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 = 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 1.97014
Variance 1.61044
Median 1.63296
Mode 1.15145
First quartile 1.11853
Third quartile 2.44144
Minimum 0.34155
Maximum 11.64675
Skewness 2.16882
Kurtosis 10.48070
Coef-variation 0.64413
3th-order moment 4.43243
4th-order moment 27.18200
5th-order moment 173.53092
6th-order moment 1275.95092
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] 1.8764
R> moment(mod, at = s , center = TRUE , order = 2) ## variance
[1] 1.5393
R> Median(mod, at = s)
[1] 1.5457
R> Mode(mod, at = s)
[1] 1.1186
R> quantile(mod , at = s)
0% 25% 50% 75% 100%
0.27414 1.06314 1.54569 2.29849 15.97073
R> kurtosis(mod , at = s)
[1] 15.785
R> skewness(mod , at = s)
[1] 2.5237
R> cv(mod , at = s )
[1] 0.66126
R> min(mod , at = s)
[1] 0.27414
R> max(mod , at = s)
[1] 15.971
R> moment(mod, at = s , center= TRUE , order = 4)
[1] 37.417
R> moment(mod, at = s , center= FALSE , order = 4)
[1] 118.52
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 1.87642
Variance 1.53960
Median 1.54569
Mode 1.11855
First quartile 1.06314
Third quartile 2.29849
Minimum 0.27414
Maximum 15.97073
Skewness 2.52369
Kurtosis 15.78513
Coef-variation 0.66126
3th-order moment 4.82113
4th-order moment 37.41664
5th-order moment 361.05430
6th-order moment 4202.05090
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] 1.15738 2.75616 1.00823 1.37275 0.61755 1.93219 2.42634 0.46522
[9] 0.92225 1.99562
R> summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.274 1.063 1.546 1.876 2.299 15.971
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' = 0.1519
x f(x)
Min. :-0.1815 Min. :0.00000
1st Qu.: 3.9705 1st Qu.:0.00006
Median : 8.1224 Median :0.00101
Mean : 8.1224 Mean :0.06015
3rd Qu.:12.2744 3rd Qu.:0.03244
Max. :16.4263 Max. :0.55231
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) = 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 = 4995 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 4995 among 5000
X Y
Mean 0.00579 0.00270
Variance 0.02082 0.00542
Median 0.00517 0.00282
Mode 0.00236 0.00115
First quartile -0.09071 -0.04443
Third quartile 0.10146 0.05064
Minimum -0.54025 -0.31584
Maximum 0.50999 0.32230
Skewness 0.00242 -0.03023
Kurtosis 3.11940 3.41785
Coef-variation 24.90935 27.31101
3th-order moment 0.00001 -0.00001
4th-order moment 0.00135 0.00010
5th-order moment -0.00001 0.00000
6th-order moment 0.00015 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] 0.0268110 0.0081968
R> moment(mod2, at = s , center = TRUE , order = 2) ## variance
[1] 0.0196747 0.0046577
R> Median(mod2, at = s)
[1] 0.0267218 0.0074823
R> Mode(mod2, at = s)
[1] 0.0288800 0.0062648
R> quantile(mod2 , at = s)
$x
0% 25% 50% 75% 100%
-0.600498 -0.066184 0.026722 0.119623 0.578464
$y
0% 25% 50% 75% 100%
-0.2641281 -0.0358245 0.0074823 0.0515105 0.3077726
R> kurtosis(mod2 , at = s)
[1] 3.1131 3.3341
R> skewness(mod2 , at = s)
[1] -0.010913 0.025349
R> cv(mod2 , at = s )
[1] 5.2322 8.3269
R> min(mod2 , at = s)
[1] -0.60050 -0.26413
R> max(mod2 , at = s)
[1] 0.57846 0.30777
R> moment(mod2 , at = s , center= TRUE , order = 4)
[1] 0.00120555 0.00007236
R> moment(mod2 , at = s , center= FALSE , order = 4)
[1] 0.001287690 0.000074507
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 4995 among 5000
X Y
Mean 0.02681 0.00820
Variance 0.01968 0.00466
Median 0.02672 0.00748
Mode 0.02888 0.00626
First quartile -0.06618 -0.03582
Third quartile 0.11962 0.05151
Minimum -0.60050 -0.26413
Maximum 0.57846 0.30777
Skewness -0.01091 0.02535
Kurtosis 3.11311 3.33413
Coef-variation 5.23219 8.32694
3th-order moment -0.00003 0.00001
4th-order moment 0.00121 0.00007
5th-order moment -0.00002 0.00000
6th-order moment 0.00013 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 0.164817 -0.0125831
2 -0.045129 0.1117686
3 0.162660 -0.0555036
4 -0.138086 0.1422665
5 0.067019 0.0499271
6 -0.059958 0.0471664
7 0.074365 0.0739886
8 0.045395 0.0499711
9 -0.147753 0.0311994
10 -0.121055 0.0040338
R> summary(x2)
x y
Min. :-0.6005 Min. :-0.26413
1st Qu.:-0.0662 1st Qu.:-0.03583
Median : 0.0267 Median : 0.00748
Mean : 0.0268 Mean : 0.00820
3rd Qu.: 0.1196 3rd Qu.: 0.05151
Max. : 0.5785 Max. : 0.30777
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 (4995 obs.); Bandwidth 'bw' = 0.02272
x f(x)
Min. :-0.66867 Min. :0.00004
1st Qu.:-0.33984 1st Qu.:0.00440
Median :-0.01102 Median :0.19373
Mean :-0.01102 Mean :0.75954
3rd Qu.: 0.31781 3rd Qu.:1.38081
Max. : 0.64664 Max. :2.94235
Marginal density of Y(t-t0)|Y(t0) = -0.5, Y(T) = 0.5 at time t = 6.75
Data: y (4995 obs.); Bandwidth 'bw' = 0.01068
y f(y)
Min. :-0.29617 Min. :0.0001
1st Qu.:-0.13717 1st Qu.:0.0329
Median : 0.02182 Median :0.3736
Mean : 0.02182 Mean :1.5708
3rd Qu.: 0.18082 3rd Qu.:2.7173
Max. : 0.33982 Max. :6.1614
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 4995 obs.);
x y f(x,y)
Min. :-0.60050 Min. :-0.264128 Min. : 0.0000
1st Qu.:-0.30576 1st Qu.:-0.121153 1st Qu.: 0.0003
Median :-0.01102 Median : 0.021822 Median : 0.0787
Mean :-0.01102 Mean : 0.021822 Mean : 1.4531
3rd Qu.: 0.28372 3rd Qu.: 0.164797 3rd Qu.: 0.9620
Max. : 0.57846 Max. : 0.307773 Max. :19.3926
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 = 4995 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 4995 among 5000
X Y Z
Mean 0.65797 0.51141 0.17736
Variance 0.01386 0.00720 0.02134
Median 0.66262 0.51190 0.17922
Mode 0.65703 0.51280 0.17437
First quartile 0.58249 0.45456 0.08095
Third quartile 0.73840 0.56694 0.27446
Minimum 0.19084 0.21476 -0.38052
Maximum 1.04900 0.90282 0.66644
Skewness -0.25687 0.10812 -0.05765
Kurtosis 3.18596 3.52316 3.09026
Coef-variation 0.17892 0.16596 0.82358
3th-order moment -0.00042 0.00007 -0.00018
4th-order moment 0.00061 0.00018 0.00141
5th-order moment -0.00005 0.00001 -0.00004
6th-order moment 0.00005 0.00001 0.00015
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] 1.99713 0.12318 -0.50129
R> moment(mod3, at = s , center = TRUE , order = 2) ## variance
[1] 0.0117604 0.0045534 0.0312532
R> Median(mod3, at = s)
[1] 1.99853 0.12572 -0.49745
R> Mode(mod3, at = s)
[1] 1.99628 0.13785 -0.48107
R> quantile(mod3 , at = s)
$x
0% 25% 50% 75% 100%
1.5818 1.9255 1.9985 2.0699 2.4237
$y
0% 25% 50% 75% 100%
-0.120062 0.077326 0.125715 0.169267 0.358439
$z
0% 25% 50% 75% 100%
-1.11784 -0.61594 -0.49745 -0.37755 0.13066
R> kurtosis(mod3 , at = s)
[1] 3.0572 2.9313 3.0166
R> skewness(mod3 , at = s)
[1] -0.011997 -0.122685 -0.100543
R> cv(mod3 , at = s )
[1] 0.054306 0.547853 -0.352697
R> min(mod3 , at = s)
[1] 1.58185 -0.12006 -1.11784
R> max(mod3 , at = s)
[1] 2.42369 0.35844 0.13066
R> moment(mod3 , at = s , center= TRUE , order = 4)
[1] 0.000422998 0.000060802 0.002947666
R> moment(mod3 , at = s , center= FALSE , order = 4)
[1] 16.19006059 0.00068703 0.11433115
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 4995 among 5000
X Y Z
Mean 1.99713 0.12318 -0.50129
Variance 0.01176 0.00455 0.03126
Median 1.99853 0.12572 -0.49745
Mode 1.99628 0.13785 -0.48107
First quartile 1.92546 0.07733 -0.61594
Third quartile 2.06991 0.16927 -0.37755
Minimum 1.58185 -0.12006 -1.11784
Maximum 2.42369 0.35844 0.13066
Skewness -0.01200 -0.12268 -0.10054
Kurtosis 3.05718 2.93132 3.01659
Coef-variation 0.05431 0.54785 -0.35270
3th-order moment -0.00002 -0.00004 -0.00056
4th-order moment 0.00042 0.00006 0.00295
5th-order moment 0.00000 0.00000 -0.00013
6th-order moment 0.00003 0.00000 0.00046
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 2.0150 0.1565508 -0.20627
2 2.0734 0.1482807 -0.29021
3 1.9524 0.2395013 -0.33978
4 2.0970 0.2143750 -0.54232
5 1.8791 0.1599987 -0.44235
6 1.8492 0.0424117 -0.77723
7 2.0421 0.0974784 -0.44946
8 2.1031 0.2475757 -0.44326
9 2.1317 0.0099146 -0.46271
10 2.0854 0.1285737 -0.60272
R> summary(x3)
x y z
Min. :1.58 Min. :-0.1201 Min. :-1.118
1st Qu.:1.93 1st Qu.: 0.0773 1st Qu.:-0.616
Median :2.00 Median : 0.1257 Median :-0.497
Mean :2.00 Mean : 0.1232 Mean :-0.501
3rd Qu.:2.07 3rd Qu.: 0.1693 3rd Qu.:-0.378
Max. :2.42 Max. : 0.3584 Max. : 0.131
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 (4995 obs.); Bandwidth 'bw' = 0.01767
x f(x)
Min. :1.5289 Min. :0.0001
1st Qu.:1.7658 1st Qu.:0.0225
Median :2.0028 Median :0.3520
Mean :2.0028 Mean :1.0540
3rd Qu.:2.2397 3rd Qu.:2.0086
Max. :2.4767 Max. :3.7813
Marginal density of Y(t-t0)|Y(t0) = -1, Y(T) = -2 at time t = 0.75
Data: y (4995 obs.); Bandwidth 'bw' = 0.01106
y f(y)
Min. :-0.15324 Min. :0.0001
1st Qu.:-0.01703 1st Qu.:0.0611
Median : 0.11919 Median :0.8573
Mean : 0.11919 Mean :1.8335
3rd Qu.: 0.25540 3rd Qu.:3.6108
Max. : 0.39162 Max. :5.8408
Marginal density of Z(t-t0)|Z(t0) = 0.5, Z(T) = 0.5 at time t = 0.75
Data: z (4995 obs.); Bandwidth 'bw' = 0.02898
z f(z)
Min. :-1.20477 Min. :0.00003
1st Qu.:-0.84918 1st Qu.:0.02555
Median :-0.49359 Median :0.31560
Mean :-0.49359 Mean :0.70237
3rd Qu.:-0.13800 3rd Qu.:1.35972
Max. : 0.21759 Max. :2.24703
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.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 (2017). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.0, 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)↩