snssde1d()
Assume that we want to describe the following SDE:
Itô form3:
\[\begin{equation}\label{eq:05} dX_{t} = \frac{1}{2}\theta^{2} X_{t} dt + \theta X_{t} dW_{t},\qquad X_{0}=x_{0} > 0 \end{equation}\] Stratonovich form: \[\begin{equation}\label{eq:06} dX_{t} = \frac{1}{2}\theta^{2} X_{t} dt +\theta X_{t} \circ dW_{t},\qquad X_{0}=x_{0} > 0 \end{equation}\]In the above \(f(t,x)=\frac{1}{2}\theta^{2} x\) and \(g(t,x)= \theta x\) (\(\theta > 0\)), \(W_{t}\) is a standard Wiener process. To simulate this models using snssde1d()
function we need to specify:
drift
and diffusion
coefficients as R expressions that depend on the state variable x
and time variable t
.N=1000
(by default: N=1000
).M=10000
(by default: M=1
).t0=0
, x0=10
and end time T=1
(by default: t0=0
, x0=0
and T=1
).Dt=0.001
(by default: Dt=(T-t0)/N
).type="ito"
for Ito or type="str"
for Stratonovich (by default type="ito"
).method
(by default method="euler"
).R> theta = 0.5
R> f <- expression( (0.5*theta^2*x) )
R> g <- expression( theta*x )
R> mod1 <- snssde1d(drift=f,diffusion=g,x0=10,M=10000,type="ito") # Using Itô
R> mod2 <- snssde1d(drift=f,diffusion=g,x0=10,M=10000,type="str") # Using Stratonovich
R> mod1
Itô Sde 1D:
| dX(t) = (0.5 * theta^2 * X(t)) * dt + theta * X(t) * dW(t)
Method:
| Euler scheme with order 0.5
Summary:
| Size of process | N = 1001.
| Number of simulation | M = 10000.
| Initial value | x0 = 10.
| Time of process | t in [0,1].
| Discretization | Dt = 0.001.
R> mod2
Stratonovich Sde 1D:
| dX(t) = (0.5 * theta^2 * X(t)) * dt + theta * X(t) o dW(t)
Method:
| Euler scheme with order 0.5
Summary:
| Size of process | N = 1001.
| Number of simulation | M = 10000.
| Initial value | x0 = 10.
| Time of process | t in [0,1].
| Discretization | Dt = 0.001.
Using Monte-Carlo simulations, the following statistical measures (S3 method
) for class snssde1d()
can be approximated for the \(X_{t}\) process at any time \(t\):
mean
.moment
with order=2
and center=TRUE
.Median
.Mode
.quantile
.min
and max
.skewness
and kurtosis
.cv
.moment
.bconfint
.summary
.The following statistical measures (S3 method
) for class snssde1d()
can be approximated for the \(X_{t}\) process at any time \(t\), for example at=1
:
R> s = 1
R> mean(mod1, at = s)
[1] 11.363
R> moment(mod1, at = s , center = TRUE , order = 2) ## variance
[1] 37.589
R> Median(mod1, at = s)
[1] 9.9825
R> Mode(mod1, at =s)
[1] 7.9492
R> quantile(mod1 , at = s)
0% 25% 50% 75% 100%
1.6273 7.0831 9.9825 14.0993 75.2783
R> kurtosis(mod1 , at = s)
[1] 8.6708
R> skewness(mod1 , at = s)
[1] 1.7559
R> cv(mod1 , at = s )
[1] 0.5396
R> min(mod1 , at = s)
[1] 1.6273
R> max(mod1 , at = s)
[1] 75.278
R> moment(mod1, at = s , center= TRUE , order = 4)
[1] 12254
R> moment(mod1, at = s , center= FALSE , order = 4)
[1] 76435
The summary of the results of mod1
and mod2
at time \(t=1\) of class snssde1d()
is given by:
R> summary(mod1, at = 1)
Monte-Carlo Statistics for X(t) at time t = 1
Mean 11.3626
Variance 37.5926
Median 9.9825
Mode 7.9492
First quartile 7.0831
Third quartile 14.0993
Minimum 1.6273
Maximum 75.2783
Skewness 1.7559
Kurtosis 8.6708
Coef-variation 0.5396
3th-order moment 404.7246
4th-order moment 12253.5885
5th-order moment 383645.0793
6th-order moment 15583336.9981
R> summary(mod2, at = 1)
Monte-Carlo Statistics for X(t) at time t = 1
Mean 10.03733
Variance 27.63061
Median 8.89745
Mode 7.46607
First quartile 6.39703
Third quartile 12.44638
Minimum 1.42120
Maximum 49.15364
Skewness 1.57450
Kurtosis 7.03141
Coef-variation 0.52369
3th-order moment 228.68076
4th-order moment 5368.13144
5th-order moment 113997.58809
6th-order moment 2980394.32500
Hence we can just make use of the rsde1d()
function to build our random number generator for the conditional density of the \(X_{t}|X_{0}\) (\(X_{t}^{\text{mod1}}| X_{0}\) and \(X_{t}^{\text{mod2}}|X_{0}\)) at time \(t = 1\).
R> x1 <- rsde1d(object = mod1, at = 1) # X(t=1) | X(0)=x0 (Itô SDE)
R> x2 <- rsde1d(object = mod2, at = 1) # X(t=1) | X(0)=x0 (Stratonovich SDE)
R> head(x1,n=10)
[1] 17.1014 20.0566 10.2885 14.1245 19.7357 9.8151 11.0635 8.9138
[9] 17.8666 9.1985
R> head(x2,n=10)
[1] 8.5531 6.7120 4.2750 4.9903 6.3169 15.3767 14.0683 7.8177
[9] 5.9116 5.6135
R> summary(data.frame(x1,x2))
x1 x2
Min. : 1.63 Min. : 1.42
1st Qu.: 7.08 1st Qu.: 6.40
Median : 9.98 Median : 8.90
Mean :11.36 Mean :10.04
3rd Qu.:14.10 3rd Qu.:12.45
Max. :75.28 Max. :49.15
The function dsde1d()
can be used to show the Approximate transitional density for \(X_{t}|X_{0}\) at time \(t-s=1\) with log-normal curves:
R> mu1 = log(10); sigma1= sqrt(theta^2) # log mean and log variance for mod1
R> mu2 = log(10)-0.5*theta^2 ; sigma2 = sqrt(theta^2) # log mean and log variance for mod2
R> AppdensI <- dsde1d(mod1, at = 1)
R> AppdensS <- dsde1d(mod2, at = 1)
R> plot(AppdensI , dens = function(x) dlnorm(x,meanlog=mu1,sdlog = sigma1))
R> plot(AppdensS , dens = function(x) dlnorm(x,meanlog=mu2,sdlog = sigma2))
In Figure 2, we present the flow of trajectories, the mean path (red lines) of solution of and , with their empirical \(95\%\) confidence bands, that is to say from the \(2.5th\) to the \(97.5th\) percentile for each observation at time \(t\) (blue lines):
R> ## Itô
R> plot(mod1,ylab=expression(X^mod1))
R> lines(time(mod1),apply(mod1$X,1,mean),col=2,lwd=2)
R> lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[1,],col=4,lwd=2)
R> lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[2,],col=4,lwd=2)
R> legend("topleft",c("mean path",paste("bound of", 95,"% confidence")),inset = .01,col=c(2,4),lwd=2,cex=0.8)
R> ## Stratonovich
R> plot(mod2,ylab=expression(X^mod2))
R> lines(time(mod2),apply(mod2$X,1,mean),col=2,lwd=2)
R> lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[1,],col=4,lwd=2)
R> lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[2,],col=4,lwd=2)
R> legend("topleft",c("mean path",paste("bound of",95,"% confidence")),col=c(2,4),inset =.01,lwd=2,cex=0.8)
mod1: Itô and mod2: Stratonovich
snssde2d()
The following \(2\)-dimensional SDE’s with a vector of drift and a diagonal matrix of diffusion coefficients:
Itô form: \[\begin{equation}\label{eq:09} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) dW_{2,t} \end{cases} \end{equation}\] Stratonovich form: \[\begin{equation}\label{eq:10} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) \circ dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) \circ dW_{2,t} \end{cases} \end{equation}\]\(W_{1,t}\) and \(W_{2,t}\) is a two independent standard Wiener process. To simulate \(2d\) models using snssde2d()
function we need to specify:
drift
(2d) and diffusion
(2d) coefficients as R expressions that depend on the state variable x
, y
and time variable t
.N
(default: N=1000
).M
(default: M=1
).t0
, x0
and end time T
(default: t0=0
, x0=c(0,0)
and T=1
).Dt
(default: Dt=(T-t0)/N
).type="ito"
for Ito or type="str"
for Stratonovich (default type="ito"
).method
(default method="euler"
).We simulate a flow of \(10000\) trajectories of \((X_{t},Y_{t})\), with integration step size \(\Delta t = 0.01\), and using second Milstein method.
R> x0=5;y0=0
R> mu=3;sigma=0.5
R> fx <- expression(-(x/mu),x)
R> gx <- expression(sqrt(sigma),0)
R> mod2d <- snssde2d(drift=fx,diffusion=gx,Dt=0.01,M=10000,x0=c(x0,y0),method="smilstein")
R> mod2d
Itô Sde 2D:
| dX(t) = -(X(t)/mu) * dt + sqrt(sigma) * dW1(t)
| dY(t) = X(t) * dt + 0 * dW2(t)
Method:
| Second-order Milstein scheme
Summary:
| Size of process | N = 1001.
| Number of simulation | M = 10000.
| Initial values | (x0,y0) = (5,0).
| Time of process | t in [0,10].
| Discretization | Dt = 0.01.
The following statistical measures (S3 method
) for class snssde2d()
can be approximated for the \((X_{t},Y_{t})\) process at any time \(t\), for example at=5
:
R> s = 5
R> mean(mod2d, at = s)
[1] 0.93766 12.14376
R> moment(mod2d, at = s , center = TRUE , order = 2) ## variance
[1] 0.72106 7.16658
R> Median(mod2d, at = s)
[1] 0.93424 12.17686
R> Mode(mod2d, at = s)
[1] 0.9398 12.3973
R> quantile(mod2d , at = s)
$x
0% 25% 50% 75% 100%
-2.34076 0.36344 0.93424 1.51820 4.23466
$y
0% 25% 50% 75% 100%
2.0998 10.3602 12.1769 13.9129 21.6283
R> kurtosis(mod2d , at = s)
[1] 3.0216 3.0434
R> skewness(mod2d , at = s)
[1] 0.008700 -0.013332
R> cv(mod2d , at = s )
[1] 0.90565 0.22046
R> min(mod2d , at = s)
[1] -2.3408 2.0998
R> max(mod2d , at = s)
[1] 4.2347 21.6283
R> moment(mod2d, at = s , center= TRUE , order = 4)
[1] 1.5713 156.3417
R> moment(mod2d, at = s , center= FALSE , order = 4)
[1] 6.168 28232.729
The summary of the results of mod2d
at time \(t=5\) of class snssde2d()
is given by:
R> summary(mod2d, at = s)
Monte-Carlo Statistics for (X(t),Y(t)) at time t = 5
X Y
Mean 0.93766 12.14376
Variance 0.72113 7.16730
Median 0.93424 12.17686
Mode 0.93980 12.39728
First quartile 0.36344 10.36019
Third quartile 1.51820 13.91290
Minimum -2.34076 2.09978
Maximum 4.23466 21.62835
Skewness 0.00870 -0.01333
Kurtosis 3.02159 3.04343
Coef-variation 0.90565 0.22046
3th-order moment 0.00533 -0.25582
4th-order moment 1.57131 156.34170
5th-order moment 0.05265 -25.90355
6th-order moment 5.65976 5559.77008
For plotting (back in time) using the command plot
, the results of the simulation are shown in Figure 3.
R> plot(mod2d)
Ornstein-Uhlenbeck process and its integral
Take note of the well known result, which can be derived from either this equations. That for any \(t > 0\) the OU process \(X_t\) and its integral \(Y_t\) will be the normal distribution with mean and variance given by: \[ \begin{cases} \text{E}(X_{t}) =x_{0} e^{-t/\mu} &\text{and}\quad\text{Var}(X_{t})=\frac{\sigma \mu}{2} \left (1-e^{-2t/\mu}\right )\\ \text{E}(Y_{t}) = y_{0}+x_{0}\mu \left (1-e^{-t/\mu}\right ) &\text{and}\quad\text{Var}(Y_{t})=\sigma\mu^{3}\left (\frac{t}{\mu}-2\left (1-e^{-t/\mu}\right )+\frac{1}{2}\left (1-e^{-2t/\mu}\right )\right ) \end{cases} \]
Hence we can just make use of the rsde2d()
function to build our random number for \((X_{t},Y_{t})\) at time \(t = 10\).
R> out <- rsde2d(object = mod2d, at = 10)
R> head(out,n=10)
x y
1 -0.125743 20.720
2 0.198664 13.106
3 -1.249980 17.829
4 -0.360778 18.707
5 0.278290 11.122
6 -0.692971 22.814
7 0.752602 23.364
8 -0.087329 14.902
9 0.724739 12.780
10 0.398091 11.419
R> summary(out)
x y
Min. :-3.524 Min. :-5.69
1st Qu.:-0.375 1st Qu.:11.11
Median : 0.197 Median :14.49
Mean : 0.199 Mean :14.48
3rd Qu.: 0.783 3rd Qu.:17.90
Max. : 3.680 Max. :31.74
R> cov(out)
x y
x 0.75136 2.0579
y 2.05786 25.5791
Figure 4, show simulation results for moments of system :
R> mx <- apply(mod2d$X,1,mean)
R> my <- apply(mod2d$Y,1,mean)
R> Sx <- apply(mod2d$X,1,var)
R> Sy <- apply(mod2d$Y,1,var)
R> Cxy <- sapply(1:1001,function(i) cov(mod2d$X[i,],mod2d$Y[i,]))
R> out_b <- data.frame(mx,my,Sx,Sy,Cxy)
R> matplot(time(mod2d), out_b, type = "l", xlab = "time", ylab = "",col=2:6,lwd=2,lty=2:6,las=1)
R> legend("topleft",c(expression(hat(E)(X[t]),hat(E)(Y[t]),hat(Var)(X[t]),hat(Var)(Y[t]),hat(Cov)(X[t],Y[t]))),inset = .05,col=2:6,lty=2:6,lwd=2,cex=0.9)
For each SDE type and for each numerical scheme, the density of \(X_t\) and \(Y_t\) at time \(t=10\) are reported using dsde2d()
function, see e.g. Figure 5: the marginal density of \(X_t\) and \(Y_t\) at time \(t=10\).
R> denM <- dsde2d(mod2d,pdf="M",at =10)
R> denM
Marginal density of X(t-t0)|X(t0)=5 at time t = 10
Data: x (10000 obs.); Bandwidth 'bw' = 0.1232
x f(x)
Min. :-3.8940 Min. :0.00000
1st Qu.:-1.9081 1st Qu.:0.00153
Median : 0.0778 Median :0.03565
Mean : 0.0778 Mean :0.12576
3rd Qu.: 2.0637 3rd Qu.:0.24133
Max. : 4.0496 Max. :0.47596
Marginal density of Y(t-t0)|Y(t0)=0 at time t = 10
Data: y (10000 obs.); Bandwidth 'bw' = 0.7214
y f(y)
Min. :-7.853 Min. :0.000001
1st Qu.: 2.586 1st Qu.:0.000739
Median :13.026 Median :0.009363
Mean :13.026 Mean :0.023925
3rd Qu.:23.465 3rd Qu.:0.047221
Max. :33.904 Max. :0.078524
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 system \((X_{t},Y_{t})\) at time t=10
.
R> denJ <- dsde2d(mod2d, pdf="J", n=100,at =10)
R> denJ
Joint density of (X(t-t0),Y(t-t0)|X(t0)=5,Y(t0)=0) at time t = 10
Data: (x,y) (2 x 10000 obs.);
x y f(x,y)
Min. :-3.5243 Min. :-5.689 Min. :0.000000
1st Qu.:-1.7233 1st Qu.: 3.668 1st Qu.:0.000000
Median : 0.0778 Median :13.026 Median :0.000194
Mean : 0.0778 Mean :13.026 Mean :0.003634
3rd Qu.: 1.8788 3rd Qu.:22.383 3rd Qu.:0.002543
Max. : 3.6799 Max. :31.740 Max. :0.042365
R> plot(denJ,display="contour",main="Bivariate Transition Density at time t=10")
R> plot(denJ,display="image",main="Bivariate Transition Density at time t=10")
A \(3\)D plot of the transition density at \(t=10\) obtained with:
R> plot(denJ,main="Bivariate Transition Density at time t=10")
We approximate the bivariate transition density over the set transition horizons \(t\in [1,10]\) by \(\Delta t = 0.005\) using the code:
R> for (i in seq(1,10,by=0.005)){
+ plot(dsde2d(mod2d, at = i,n=100),display="contour",main=paste0('Transition Density \n t = ',i))
+ }
Implemente in R as follows, with integration step size \(\Delta t = 0.01\) and using stochastic Runge-Kutta methods 1-stage.
R> mu = 4; sigma=0.1
R> fx <- expression( y , (mu*( 1-x^2 )* y - x))
R> gx <- expression( 0 ,2*sigma)
R> mod2d <- snssde2d(drift=fx,diffusion=gx,N=10000,Dt=0.01,type="str",method="rk1")
R> mod2d
Stratonovich Sde 2D:
| dX(t) = Y(t) * dt + 0 o dW1(t)
| dY(t) = (mu * (1 - X(t)^2) * Y(t) - X(t)) * dt + 2 * sigma o dW2(t)
Method:
| Runge-Kutta method with order 1
Summary:
| Size of process | N = 10001.
| Number of simulation | M = 1.
| Initial values | (x0,y0) = (0,0).
| Time of process | t in [0,100].
| Discretization | Dt = 0.01.
For plotting (back in time) using the command plot
, and plot2d
in plane the results of the simulation are shown in Figure 8.
R> plot2d(mod2d) ## in plane (O,X,Y)
R> plot(mod2d) ## back in time
snssde3d()
The following \(3\)-dimensional SDE’s with a vector of drift and a diagonal matrix of diffusion coefficients:
Itô form: \[\begin{equation}\label{eq17} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) dW_{2,t}\\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) dW_{3,t} \end{cases} \end{equation}\] Stratonovich form: \[\begin{equation}\label{eq18} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) \circ dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) \circ dW_{2,t}\\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) \circ dW_{3,t} \end{cases} \end{equation}\]\(W_{1,t}\), \(W_{2,t}\) and \(W_{3,t}\) is a 3 independent standard Wiener process. To simulate this system using snssde3d()
function we need to specify:
drift
(3d) and diffusion
(3d) coefficients as R expressions that depend on the state variables x
, y
, z
and time variable t
.N
(default: N=1000
).M
(default: M=1
).t0
, x0
and end time T
(default: t0=0
, x0=c(0,0,0)
and T=1
).Dt
(default: Dt=(T-t0)/N
).type="ito"
for Ito or type="str"
for Stratonovich (default type="ito"
).method
(default method="euler"
).We simulate a flow of \(10000\) 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> mod3d <- snssde3d(x0=c(x=2,y=-2,z=-2),drift=fx,diffusion=gx,M=10000)
R> mod3d
Itô 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.
| Number of simulation | M = 10000.
| Initial values | (x0,y0,z0) = (2,-2,-2).
| Time of process | t in [0,1].
| Discretization | Dt = 0.001.
The following statistical measures (S3 method
) for class snssde3d()
can be approximated for the \((X_{t},Y_{t},Z_{t})\) process at any time \(t\), for example at=1
:
R> s = 1
R> mean(mod3d, at = s)
[1] -0.79605 0.87793 0.79497
R> moment(mod3d, at = s , center = TRUE , order = 2) ## variance
[1] 0.009681 0.103446 0.010391
R> Median(mod3d, at = s)
[1] -0.80262 0.85968 0.80243
R> Mode(mod3d, at = s)
[1] -0.81559 0.83478 0.81109
R> quantile(mod3d , at = s)
$x
0% 25% 50% 75% 100%
-1.10478 -0.86437 -0.80262 -0.73359 -0.35522
$y
0% 25% 50% 75% 100%
-0.11828 0.65401 0.85968 1.07801 2.31093
$z
0% 25% 50% 75% 100%
0.25421 0.73398 0.80243 0.86626 1.08362
R> kurtosis(mod3d , at = s)
[1] 3.1122 3.2932 3.5399
R> skewness(mod3d , at = s)
[1] 0.34678 0.38076 -0.53128
R> cv(mod3d , at = s )
[1] -0.12361 0.36637 0.12823
R> min(mod3d , at = s)
[1] -1.10478 -0.11828 0.25421
R> max(mod3d , at = s)
[1] -0.35522 2.31093 1.08362
R> moment(mod3d, at = s , center= TRUE , order = 4)
[1] 0.00029174 0.03524778 0.00038230
R> moment(mod3d, at = s , center= FALSE , order = 4)
[1] 0.43762 1.15222 0.43740
The summary of the results of mod3d
at time \(t=1\) of class snssde3d()
is given by:
R> summary(mod3d, at = s)
Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 1
X Y Z
Mean -0.79605 0.87793 0.79497
Variance 0.00968 0.10346 0.01039
Median -0.80262 0.85968 0.80243
Mode -0.81559 0.83478 0.81109
First quartile -0.86437 0.65401 0.73398
Third quartile -0.73359 1.07801 0.86626
Minimum -1.10478 -0.11828 0.25421
Maximum -0.35522 2.31093 1.08362
Skewness 0.34678 0.38076 -0.53128
Kurtosis 3.11222 3.29316 3.53988
Coef-variation -0.12361 0.36637 0.12823
3th-order moment 0.00033 0.01267 -0.00056
4th-order moment 0.00029 0.03525 0.00038
5th-order moment 0.00003 0.01353 -0.00006
6th-order moment 0.00002 0.02214 0.00003
For plotting (back in time) using the command plot
, and plot3D
in space the results of the simulation are shown in Figure 9.
R> plot(mod3d,union = TRUE) ## back in time
R> plot3D(mod3d,display="persp") ## in space (O,X,Y,Z)
3D SDEs
Hence we can just make use of the rsde3d()
function to build our random number for \((X_{t},Y_{t},Z_{t})\) at time \(t = 1\).
R> out <- rsde3d(object = mod3d, at = s)
R> head(out,n=10)
x y z
1 -0.86795 1.03414 0.77641
2 -0.80683 0.78026 0.78566
3 -0.86290 1.49416 0.82761
4 -0.70053 0.45556 0.74760
5 -0.74101 0.82970 0.68647
6 -0.98061 0.42199 0.66585
7 -0.79099 0.56426 0.76548
8 -0.84643 0.78707 0.71232
9 -0.69390 0.44960 0.64746
10 -0.61427 0.68495 0.56993
R> summary(out)
x y z
Min. :-1.105 Min. :-0.118 Min. :0.254
1st Qu.:-0.864 1st Qu.: 0.654 1st Qu.:0.734
Median :-0.803 Median : 0.860 Median :0.802
Mean :-0.796 Mean : 0.878 Mean :0.795
3rd Qu.:-0.734 3rd Qu.: 1.078 3rd Qu.:0.866
Max. :-0.355 Max. : 2.311 Max. :1.084
R> cov(out)
x y z
x 0.009682 -0.018004 -0.004178
y -0.018004 0.103457 0.019585
z -0.004178 0.019585 0.010392
For each SDE type and for each numerical scheme, the marginal density of \(X_t\), \(Y_t\) and \(Z_t\) at time \(t=1\) are reported using dsde3d()
function, see e.g. Figure 10.
R> den <- dsde3d(mod3d,pdf="M",at =1)
R> den
Marginal density of X(t-t0)|X(t0)=2 at time t = 1
Data: x (10000 obs.); Bandwidth 'bw' = 0.01392
x f(x)
Min. :-1.14654 Min. :0.0000
1st Qu.:-0.93827 1st Qu.:0.0173
Median :-0.73000 Median :0.4388
Mean :-0.73000 Mean :1.1992
3rd Qu.:-0.52173 3rd Qu.:2.2751
Max. :-0.31346 Max. :4.1212
Marginal density of Y(t-t0)|Y(t0)=-2 at time t = 1
Data: y (10000 obs.); Bandwidth 'bw' = 0.04513
y f(y)
Min. :-0.25368 Min. :0.00001
1st Qu.: 0.42132 1st Qu.:0.01324
Median : 1.09633 Median :0.13243
Mean : 1.09633 Mean :0.37001
3rd Qu.: 1.77133 3rd Qu.:0.71624
Max. : 2.44633 Max. :1.28773
Marginal density of Z(t-t0)|Z(t0)=-2 at time t = 1
Data: z (10000 obs.); Bandwidth 'bw' = 0.01408
z f(z)
Min. :0.21196 Min. :0.0000
1st Qu.:0.44044 1st Qu.:0.0162
Median :0.66891 Median :0.2836
Mean :0.66891 Mean :1.0931
3rd Qu.:0.89739 3rd Qu.:1.9952
Max. :1.12586 Max. :4.0831
R> plot(den, main="Marginal Density")
For an approximate joint transition density for \((X_t,Y_t,Z_t)\) (for more details, see package sm or ks.)
R> denJ <- dsde3d(mod3d,pdf="J")
R> plot(denJ,display="rgl")
with initial conditions \((X_{0},Y_{0},Z_{0})=(1,1,1)\), by specifying the drift and diffusion coefficients of three processes \(X_{t}\), \(Y_{t}\) and \(Z_{t}\) as R expressions which depends on the three state variables (x,y,z)
and time variable t
, with integration step size Dt=0.0001
.
R> K = 4; s = 1; sigma = 0.2
R> fx <- expression( (-K*x/sqrt(x^2+y^2+z^2)) , (-K*y/sqrt(x^2+y^2+z^2)) , (-K*z/sqrt(x^2+y^2+z^2)) )
R> gx <- rep(expression(sigma),3)
R> mod3d <- snssde3d(drift=fx,diffusion=gx,N=10000,x0=c(x=1,y=1,z=1))
R> mod3d
Itô Sde 3D:
| dX(t) = (-K * X(t)/sqrt(X(t)^2 + Y(t)^2 + Z(t)^2)) * dt + sigma * dW1(t)
| dY(t) = (-K * Y(t)/sqrt(X(t)^2 + Y(t)^2 + Z(t)^2)) * dt + sigma * dW2(t)
| dZ(t) = (-K * Z(t)/sqrt(X(t)^2 + Y(t)^2 + Z(t)^2)) * dt + sigma * dW3(t)
Method:
| Euler scheme with order 0.5
Summary:
| Size of process | N = 10001.
| Number of simulation | M = 1.
| Initial values | (x0,y0,z0) = (1,1,1).
| Time of process | t in [0,1].
| Discretization | Dt = 0.0001.
The results of simulation are shown:
R> plot3D(mod3d,display="persp",col="blue")
run by calling the function snssde3d()
to produce a simulation of the solution, with \(\mu = 1\) and \(\sigma = 1\).
R> fx <- expression(y,0,0)
R> gx <- expression(z,1,1)
R> modtra <- snssde3d(drift=fx,diffusion=gx,M=10000,type="str")
R> modtra
Stratonovich Sde 3D:
| dX(t) = Y(t) * dt + Z(t) o dW1(t)
| dY(t) = 0 * dt + 1 o dW2(t)
| dZ(t) = 0 * dt + 1 o dW3(t)
Method:
| Euler scheme with order 0.5
Summary:
| Size of process | N = 1001.
| Number of simulation | M = 10000.
| Initial values | (x0,y0,z0) = (0,0,0).
| Time of process | t in [0,1].
| Discretization | Dt = 0.001.
R> summary(modtra)
Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 1
X Y Z
Mean -0.01044 -0.01647 -0.01930
Variance 0.78983 1.01403 0.96885
Median -0.01837 -0.01114 -0.01814
Mode -0.13435 0.02920 -0.01126
First quartile -0.55623 -0.68685 -0.68549
Third quartile 0.53909 0.65842 0.65707
Minimum -4.98033 -4.31390 -3.40609
Maximum 4.33912 3.99552 3.45899
Skewness 0.00603 -0.02683 -0.02529
Kurtosis 4.07151 3.07348 2.96449
Coef-variation -85.13068 -61.15766 -51.00351
3th-order moment 0.00424 -0.02740 -0.02412
4th-order moment 2.53991 3.16031 2.78267
5th-order moment -0.01685 -0.37694 -0.15523
6th-order moment 17.68542 16.73578 13.12140
the following code produces the result in Figure 12.
R> plot(modtra$X,plot.type="single",ylab=expression(X[t]))
R> lines(time(modtra),apply(modtra$X,1,mean),col=2,lwd=2)
R> legend("topleft",c("mean path"),col=2,lwd=2,cex=0.8)
Simulation of \(X_t\)
The histogram and kernel density of \(X_t\) at time \(t=1\) are reported using dsde3d()
function, see e.g. Figure 13.
R> den <- dsde3d(modtra,pdf="Marginal",at=1)
R> den$resx
Call:
density.default(x = x, na.rm = TRUE)
Data: x (10000 obs.); Bandwidth 'bw' = 0.1166
x y
Min. :-5.330 Min. :0.00000
1st Qu.:-2.825 1st Qu.:0.00122
Median :-0.321 Median :0.01272
Mean :-0.321 Mean :0.09971
3rd Qu.: 2.184 3rd Qu.:0.14684
Max. : 4.689 Max. :0.49431
R> MASS::truehist(den$ech$x,xlab = expression(X[t==1]));box()
R> lines(den$resx,col="red",lwd=2)
R> legend("topleft",c("Distribution histogram","Kernel Density"),inset =.01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"), lwd=2,cex=0.8)
Figure 14 and 15, show approximation results for \(m_{1}(t)= \text{E}(X_{t})\), \(S_{1}(t)=\text{V}(X_{t})\) and \(C(s,t)=\text{Cov}(X_{s},X_{t})\):
R> m1 <- apply(modtra$X,1,mean) ## m1(t)
R> S1 <- apply(modtra$X,1,var) ## s1(t)
R> out_a <- data.frame(m1,S1)
R> matplot(time(modtra), out_a, type = "l", xlab = "time", ylab = "", col=2:3,lwd=2,lty=2:3,las=1)
R> legend("topleft",c(expression(m[1](t),S[1](t))),inset = .09,col=2:3,lty=2:3,lwd=2,cex=0.9)
R> color.palette=colorRampPalette(c('white','green','blue','red'))
R> filled.contour(time(modtra), time(modtra), cov(t(modtra$X)), color.palette=color.palette,plot.title = title(main = expression(paste("Covariance empirique:",cov(X[s],X[t]))),xlab = "time", ylab = "time"),key.title = title(main = ""))
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.Boukhetala K (1996). Modelling and Simulation of a Dispersion Pollutant with Attractive Centre, volume 3, pp. 245-252. Computer Methods and Water Resources, Computational Mechanics Publications, Boston, USA.
Guidoum AC, Boukhetala K (2017). Performing Parallel Monte Carlo and Moment Equations Methods for Itô and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc. Preprint submitted to Journal of Statistical Software.
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)↩
The equivalently of \(X_{t}^{\text{mod1}}\) the following Stratonovich SDE: \(dX_{t} = \theta X_{t} \circ dW_{t}\).↩