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.
bridgesdekd()
functionsThe (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:
drift
and diffusion
coefficients as R expressions that depend on the state variable x
(y
and z
) and time variable t
.N
.M
(default: M=1
).x0
at initial time t0
.y
final time T
Dt
(default: Dt=(T-t0)/N
).type="ito"
for Ito or type="str"
for Stratonovich (by default type="ito"
).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
):
mean
.moment
with order=2
and center=TRUE
.Median
.Mode
.quantile
.min
and max
.skewness
and kurtosis
.cv
.moment
.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 \(1000\) 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")
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] 1.9133
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> 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 \(t-s = 0.25,0.75\)
bridgesde2d()
Assume that we want to describe the following \(2\)-dimensional bridge SDE’s in Stratonovich form:
\[\begin{equation}\label{eq:09} \begin{cases} dX_t = -(1+Y_{t}) X_{t} dt + 0.2 (1-Y_{t})\circ dW_{1,t},\quad X_{t_{0}}=1 \quad\text{and}\quad X_{T}=1\\ dY_t = -(1+X_{t}) Y_{t} dt + 0.1 (1-X_{t}) \circ dW_{2,t},\quad Y_{t_{0}}=-0.5 \quad\text{and}\quad Y_{T}=0.5 \end{cases} \end{equation}\]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'))
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] 0.042307 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:
\[\begin{equation} \begin{cases} dX_t = -4 (1+X_{t}) Y_{t} dt + 0.2 dW_{1,t},\quad X_{t_{0}}=0 \quad\text{and}\quad X_{T}=0\\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t},\quad Y_{t_{0}}=-1 \quad\text{and}\quad Y_{T}=-2\\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t},\quad Z_{t_{0}}=0.5 \quad\text{and}\quad Z_{T}=0.5 \end{cases} \end{equation}\]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
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] 1.99336 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")
snssdekd()
& dsdekd()
& rsdekd()
- Monte-Carlo Simulation and Analysis of Stochastic Differential Equations.bridgesdekd()
& dsdekd()
& rsdekd()
- Constructs and Analysis of Bridges Stochastic Differential Equations.fptsdekd()
& dfptsdekd()
- Monte-Carlo Simulation and Kernel Density Estimation of First passage time.MCM.sde()
& MEM.sde()
- Parallel Monte-Carlo and Moment Equations for SDEs.TEX.sde()
- Converting Sim.DiffProc Objects to LaTeX.fitsde()
- Parametric Estimation of 1-D Stochastic Differential Equation.Bladt, M. and Sorensen, M. (2007). Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Working Paper, University of Copenhagen.
Guidoum AC, Boukhetala K (2018). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.2, URL https://cran.r-project.org/package=Sim.DiffProc.
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)↩
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)↩