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 \(5000\) 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 = 4887 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 4887 among 5000
Mean 1.99047
Variance 1.68836
Median 1.66638
Mode 1.20559
First quartile 1.14759
Third quartile 2.43640
Minimum 0.26261
Maximum 21.95516
Skewness 2.89675
Kurtosis 23.22497
Coef-variation 0.65279
3th-order moment 6.35488
4th-order moment 66.20395
5th-order moment 927.76759
6th-order moment 15762.45036
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] 1.8889
R> moment(mod, at = s , center = TRUE , order = 2) ## variance
[1] 1.6047
R> Median(mod, at = s)
[1] 1.5553
R> Mode(mod, at = s)
[1] 1.1819
R> quantile(mod , at = s)
0% 25% 50% 75% 100%
0.24517 1.08343 1.55526 2.28799 19.51415
R> kurtosis(mod , at = s)
[1] 22.48
R> skewness(mod , at = s)
[1] 3.0465
R> cv(mod , at = s )
[1] 0.6707
R> min(mod , at = s)
[1] 0.24517
R> max(mod , at = s)
[1] 19.514
R> moment(mod, at = s , center= TRUE , order = 4)
[1] 57.911
R> moment(mod, at = s , center= FALSE , order = 4)
[1] 151.8
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 4887 among 5000
Mean 1.88892
Variance 1.60504
Median 1.55526
Mode 1.18187
First quartile 1.08343
Third quartile 2.28799
Minimum 0.24517
Maximum 19.51415
Skewness 3.04647
Kurtosis 22.47991
Coef-variation 0.67070
3th-order moment 6.19477
4th-order moment 57.91149
5th-order moment 681.47492
6th-order moment 9565.26028
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] 1.21014 3.09950 2.17252 1.03385 2.81539 0.69328 0.97909 2.76190
[9] 1.09426 1.34471
R> summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.245 1.083 1.555 1.889 2.288 19.514
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 (4887 obs.); Bandwidth 'bw' = 0.148
x f(x)
Min. :-0.1987 Min. :0.00000
1st Qu.: 4.8405 1st Qu.:0.00001
Median : 9.8797 Median :0.00084
Mean : 9.8797 Mean :0.04956
3rd Qu.:14.9188 3rd Qu.:0.01847
Max. :19.9580 Max. :0.53919
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 \(5000\) 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) = -(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 = 4996 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 4996 among 5000
X Y
Mean 0.00652 0.00048
Variance 0.02041 0.00519
Median 0.00655 0.00111
Mode 0.01153 0.00584
First quartile -0.08829 -0.04528
Third quartile 0.10336 0.04699
Minimum -0.54448 -0.41082
Maximum 0.55540 0.31805
Skewness -0.00272 -0.06595
Kurtosis 3.00850 3.57324
Coef-variation 21.92481 149.37067
3th-order moment -0.00001 -0.00002
4th-order moment 0.00125 0.00010
5th-order moment 0.00000 0.00000
6th-order moment 0.00013 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] 0.0276908 0.0084453
R> moment(mod2, at = s , center = TRUE , order = 2) ## variance
[1] 0.0195837 0.0045949
R> Median(mod2, at = s)
[1] 0.0274272 0.0080058
R> Mode(mod2, at = s)
[1] 0.0058589 0.0043362
R> quantile(mod2 , at = s)
$x
0% 25% 50% 75% 100%
-0.569030 -0.067677 0.027427 0.122797 0.529930
$y
0% 25% 50% 75% 100%
-0.2279534 -0.0370276 0.0080058 0.0524257 0.3335869
R> kurtosis(mod2 , at = s)
[1] 2.9910 3.3644
R> skewness(mod2 , at = s)
[1] 0.016196 0.083420
R> cv(mod2 , at = s )
[1] 5.0542 8.0273
R> min(mod2 , at = s)
[1] -0.56903 -0.22795
R> max(mod2 , at = s)
[1] 0.52993 0.33359
R> moment(mod2 , at = s , center= TRUE , order = 4)
[1] 0.001147577 0.000071063
R> moment(mod2 , at = s , center= FALSE , order = 4)
[1] 0.001243181 0.000073912
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 4996 among 5000
X Y
Mean 0.02769 0.00845
Variance 0.01959 0.00460
Median 0.02743 0.00801
Mode 0.00586 0.00434
First quartile -0.06768 -0.03703
Third quartile 0.12280 0.05243
Minimum -0.56903 -0.22795
Maximum 0.52993 0.33359
Skewness 0.01620 0.08342
Kurtosis 2.99101 3.36443
Coef-variation 5.05423 8.02727
3th-order moment 0.00004 0.00003
4th-order moment 0.00115 0.00007
5th-order moment 0.00000 0.00000
6th-order moment 0.00011 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 0.052737 -0.051129
2 -0.059465 0.027146
3 -0.032389 0.099025
4 0.154529 -0.139782
5 0.174460 0.055512
6 0.126719 -0.074817
7 0.175904 -0.020664
8 -0.069990 -0.034521
9 -0.197377 -0.084844
10 -0.087435 0.076491
R> summary(x2)
x y
Min. :-0.5690 Min. :-0.22795
1st Qu.:-0.0677 1st Qu.:-0.03703
Median : 0.0274 Median : 0.00801
Mean : 0.0277 Mean : 0.00845
3rd Qu.: 0.1228 3rd Qu.: 0.05243
Max. : 0.5299 Max. : 0.33359
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 (4996 obs.); Bandwidth 'bw' = 0.02294
x f(x)
Min. :-0.63784 Min. :0.00004
1st Qu.:-0.32869 1st Qu.:0.01019
Median :-0.01955 Median :0.25508
Mean :-0.01955 Mean :0.80790
3rd Qu.: 0.28959 3rd Qu.:1.56059
Max. : 0.59874 Max. :2.75304
Marginal density of Y(t-t0)|Y(t0) = -0.5, Y(T) = 0.5 at time t = 6.75
Data: y (4996 obs.); Bandwidth 'bw' = 0.01094
y f(y)
Min. :-0.26077 Min. :0.0001
1st Qu.:-0.10398 1st Qu.:0.0396
Median : 0.05282 Median :0.4317
Mean : 0.05282 Mean :1.5929
3rd Qu.: 0.20961 3rd Qu.:2.9777
Max. : 0.36641 Max. :5.8921
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 4996 obs.);
x y f(x,y)
Min. :-0.56903 Min. :-0.22795 Min. : 0.0000
1st Qu.:-0.29429 1st Qu.:-0.08757 1st Qu.: 0.0006
Median :-0.01955 Median : 0.05282 Median : 0.1155
Mean :-0.01955 Mean : 0.05282 Mean : 1.5873
3rd Qu.: 0.25519 3rd Qu.: 0.19320 3rd Qu.: 1.2435
Max. : 0.52993 Max. : 0.33359 Max. :17.5880
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 \(5000\) 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 = 4997 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 4997 among 5000
X Y Z
Mean 0.68476 0.50814 0.13153
Variance 0.01096 0.00699 0.01858
Median 0.68537 0.51003 0.13334
Mode 0.68359 0.51193 0.15272
First quartile 0.61898 0.45329 0.04146
Third quartile 0.75491 0.56244 0.21918
Minimum 0.20120 0.19913 -0.38649
Maximum 1.07982 0.87981 0.62700
Skewness -0.22328 -0.04394 0.03937
Kurtosis 3.52565 3.22149 3.17931
Coef-variation 0.15288 0.16455 1.03628
3th-order moment -0.00026 -0.00003 0.00010
4th-order moment 0.00042 0.00016 0.00110
5th-order moment -0.00004 0.00000 0.00002
6th-order moment 0.00003 0.00001 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] 1.99606 0.12313 -0.49911
R> moment(mod3, at = s , center = TRUE , order = 2) ## variance
[1] 0.0114860 0.0046408 0.0313156
R> Median(mod3, at = s)
[1] 1.99583 0.12392 -0.49743
R> Mode(mod3, at = s)
[1] 2.01020 0.13022 -0.50230
R> quantile(mod3 , at = s)
$x
0% 25% 50% 75% 100%
1.6492 1.9231 1.9958 2.0686 2.3415
$y
0% 25% 50% 75% 100%
-0.146289 0.078409 0.123924 0.168586 0.369169
$z
0% 25% 50% 75% 100%
-1.16145 -0.61586 -0.49743 -0.37919 0.14239
R> kurtosis(mod3 , at = s)
[1] 2.8462 3.1230 2.9764
R> skewness(mod3 , at = s)
[1] -0.00007621 -0.08858797 -0.05841234
R> cv(mod3 , at = s )
[1] 0.053698 0.553330 -0.354594
R> min(mod3 , at = s)
[1] 1.64921 -0.14629 -1.16145
R> max(mod3 , at = s)
[1] 2.34153 0.36917 0.14239
R> moment(mod3 , at = s , center= TRUE , order = 4)
[1] 0.000375651 0.000067288 0.002920024
R> moment(mod3 , at = s , center= FALSE , order = 4)
[1] 16.14921098 0.00070548 0.11242655
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 4997 among 5000
X Y Z
Mean 1.99606 0.12313 -0.49911
Variance 0.01149 0.00464 0.03132
Median 1.99583 0.12392 -0.49743
Mode 2.01020 0.13022 -0.50230
First quartile 1.92314 0.07841 -0.61586
Third quartile 2.06861 0.16859 -0.37919
Minimum 1.64921 -0.14629 -1.16145
Maximum 2.34153 0.36917 0.14239
Skewness -0.00008 -0.08859 -0.05841
Kurtosis 2.84623 3.12302 2.97639
Coef-variation 0.05370 0.55333 -0.35459
3th-order moment 0.00000 -0.00003 -0.00032
4th-order moment 0.00038 0.00007 0.00292
5th-order moment 0.00000 0.00000 -0.00009
6th-order moment 0.00002 0.00000 0.00043
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 1.9177 0.077976 -0.17318
2 2.0182 -0.012863 -0.55599
3 2.1491 0.134472 -0.30349
4 2.1932 0.031635 -0.36818
5 2.0641 0.259634 -0.42586
6 2.1191 0.181361 -0.67260
7 2.0746 0.083116 -0.68507
8 2.0577 0.167392 -0.51550
9 2.0138 0.081222 -0.67765
10 1.9741 0.128047 -0.57706
R> summary(x3)
x y z
Min. :1.65 Min. :-0.1463 Min. :-1.161
1st Qu.:1.92 1st Qu.: 0.0784 1st Qu.:-0.616
Median :2.00 Median : 0.1239 Median :-0.497
Mean :2.00 Mean : 0.1231 Mean :-0.499
3rd Qu.:2.07 3rd Qu.: 0.1686 3rd Qu.:-0.379
Max. :2.34 Max. : 0.3692 Max. : 0.142
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 (4997 obs.); Bandwidth 'bw' = 0.01756
x f(x)
Min. :1.5965 Min. :0.0001
1st Qu.:1.7959 1st Qu.:0.0622
Median :1.9954 Median :0.7275
Mean :1.9954 Mean :1.2524
3rd Qu.:2.1948 3rd Qu.:2.3771
Max. :2.3942 Max. :3.6315
Marginal density of Y(t-t0)|Y(t0) = -1, Y(T) = -2 at time t = 0.75
Data: y (4997 obs.); Bandwidth 'bw' = 0.01103
y f(y)
Min. :-0.17937 Min. :0.0001
1st Qu.:-0.03397 1st Qu.:0.0382
Median : 0.11144 Median :0.6067
Mean : 0.11144 Mean :1.7176
3rd Qu.: 0.25685 3rd Qu.:3.2878
Max. : 0.40225 Max. :5.9221
Marginal density of Z(t-t0)|Z(t0) = 0.5, Z(T) = 0.5 at time t = 0.75
Data: z (4997 obs.); Bandwidth 'bw' = 0.02894
z f(z)
Min. :-1.24828 Min. :0.00003
1st Qu.:-0.87890 1st Qu.:0.01448
Median :-0.50953 Median :0.28973
Mean :-0.50953 Mean :0.67616
3rd Qu.:-0.14016 3rd Qu.:1.26493
Max. : 0.22922 Max. :2.26311
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 4.1, 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)↩