(t-t0)|X(t0) = 3, X(T) = 1 at time t = 0.55

Data: x (969 obs.); Bandwidth 'bw' = 0.219

x                f(x)
Min.   :-0.2719   Min.   :0.00000
1st Qu.: 4.4166   1st Qu.:0.00000
Median : 9.1050   Median :0.00150
Mean   : 9.1050   Mean   :0.05327
3rd Qu.:13.7934   3rd Qu.:0.03041
Max.   :18.4819   Max.   :0.50401  
R> plot(dens,hist=TRUE) ## histgramme
R> plot(dens,add=TRUE)  ## kernel density

Approximate the transitional densitie of $$X_{t}|X_{0}=3,X_{T}=1$$ at $$t-s = \{0.25,0.75\}$$:

R> plot(dsde1d(mod,at=0.75))
R> legend('topright',col=c('#0000FF4B','#FF00004B'),pch=15,legend=c("t-s=0.25","t-s=0.75"),bty = 'n')

# bridgesde2d()

Assume that we want to describe the following $$2$$-dimensional bridge SDE’s in Stratonovich form:

$$$\label{eq:09} \begin{cases} dX_t = -(1+Y_{t}) X_{t} dt + 0.2 (1-Y_{t})\circ dW_{1,t},\quad X_{t_{0}}=1 \quad\text{and}\quad X_{T}=1\\ dY_t = -(1+X_{t}) Y_{t} dt + 0.1 (1-X_{t}) \circ dW_{2,t},\quad Y_{t_{0}}=-0.5 \quad\text{and}\quad Y_{T}=0.5 \end{cases}$$$

We simulate a flow of $$1000$$ trajectories, with integration step size $$\Delta t = 0.01$$, and using Runge-Kutta method order 1:

R> fx <- expression(-(1+y)*x , -(1+x)*y)
R> gx <- expression(0.2*(1-y),0.1*(1-x))
R> mod2 <- bridgesde2d(drift=fx,diffusion=gx,x0=c(1,-0.5),y=c(1,0.5),Dt=0.01,M=1000,type="str",method="rk1")
R> mod2
Stratonovich Bridge Sde 2D:
| dX(t) = -(1 + Y(t)) * X(t) * dt + 0.2 * (1 - Y(t)) o dW1(t)
| dY(t) = -(1 + X(t)) * Y(t) * dt + 0.1 * (1 - X(t)) o dW2(t)
Method:
| Runge-Kutta method with order 1
Summary:
| Size of process   | N = 1001.
| Crossing realized | C = 1000 among 1000.
| Initial values    | x0 = (1,-0.5).
| Ending values     | y = (1,0.5).
| Time of process   | t in [0,10].
| Discretization    | Dt = 0.01.
R> summary(mod2) ## default: summary at time = (T-t0)/2

Monte-Carlo Statistics for (X(t),Y(t)) at time t = 5
| Crossing realized 1000 among 1000
X         Y
Mean              0.00615  -0.00501
Variance          0.02257   0.00489
Median            0.00622  -0.00631
Mode              0.00224  -0.01022
First quartile   -0.09538  -0.05133
Third quartile    0.10514   0.03854
Minimum          -0.44747  -0.24572
Maximum           0.52315   0.26916
Skewness          0.09847   0.10447
Kurtosis          3.20067   3.28901
Coef-variation   24.44751 -13.97160
3th-order moment  0.00033   0.00004
4th-order moment  0.00163   0.00008
5th-order moment  0.00008   0.00000
6th-order moment  0.00020   0.00000

In Figure 6, we present the flow of trajectories of $$X_{t}|X_{0}=1,X_{T}=1$$ and $$Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$$:

R> plot(mod2,col=c('#FF00004B','#0000FF82'))

Figure 7, show approximation results for $$m_{1}(t)=\text{E}(X_{t}|X_{0}=1,X_{T}=1)$$, $$m_{2}(t)=\text{E}(Y_{t}|Y_{0}=-0.5,Y_{T}=0.5)$$,and $$S_{1}(t)=\text{V}(X_{t}|X_{0}=1,X_{T}=1)$$, $$S_{2}(t)=\text{V}(Y_{t}|Y_{0}=-0.5,Y_{T}=0.5)$$, and $$C_{12}(t)=\text{COV}(X_{t},Y_{t}|X_{0}=1,Y_{0}=-0.5,X_{T}=1,Y_{T}=0.5)$$:

R> m1  <- apply(mod2$X,1,mean) R> m2 <- apply(mod2$Y,1,mean)
R> S1  <- apply(mod2$X,1,var) R> S2 <- apply(mod2$Y,1,var)
R> C12 <- sapply(1:dim(mod2$X)[1],function(i) cov(mod2$X[i,],mod2$Y[i,])) R> out2 <- data.frame(m1,m2,S1,S2,C12) R> matplot(time(mod2), out2, type = "l", xlab = "time", ylab = "", col=2:6,lwd=2,lty=2:6,las=1) R> legend("top",c(expression(m[1](t),m[2](t),S[1](t),S[2](t),C[12](t))),col=2:6,lty=2:6,lwd=2,bty="n") The following statistical measures (S3 method) for class bridgesde2d() can be approximated for the $$X_{t}|X_{0}=1,X_{T}=1$$ and $$Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$$ process at any time $$t$$, for example at=6.75: R> s = 6.75 R> mean(mod2, at = s) [1] 0.042307 0.010663 R> moment(mod2, at = s , center = TRUE , order = 2) ## variance [1] 0.0181854 0.0046176 R> Median(mod2, at = s) [1] 0.043937 0.012868 R> Mode(mod2, at = s) [1] 0.053067 0.020458 R> quantile(mod2 , at = s) $x
0%       25%       50%       75%      100%
-0.350029 -0.050854  0.043937  0.131716  0.473898

$y 0% 25% 50% 75% 100% -0.223561 -0.034450 0.012868 0.054959 0.227391  R> kurtosis(mod2 , at = s) [1] 2.8539 3.2925 R> skewness(mod2 , at = s) [1] 0.040691 -0.056701 R> cv(mod2 , at = s ) [1] 3.1891 6.3757 R> min(mod2 , at = s) [1] -0.35003 -0.22356 R> max(mod2 , at = s) [1] 0.47390 0.22739 R> moment(mod2 , at = s , center= TRUE , order = 4) [1] 0.000945700 0.000070342 R> moment(mod2 , at = s , center= FALSE , order = 4) [1] 0.001161118 0.000072746 The result summaries of the $$X_{t}|X_{0}=1,X_{T}=1$$ and $$Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$$ process at time $$t=6.75$$: R> summary(mod2, at = 6.75)  Monte-Carlo Statistics for (X(t),Y(t)) at time t = 6.75 | Crossing realized 1000 among 1000 X Y Mean 0.04231 0.01066 Variance 0.01820 0.00462 Median 0.04394 0.01287 Mode 0.05307 0.02046 First quartile -0.05085 -0.03445 Third quartile 0.13172 0.05496 Minimum -0.35003 -0.22356 Maximum 0.47390 0.22739 Skewness 0.04069 -0.05670 Kurtosis 2.85390 3.29247 Coef-variation 3.18906 6.37567 3th-order moment 0.00010 -0.00002 4th-order moment 0.00095 0.00007 5th-order moment 0.00002 0.00000 6th-order moment 0.00008 0.00000 Hence we can just make use of the rsde2d() function to build our random number generator for the couple $$X_{t},Y_{t}|X_{0}=1,Y_{0}=-0.5,X_{T}=1,Y_{T}=0.5$$ at time $$t=6.75$$: R> x2 <- rsde2d(object = mod2, at = s) R> head(x2, n = 10)  x y 1 -0.019149 0.0453936 2 0.079367 0.0907089 3 0.080259 -0.0062758 4 -0.010031 0.0250397 5 -0.046938 0.0663944 6 -0.099898 -0.0410747 7 0.269731 -0.0367088 8 0.152037 0.0085386 9 -0.090118 -0.0115128 10 0.139334 0.0325991 R> summary(x2)  x y Min. :-0.3500 Min. :-0.2236 1st Qu.:-0.0508 1st Qu.:-0.0345 Median : 0.0439 Median : 0.0129 Mean : 0.0423 Mean : 0.0107 3rd Qu.: 0.1317 3rd Qu.: 0.0550 Max. : 0.4739 Max. : 0.2274  Display the random number generator for the couple $$X_{t},Y_{t}|X_{0}=1,Y_{0}=-0.5,X_{T}=1,Y_{T}=0.5$$, see Figure 8: R> plot(ts.union(mod2$X[,1],mod2$Y[,1]),col=1:2,lty=3,plot.type="single",type="l",ylab= "",xlab="time",axes=F) R> points(s,x2$x[1],pch=19,col=3,cex=0.8)
R> points(s,x2$y[1],pch=19,col=4,cex=0.8) R> lines(c(s,s),c(-10,x2$x[1]),lty=2,col=6)
R> lines(c(0,s),c(x2$x[1],x2$x[1]),lty=2,col=3)
R> lines(c(0,s),c(x2$y[1],x2$y[1]),lty=2,col=4)
R> axis(1, s, bquote(at==.(s)),col=6,col.ticks=6)
R> axis(2, x2$x[1], bquote(X[t==.(s)]),col=3,col.ticks=3) R> axis(2, x2$y[1], bquote(Y[t==.(s)]),col=4,col.ticks=4)
R> legend('topright',legend=bquote(c(X[t==.(s)]==.(x2$x[1]),Y[t==.(s)]==.(x2$y[1]))),bty = 'n')
R> box()

For each SDE type and for each numerical scheme, the density of $$X_{t}|X_{0}=1,X_{T}=1$$ and $$Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$$ at time $$t=6.75$$ are reported using dsde2d() function, see e.g. Figure 9:

R> denM <- dsde2d(mod2,pdf="M",at =s)
R> denM

Marginal density of X(t-t0)|X(t0) = 1, X(T) = 1 at time t = 6.75

Data: x (1000 obs.);    Bandwidth 'bw' = 0.0305

x                 f(x)
Min.   :-0.44153   Min.   :0.00015
1st Qu.:-0.18980   1st Qu.:0.07301
Median : 0.06193   Median :0.60842
Mean   : 0.06193   Mean   :0.99214
3rd Qu.: 0.31367   3rd Qu.:1.79505
Max.   : 0.56540   Max.   :3.13731

Marginal density of Y(t-t0)|Y(t0) = -0.5, Y(T) = 0.5 at time t = 6.75

Data: y (1000 obs.);    Bandwidth 'bw' = 0.01508

y                  f(y)
Min.   :-0.268813   Min.   :0.0003
1st Qu.:-0.133449   1st Qu.:0.1294
Median : 0.001915   Median :0.8270
Mean   : 0.001915   Mean   :1.8451
3rd Qu.: 0.137279   3rd Qu.:3.6089
Max.   : 0.272643   Max.   :6.1753  
R> plot(denM, main="Marginal Density")

Created using dsde2d() plotted in (x, y)-space with dim = 2. A contour and image plot of density obtained from a realization of the couple $$X_{t},Y_{t}|X_{0}=1,Y_{0}=-0.5,X_{T}=1,Y_{T}=0.5$$ at time t=6.75.

R> denJ <- dsde2d(mod2, pdf="J", n=100,at =s)
R> denJ

Joint density of (X(t-t0),Y(t-t0)|X(t0)=1,Y(t0)=-0.5,X(T)=1,Y(T)=0.5) at time t = 6.75

Data: (x,y) (2 x 1000 obs.);

x                  y                 f(x,y)
Min.   :-0.35003   Min.   :-0.223561   Min.   : 0.0000
1st Qu.:-0.14405   1st Qu.:-0.110823   1st Qu.: 0.1623
Median : 0.06193   Median : 0.001915   Median : 0.7651
Mean   : 0.06193   Mean   : 0.001915   Mean   : 2.6280
3rd Qu.: 0.26792   3rd Qu.: 0.114653   3rd Qu.: 3.4127
Max.   : 0.47390   Max.   : 0.227391   Max.   :20.2096  
R> plot(denJ,display="contour",main="Bivariate Transition Density at time t=6.755")
R> plot(denJ,display="image",main="Bivariate Transition Density at time t=6.755")

A $$3$$D plot of the transition density at $$t=6.75$$ obtained with:

R> plot(denJ,main="Bivariate Transition Density at time t=6.75")

We approximate the bivariate transition density over the set transition horizons $$t\in [1,9]$$ with $$\Delta t = 0.005$$ using the code:

R> for (i in seq(1,9,by=0.005)){
+ plot(dsde2d(mod2, at = i,n=100),display="contour",main=paste0('Transition Density \n t = ',i))
+ }

# bridgesde3d()

Assume that we want to describe the following bridges SDE’s (3D) in Itô form:

$$$\begin{cases} dX_t = -4 (1+X_{t}) Y_{t} dt + 0.2 dW_{1,t},\quad X_{t_{0}}=0 \quad\text{and}\quad X_{T}=0\\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t},\quad Y_{t_{0}}=-1 \quad\text{and}\quad Y_{T}=-2\\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t},\quad Z_{t_{0}}=0.5 \quad\text{and}\quad Z_{T}=0.5 \end{cases}$$$

We simulate a flow of $$1000$$ trajectories, with integration step size $$\Delta t = 0.001$$.

R> fx <- expression(-4*(1+x)*y, 4*(1-y)*x, 4*(1-z)*y)
R> gx <- rep(expression(0.2),3)
R> mod3 <- bridgesde3d(x0=c(0,-1,0.5),y=c(0,-2,0.5),drift=fx,diffusion=gx,M=1000)
R> mod3
Itô Bridge 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)
Method:
| Euler scheme with order 0.5
Summary:
| Size of process   | N = 1001.
| Crossing realized | C = 999 among 1000.
| Initial values    | x0 = (0,-1,0.5).
| Ending values     | y  = (0,-2,0.5).
| Time of process   | t in [0,1].
| Discretization    | Dt = 0.001.
R> summary(mod3) ## default: summary at time = (T-t0)/2

Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 0.5
| Crossing realized 999 among 1000
X       Y        Z
Mean              0.68355 0.50528  0.11996
Variance          0.00959 0.00694  0.01694
Median            0.68231 0.50458  0.12185
Mode              0.67204 0.49738  0.12774
First quartile    0.61941 0.44655  0.03952
Third quartile    0.74895 0.55884  0.20305
Minimum           0.29478 0.26058 -0.39210
Maximum           1.04413 0.73765  0.59136
Skewness         -0.09931 0.00026 -0.19294
Kurtosis          3.56520 2.73325  3.47568
Coef-variation    0.14328 0.16489  1.08502
3th-order moment -0.00009 0.00000 -0.00043
4th-order moment  0.00033 0.00013  0.00100
5th-order moment -0.00001 0.00000 -0.00010
6th-order moment  0.00002 0.00000  0.00011

For plotting (back in time) using the command plot, and plot3D in space the results of the simulation are shown in Figure 12:

R> plot(mod3) ## in time
R> plot3D(mod3,display = "persp",main="3D Bridge SDE's") ## in space 

Figure 13, show approximation results for $$m_{1}(t)=\text{E}(X_{t}|X_{0}=0,X_{T}=0)$$, $$m_{2}(t)=\text{E}(Y_{t}|Y_{0}=-1,Y_{T}=-2)$$, $$m_{3}(t)=\text{E}(Z_{t}|Z_{0}=0.5,Z_{T}=0.5)$$ and $$S_{1}(t)=\text{V}(X_{t}|X_{0}=0,X_{T}=0)$$, $$S_{2}(t)=\text{V}(Y_{t}|Y_{0}=-1,Y_{T}=-2)$$, $$S_{3}(t)=\text{V}(Z_{t}|Z_{0}=0.5,Z_{T}=0.5)$$,

R> m1  <- apply(mod3$X,1,mean) R> m2 <- apply(mod3$Y,1,mean)
R> m3  <- apply(mod3$Z,1,mean) R> S1 <- apply(mod3$X,1,var)
R> S2  <- apply(mod3$Y,1,var) R> S3 <- apply(mod3$Z,1,var)
R> out3 <- data.frame(m1,m2,m3,S1,S2,S3)
R> matplot(time(mod3), out3, type = "l", xlab = "time", ylab = "", col=2:7,lwd=2,lty=2:7,las=1)
R> legend("bottom",c(expression(m[1](t),m[2](t),m[3](t),S[1](t),S[2](t),S[3](t))),col=2:7,lty=2:7,lwd=2,bty="n")

The following statistical measures (S3 method) for class bridgesde3d() can be approximated for the $$X_{t}|X_{0}=0,X_{T}=0$$, $$Y_{t}|Y_{0}=-1,Y_{T}=-2$$ and $$Z_{t}|Z_{0}=0.5,Z_{T}=0.5$$ process at any time $$t$$, for example at=0.75:

R> s = 0.75
R> mean(mod3, at = s)
[1]  1.99336  0.11872 -0.51284
R> moment(mod3, at = s , center = TRUE , order = 2) ## variance
[1] 0.0099619 0.0043353 0.0333037
R> Median(mod3, at = s)
[1]  1.99438  0.11787 -0.50811
R> Mode(mod3, at = s)
[1]  2.01312  0.10984 -0.47086
R> quantile(mod3 , at = s)
$x 0% 25% 50% 75% 100% 1.6693 1.9246 1.9944 2.0636 2.3853$y
0%       25%       50%       75%      100%
-0.096136  0.076739  0.117868  0.165799  0.352688

$z 0% 25% 50% 75% 100% -1.07575 -0.64087 -0.50811 -0.39409 0.16610  R> kurtosis(mod3 , at = s) [1] 3.0995 3.2287 3.1300 R> skewness(mod3 , at = s) [1] 0.038879 -0.036829 -0.024780 R> cv(mod3 , at = s ) [1] 0.050096 0.554881 -0.356025 R> min(mod3 , at = s) [1] 1.669258 -0.096136 -1.075755 R> max(mod3 , at = s) [1] 2.38531 0.35269 0.16610 R> moment(mod3 , at = s , center= TRUE , order = 4) [1] 0.000308211 0.000060805 0.003478563 R> moment(mod3 , at = s , center= FALSE , order = 4) [1] 16.02659619 0.00062109 0.12551519 The result summaries of the $$X_{t}|X_{0}=0,X_{T}=0$$, $$Y_{t}|Y_{0}=-1,Y_{T}=-2$$ and $$Z_{t}|Z_{0}=0.5,Z_{T}=0.5$$ process at time $$t=0.75$$: R> summary(mod3, at = 0.75)  Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 0.75 | Crossing realized 999 among 1000 X Y Z Mean 1.99336 0.11872 -0.51284 Variance 0.00997 0.00434 0.03334 Median 1.99438 0.11787 -0.50811 Mode 2.01312 0.10984 -0.47086 First quartile 1.92464 0.07674 -0.64087 Third quartile 2.06361 0.16580 -0.39409 Minimum 1.66926 -0.09614 -1.07575 Maximum 2.38531 0.35269 0.16610 Skewness 0.03888 -0.03683 -0.02478 Kurtosis 3.09954 3.22872 3.13000 Coef-variation 0.05010 0.55488 -0.35602 3th-order moment 0.00004 -0.00001 -0.00015 4th-order moment 0.00031 0.00006 0.00348 5th-order moment 0.00001 0.00000 0.00006 6th-order moment 0.00002 0.00000 0.00060 Hence we can just make use of the rsde3d() function to build our random number generator for the triplet $$X_{t},Y_{t},Z_{t}|X_{0}=0,Y_{0}=-1,Z_{0}=0.5,X_{T}=0,Y_{T}=-2,Z_{T}=0.5$$ at time $$t=0.75$$: R> x3 <- rsde3d(object = mod3, at = s) R> head(x3, n = 10)  x y z 1 1.8393 0.103766 -0.33912 2 1.9585 0.167583 -0.35105 3 2.0859 0.016888 -0.29588 4 2.1415 0.140960 -0.25574 5 1.7273 0.215792 -0.39200 6 2.0653 0.020544 -0.58251 7 2.1246 0.142620 -0.73805 8 1.9357 0.124305 -0.80494 9 1.9637 0.124545 -0.80851 10 1.9665 0.199791 -0.67566 R> summary(x3)  x y z Min. :1.67 Min. :-0.0961 Min. :-1.076 1st Qu.:1.92 1st Qu.: 0.0767 1st Qu.:-0.641 Median :1.99 Median : 0.1179 Median :-0.508 Mean :1.99 Mean : 0.1187 Mean :-0.513 3rd Qu.:2.06 3rd Qu.: 0.1658 3rd Qu.:-0.394 Max. :2.39 Max. : 0.3527 Max. : 0.166  Display the random number generator for triplet $$X_{t},Y_{t},Z_{t}|X_{0}=0,Y_{0}=-1,Z_{0}=0.5,X_{T}=0,Y_{T}=-2,Z_{T}=0.5$$ at time $$t=0.75$$: , see Figure 14: R> plot(ts.union(mod3$X[,1],mod3$Y[,1],mod3$Z[,1]),col=1:3,lty=3,plot.type="single",type="l",ylab= "",xlab="time",axes=F)
R> points(s,x3$x[1],pch=19,col=4,cex=0.8) R> points(s,x3$y[1],pch=19,col=5,cex=0.8)
R> points(s,x3$z[1],pch=19,col=6,cex=0.8) R> lines(c(s,s),c(-10,x3$x[1]),lty=2,col=7)
R> lines(c(0,s),c(x3$x[1],x3$x[1]),lty=2,col=4)
R> lines(c(0,s),c(x3$y[1],x3$y[1]),lty=2,col=5)
R> lines(c(0,s),c(x3$z[1],x3$z[1]),lty=2,col=6)
R> axis(1, s, bquote(at==.(s)),col=7,col.ticks=7)
R> axis(2, x3$x[1], bquote(X[t==.(s)]),col=4,col.ticks=4) R> axis(2, x3$y[1], bquote(Y[t==.(s)]),col=5,col.ticks=5)
R> axis(2, x3$z[1], bquote(Z[t==.(s)]),col=6,col.ticks=6) R> legend("bottomleft",legend=bquote(c(X[t==.(s)]==.(x3$x[1]),Y[t==.(s)]==.(x3$y[1]),Z[t==.(s)]==.(x3$z[1]))),bty = 'n',cex=0.75)
R> box()

For each SDE type and for each numerical scheme, the density of $$X_{t}|X_{0}=0,X_{T}=0$$, $$Y_{t}|Y_{0}=-1,Y_{T}=-2$$ and $$Z_{t}|Z_{0}=0.5,Z_{T}=0.5$$ process at time $$t=0.75$$ are reported using dsde3d() function, see e.g. Figure 15:

R> denM <- dsde3d(mod3,pdf="M",at =s)
R> denM

Marginal density of X(t-t0)|X(t0) = 0, X(T) = 0 at time t = 0.75

Data: x (999 obs.); Bandwidth 'bw' = 0.02258

x               f(x)
Min.   :1.6015   Min.   :0.0002
1st Qu.:1.8144   1st Qu.:0.0291
Median :2.0273   Median :0.3420
Mean   :2.0273   Mean   :1.1732
3rd Qu.:2.2402   3rd Qu.:2.4774
Max.   :2.4531   Max.   :3.7658

Marginal density of Y(t-t0)|Y(t0) = -1, Y(T) = -2 at time t = 0.75

Data: y (999 obs.); Bandwidth 'bw' = 0.0149

y                 f(y)
Min.   :-0.14082   Min.   :0.0003
1st Qu.:-0.00627   1st Qu.:0.1037
Median : 0.12828   Median :0.7921
Mean   : 0.12828   Mean   :1.8562
3rd Qu.: 0.26283   3rd Qu.:3.5932
Max.   : 0.39737   Max.   :6.3759

Marginal density of Z(t-t0)|Z(t0) = 0.5, Z(T) = 0.5 at time t = 0.75

Data: z (999 obs.); Bandwidth 'bw' = 0.04129

z                 f(z)
Min.   :-1.19961   Min.   :0.00011
1st Qu.:-0.82722   1st Qu.:0.02757
Median :-0.45483   Median :0.32151
Mean   :-0.45483   Mean   :0.67068
3rd Qu.:-0.08244   3rd Qu.:1.31164
Max.   : 0.28995   Max.   :2.13085  
R> plot(denM, main="Marginal Density")

For an approximate joint density for triplet $$X_{t},Y_{t},Z_{t}|X_{0}=0,Y_{0}=-1,Z_{0}=0.5,X_{T}=0,Y_{T}=-2,Z_{T}=0.5$$ at time $$t=0.75$$ (for more details, see package sm or ks.)

R> denJ <- dsde3d(mod3,pdf="J",at=0.75)
R> plot(denJ,display="rgl")