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()

We simulate a flow of \(5000\) 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=5000,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 = 4867 among 5000.
    | 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 4867 among 5000
                            
Mean                 1.98885
Variance             1.77324
Median               1.64320
Mode                 1.17678
First quartile       1.13710
Third quartile       2.42955
Minimum              0.27403
Maximum             18.37158
Skewness             3.00933
Kurtosis            21.78707
Coef-variation       0.66955
3th-order moment     7.10590
4th-order moment    68.50650
5th-order moment   807.96130
6th-order moment 11063.58311

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")

Bridge sde 1D

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] 1.8894

R> moment(mod, at = s , center = TRUE , order = 2) ## variance

[1] 1.637

R> Median(mod, at = s)

[1] 1.5516

R> Mode(mod, at = s)

[1] 1.1551

R> quantile(mod , at = s)

      0%      25%      50%      75%     100% 
 0.26933  1.07407  1.55160  2.30822 14.93212 

R> kurtosis(mod , at = s)

[1] 16.668

R> skewness(mod , at = s)

[1] 2.7531

R> cv(mod , at = s )

[1] 0.67724

R> min(mod , at = s)

[1] 0.26933

R> max(mod , at = s)

[1] 14.932

R> moment(mod, at = s , center= TRUE , order = 4)

[1] 44.684

R> moment(mod, at = s , center= FALSE , order = 4)

[1] 136.08

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 4867 among 5000
                           
Mean                1.88942
Variance            1.63733
Median              1.55160
Mode                1.15509
First quartile      1.07407
Third quartile      2.30822
Minimum             0.26933
Maximum            14.93212
Skewness            2.75308
Kurtosis           16.66763
Coef-variation      0.67724
3th-order moment    5.76799
4th-order moment   44.68364
5th-order moment  395.80345
6th-order moment 4029.66373

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] 1.3899 1.9269 1.1417 2.5318 1.8477 1.9571 1.1669 1.1350 2.3831
[10] 2.3246

R> summary(x)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.269   1.074   1.552   1.889   2.308  14.932 

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 (4867 obs.);    Bandwidth 'bw' = 0.1517

       x                f(x)        
 Min.   :-0.1858   Min.   :0.00000  
 1st Qu.: 3.7074   1st Qu.:0.00050  
 Median : 7.6007   Median :0.00284  
 Mean   : 7.6007   Mean   :0.06415  
 3rd Qu.:11.4940   3rd Qu.:0.04416  
 Max.   :15.3873   Max.   :0.54378  

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

Transitional densitie at time \(t-s = 0.25,0.75\)

Return to bridgesde1d()

bridgesde2d()

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

We simulate a flow of \(5000\) 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=5000,type="str",method="rk1")
R> mod2

Stratonovich Bridge Sde 2D:
    | dX(t) = expression(-(1 + y) * x) * dt + expression(0.2 * (1 - y)) o dW1(t)
    | dY(t) = expression(-(1 + x) * y) * dt + expression(0.1 * (1 - x)) o dW2(t)
Method:
    | Runge-Kutta method with order 1
Summary:
    | Size of process   | N = 1001.
    | Crossing realized | C = 4998 among 5000.
    | 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 4998 among 5000
                        X         Y
Mean              0.00831  -0.00254
Variance          0.01958   0.00512
Median            0.00589  -0.00209
Mode              0.00908  -0.00237
First quartile   -0.08401  -0.04804
Third quartile    0.10458   0.04361
Minimum          -0.51235  -0.28787
Maximum           0.63896   0.28438
Skewness          0.02005  -0.03346
Kurtosis          3.05726   3.52267
Coef-variation   16.84698 -28.14040
3th-order moment  0.00005  -0.00001
4th-order moment  0.00117   0.00009
5th-order moment  0.00001   0.00000
6th-order moment  0.00012   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'))

Bridge sde 2D

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] 0.0262114 0.0081418

R> moment(mod2, at = s , center = TRUE , order = 2) ## variance

[1] 0.0189430 0.0044029

R> Median(mod2, at = s)

[1] 0.027607 0.007586

R> Mode(mod2, at = s)

[1] 0.029730 0.016605

R> quantile(mod2 , at = s)

$x
       0%       25%       50%       75%      100% 
-0.526017 -0.065815  0.027607  0.119805  0.514165 

$y
       0%       25%       50%       75%      100% 
-0.226050 -0.035261  0.007586  0.050428  0.269088 

R> kurtosis(mod2 , at = s)

[1] 3.0379 3.2958

R> skewness(mod2 , at = s)

[1] -0.029599  0.077942

R> cv(mod2 , at = s )

[1] 5.2514 8.1506

R> min(mod2 , at = s)

[1] -0.52602 -0.22605

R> max(mod2 , at = s)

[1] 0.51417 0.26909

R> moment(mod2 , at = s , center= TRUE , order = 4)

[1] 0.001090540 0.000063916

R> moment(mod2 , at = s , center= FALSE , order = 4)

[1] 0.001161006 0.000066414

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 4998 among 5000
                        X        Y
Mean              0.02621  0.00814
Variance          0.01895  0.00440
Median            0.02761  0.00759
Mode              0.02973  0.01660
First quartile   -0.06581 -0.03526
Third quartile    0.11981  0.05043
Minimum          -0.52602 -0.22605
Maximum           0.51417  0.26909
Skewness         -0.02960  0.07794
Kurtosis          3.03789  3.29578
Coef-variation    5.25142  8.15062
3th-order moment -0.00008  0.00002
4th-order moment  0.00109  0.00006
5th-order moment -0.00002  0.00000
6th-order moment  0.00010  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   0.043042  0.120642
2   0.125878 -0.035650
3  -0.077395 -0.029731
4   0.029373  0.034140
5   0.234090 -0.030233
6  -0.058481 -0.038801
7   0.082880  0.062691
8   0.161389 -0.033351
9   0.161178  0.023149
10  0.186901 -0.084523

R> summary(x2)

       x                 y           
 Min.   :-0.5260   Min.   :-0.22605  
 1st Qu.:-0.0658   1st Qu.:-0.03526  
 Median : 0.0276   Median : 0.00759  
 Mean   : 0.0262   Mean   : 0.00814  
 3rd Qu.: 0.1198   3rd Qu.: 0.05043  
 Max.   : 0.5142   Max.   : 0.26909  

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 (4998 obs.);    Bandwidth 'bw' = 0.02256

       x                 f(x)        
 Min.   :-0.59368   Min.   :0.00004  
 1st Qu.:-0.29980   1st Qu.:0.02190  
 Median :-0.00593   Median :0.32544  
 Mean   :-0.00593   Mean   :0.84986  
 3rd Qu.: 0.28795   3rd Qu.:1.66051  
 Max.   : 0.58183   Max.   :2.98021  

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

Data: y (4998 obs.);    Bandwidth 'bw' = 0.01048

       y                  f(y)       
 Min.   :-0.257486   Min.   :0.0001  
 1st Qu.:-0.117984   1st Qu.:0.0658  
 Median : 0.021519   Median :0.6257  
 Mean   : 0.021519   Mean   :1.7903  
 3rd Qu.: 0.161021   3rd Qu.:3.2947  
 Max.   : 0.300524   Max.   :6.2143  

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 4998 obs.);

       x                  y                 f(x,y)       
 Min.   :-0.52602   Min.   :-0.226050   Min.   : 0.0000  
 1st Qu.:-0.26597   1st Qu.:-0.102266   1st Qu.: 0.0187  
 Median :-0.00593   Median : 0.021519   Median : 0.2126  
 Mean   :-0.00593   Mean   : 0.021519   Mean   : 1.9019  
 3rd Qu.: 0.25412   3rd Qu.: 0.145304   3rd Qu.: 1.8044  
 Max.   : 0.51417   Max.   : 0.269088   Max.   :18.1845  

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))
+ }

Return to bridgesde2d()

bridgesde3d()

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

We simulate a flow of \(5000\) 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=5000)
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 = 4999 among 5000.
    | 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 4999 among 5000
                        X       Y        Z
Mean              0.68653 0.50785  0.12107
Variance          0.01029 0.00698  0.01816
Median            0.68864 0.50841  0.12488
Mode              0.69242 0.50334  0.12559
First quartile    0.61908 0.45142  0.03415
Third quartile    0.75518 0.56523  0.20786
Minimum           0.25672 0.16737 -0.52257
Maximum           1.04315 0.86211  0.62638
Skewness         -0.15098 0.01036 -0.11041
Kurtosis          3.38517 3.11495  3.39254
Coef-variation    0.14778 0.16452  1.11314
3th-order moment -0.00016 0.00001 -0.00027
4th-order moment  0.00036 0.00015  0.00112
5th-order moment -0.00002 0.00000 -0.00006
6th-order moment  0.00002 0.00001  0.00013

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 

Bridge sde 3D

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]  1.99799  0.12365 -0.50026

R> moment(mod3, at = s , center = TRUE , order = 2) ## variance

[1] 0.0111796 0.0046333 0.0303640

R> Median(mod3, at = s)

[1]  1.99805  0.12401 -0.49570

R> Mode(mod3, at = s)

[1]  1.99994  0.11729 -0.48679

R> quantile(mod3 , at = s)

$x
    0%    25%    50%    75%   100% 
1.5794 1.9274 1.9981 2.0685 2.4690 

$y
       0%       25%       50%       75%      100% 
-0.119346  0.077376  0.124011  0.170480  0.342551 

$z
       0%       25%       50%       75%      100% 
-1.146458 -0.616306 -0.495701 -0.380309  0.023288 

R> kurtosis(mod3 , at = s)

[1] 3.1877 2.9253 3.0427

R> skewness(mod3 , at = s)

[1]  0.024288 -0.067700 -0.159695

R> cv(mod3 , at = s )

[1]  0.052925  0.550527 -0.348362

R> min(mod3 , at = s)

[1]  1.57943 -0.11935 -1.14646

R> max(mod3 , at = s)

[1] 2.469047 0.342551 0.023288

R> moment(mod3 , at = s , center= TRUE , order = 4)

[1] 0.000398562 0.000062823 0.002806433

R> moment(mod3 , at = s , center= FALSE , order = 4)

[1] 16.20417730  0.00071111  0.11271846

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 4999 among 5000
                       X        Y        Z
Mean             1.99799  0.12365 -0.50026
Variance         0.01118  0.00463  0.03037
Median           1.99805  0.12401 -0.49570
Mode             1.99994  0.11729 -0.48679
First quartile   1.92736  0.07738 -0.61631
Third quartile   2.06850  0.17048 -0.38031
Minimum          1.57943 -0.11935 -1.14646
Maximum          2.46905  0.34255  0.02329
Skewness         0.02429 -0.06770 -0.15969
Kurtosis         3.18767  2.92532  3.04273
Coef-variation   0.05293  0.55053 -0.34836
3th-order moment 0.00003 -0.00002 -0.00085
4th-order moment 0.00040  0.00006  0.00281
5th-order moment 0.00001  0.00000 -0.00028
6th-order moment 0.00003  0.00000  0.00044

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  1.8355  0.131688 -0.69490
2  2.0015  0.089879 -0.93915
3  1.8763  0.100016 -0.21363
4  1.9862  0.096988 -0.33648
5  1.9481  0.088273 -0.51417
6  1.9816  0.091262 -0.55365
7  1.8656 -0.015809 -0.19584
8  1.9345  0.061476 -0.21002
9  2.0464  0.180440 -0.47656
10 1.8703  0.150539 -0.68193

R> summary(x3)

       x              y                 z          
 Min.   :1.58   Min.   :-0.1193   Min.   :-1.1465  
 1st Qu.:1.93   1st Qu.: 0.0774   1st Qu.:-0.6163  
 Median :2.00   Median : 0.1240   Median :-0.4957  
 Mean   :2.00   Mean   : 0.1236   Mean   :-0.5003  
 3rd Qu.:2.07   3rd Qu.: 0.1705   3rd Qu.:-0.3803  
 Max.   :2.47   Max.   : 0.3426   Max.   : 0.0233  

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 (4999 obs.);    Bandwidth 'bw' = 0.01726

       x               f(x)       
 Min.   :1.5276   Min.   :0.0001  
 1st Qu.:1.7759   1st Qu.:0.0153  
 Median :2.0242   Median :0.2646  
 Mean   :2.0242   Mean   :1.0059  
 3rd Qu.:2.2725   3rd Qu.:1.8871  
 Max.   :2.5208   Max.   :3.8189  

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

Data: y (4999 obs.);    Bandwidth 'bw' = 0.01115

       y                 f(y)       
 Min.   :-0.15281   Min.   :0.0001  
 1st Qu.:-0.02060   1st Qu.:0.0795  
 Median : 0.11160   Median :0.8635  
 Mean   : 0.11160   Mean   :1.8891  
 3rd Qu.: 0.24381   3rd Qu.:3.7539  
 Max.   : 0.37601   Max.   :5.6220  

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

Data: z (4999 obs.);    Bandwidth 'bw' = 0.02856

       z                 f(z)        
 Min.   :-1.23212   Min.   :0.00005  
 1st Qu.:-0.89685   1st Qu.:0.04242  
 Median :-0.56158   Median :0.38913  
 Mean   :-0.56158   Mean   :0.74494  
 3rd Qu.:-0.22632   3rd Qu.:1.44102  
 Max.   : 0.10895   Max.   :2.26437  

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")

Return to bridgesde3d()

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. fitsde() - Parametric Estimation of 1-D Stochastic Differential Equation.

References

1. Bladt, M. and Sorensen, M. (2007). Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Working Paper, University of Copenhagen.

2. Guidoum AC, Boukhetala K (2017). Sim.DiffProc: Simulation of Diffusion Processes. R package version 3.8, URL https://cran.r-project.org/package=Sim.DiffProc.