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] 11.294
R> moment(mod1, at = s , center = TRUE , order = 2) ## variance
[1] 35.496
R> Median(mod1, at = s)
[1] 9.9441
R> Mode(mod1, at =s)
[1] 7.8018
R> quantile(mod1 , at = s)
0% 25% 50% 75% 100%
1.6445 7.1602 9.9441 13.8679 59.7271
R> kurtosis(mod1 , at = s)
[1] 7.2934
R> skewness(mod1 , at = s)
[1] 1.621
R> cv(mod1 , at = s )
[1] 0.52756
R> min(mod1 , at = s)
[1] 1.6445
R> max(mod1 , at = s)
[1] 59.727
R> moment(mod1, at = s , center= TRUE , order = 4)
[1] 9191.3
R> moment(mod1, at = s , center= FALSE , order = 4)
[1] 68116
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 11.29388
Variance 35.49969
Median 9.94412
Mode 7.80175
First quartile 7.16021
Third quartile 13.86789
Minimum 1.64453
Maximum 59.72708
Skewness 1.62104
Kurtosis 7.29338
Coef-variation 0.52756
3th-order moment 342.87069
4th-order moment 9191.32518
5th-order moment 230302.31460
6th-order moment 7122226.23084
R> summary(mod2, at = 1)
Monte-Carlo Statistics for X(t) at time t = 1
Mean 9.97015
Variance 27.21940
Median 8.81041
Mode 7.08137
First quartile 6.33736
Third quartile 12.32062
Minimum 1.41600
Maximum 61.05984
Skewness 1.69339
Kurtosis 8.48428
Coef-variation 0.52328
3th-order moment 240.47720
4th-order moment 6285.96643
5th-order moment 168367.21002
6th-order moment 5803105.06475
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] 4.5617 15.2902 5.7172 7.8927 21.2781 15.1225 6.7414 14.9453
[9] 27.0423 14.1436
R> head(x2,n=10)
[1] 5.4088 13.2887 4.6744 6.3361 11.2292 6.5593 11.6521 10.4583
[9] 11.3059 6.9635
R> summary(data.frame(x1,x2))
x1 x2
Min. : 1.64 Min. : 1.42
1st Qu.: 7.16 1st Qu.: 6.34
Median : 9.94 Median : 8.81
Mean :11.29 Mean : 9.97
3rd Qu.:13.87 3rd Qu.:12.32
Max. :59.73 Max. :61.06
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] 0.94922 12.20826
R> moment(mod2d, at = s , center = TRUE , order = 2) ## variance
[1] 0.72484 7.16594
R> Median(mod2d, at = s)
[1] 0.93207 12.19928
R> Mode(mod2d, at = s)
[1] 0.83543 12.28881
R> quantile(mod2d , at = s)
$x
0% 25% 50% 75% 100%
-2.52964 0.38431 0.93207 1.51615 4.24452
$y
0% 25% 50% 75% 100%
2.0354 10.3969 12.1993 14.0398 21.8081
R> kurtosis(mod2d , at = s)
[1] 2.9848 2.9321
R> skewness(mod2d , at = s)
[1] 0.033821 0.014161
R> cv(mod2d , at = s )
[1] 0.89696 0.21928
R> min(mod2d , at = s)
[1] -2.5296 2.0354
R> max(mod2d , at = s)
[1] 4.2445 21.8081
R> moment(mod2d, at = s , center= TRUE , order = 4)
[1] 1.5685 150.5964
R> moment(mod2d, at = s , center= FALSE , order = 4)
[1] 6.3782 28785.4283
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 0.94922 12.20826
Variance 0.72491 7.16666
Median 0.93207 12.19928
Mode 0.83543 12.28881
First quartile 0.38431 10.39693
Third quartile 1.51615 14.03979
Minimum -2.52964 2.03544
Maximum 4.24452 21.80815
Skewness 0.03382 0.01416
Kurtosis 2.98479 2.93212
Coef-variation 0.89696 0.21928
3th-order moment 0.02087 0.27168
4th-order moment 1.56848 150.59645
5th-order moment 0.10568 4.77462
6th-order moment 5.53409 5174.30078
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 -0.025937 12.5317
2 1.089531 19.3829
3 0.790466 7.0203
4 -0.719261 19.4002
5 0.970252 27.9771
6 0.933042 16.8523
7 1.038908 18.4764
8 0.814775 21.3297
9 -1.107891 12.9421
10 0.591950 11.4679
R> summary(out)
x y
Min. :-3.567 Min. :-6.34
1st Qu.:-0.410 1st Qu.:11.04
Median : 0.190 Median :14.49
Mean : 0.180 Mean :14.53
3rd Qu.: 0.772 3rd Qu.:17.89
Max. : 3.426 Max. :30.81
R> cov(out)
x y
x 0.75571 2.1272
y 2.12724 25.9638
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' = 0.124
x f(x)
Min. :-3.9387 Min. :0.00000
1st Qu.:-2.0046 1st Qu.:0.00172
Median :-0.0705 Median :0.03835
Mean :-0.0705 Mean :0.12913
3rd Qu.: 1.8636 3rd Qu.:0.23996
Max. : 3.7977 Max. :0.45557
Marginal density of Y(t-t0)|Y(t0)=0 at time t = 10
Data: y (10000 obs.); Bandwidth 'bw' = 0.7268
y f(y)
Min. :-8.516 Min. :0.000001
1st Qu.: 1.861 1st Qu.:0.000821
Median :12.238 Median :0.009732
Mean :12.238 Mean :0.024068
3rd Qu.:22.616 3rd Qu.:0.047036
Max. :32.993 Max. :0.078066
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. :-3.5667 Min. :-6.3358 Min. :0.000000
1st Qu.:-1.8186 1st Qu.: 2.9512 1st Qu.:0.000000
Median :-0.0705 Median :12.2383 Median :0.000206
Mean :-0.0705 Mean :12.2383 Mean :0.003772
3rd Qu.: 1.6776 3rd Qu.:21.5254 3rd Qu.:0.002818
Max. : 3.4257 Max. :30.8124 Max. :0.039369
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] -0.79530 0.87844 0.79514
R> moment(mod3d, at = s , center = TRUE , order = 2) ## variance
[1] 0.0098268 0.1069379 0.0102464
R> Median(mod3d, at = s)
[1] -0.80159 0.85335 0.80060
R> Mode(mod3d, at = s)
[1] -0.81126 0.81668 0.80312
R> quantile(mod3d , at = s)
$x
0% 25% 50% 75% 100%
-1.07356 -0.86605 -0.80159 -0.73201 -0.33451
$y
0% 25% 50% 75% 100%
-0.072332 0.651842 0.853354 1.080896 2.870194
$z
0% 25% 50% 75% 100%
0.29802 0.73177 0.80060 0.86770 1.13628
R> kurtosis(mod3d , at = s)
[1] 3.2611 3.4961 3.2981
R> skewness(mod3d , at = s)
[1] 0.38751 0.45523 -0.40903
R> cv(mod3d , at = s )
[1] -0.12465 0.37229 0.12731
R> min(mod3d , at = s)
[1] -1.073563 -0.072332 0.298022
R> max(mod3d , at = s)
[1] -0.33451 2.87019 1.13628
R> moment(mod3d, at = s , center= TRUE , order = 4)
[1] 0.00031497 0.03998848 0.00034634
R> moment(mod3d, at = s , center= FALSE , order = 4)
[1] 0.43646 1.18650 0.43761
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 -0.79530 0.87844 0.79514
Variance 0.00983 0.10695 0.01025
Median -0.80159 0.85335 0.80060
Mode -0.81126 0.81668 0.80312
First quartile -0.86605 0.65184 0.73177
Third quartile -0.73201 1.08090 0.86770
Minimum -1.07356 -0.07233 0.29802
Maximum -0.33451 2.87019 1.13628
Skewness 0.38751 0.45523 -0.40903
Kurtosis 3.26113 3.49611 3.29813
Coef-variation -0.12465 0.37229 0.12731
3th-order moment 0.00038 0.01592 -0.00042
4th-order moment 0.00031 0.03999 0.00035
5th-order moment 0.00004 0.02089 -0.00005
6th-order moment 0.00002 0.03489 0.00002
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 -0.91235 1.11431 0.80904
2 -0.82417 0.78085 0.71303
3 -0.69025 0.87803 0.79209
4 -0.76321 0.49149 0.65043
5 -0.76783 0.48448 0.62280
6 -0.67763 0.39713 0.60449
7 -0.85963 1.05202 0.77641
8 -0.74741 1.04911 0.78306
9 -0.79319 0.88463 0.83531
10 -0.86931 0.87753 0.77403
R> summary(out)
x y z
Min. :-1.074 Min. :-0.0723 Min. :0.298
1st Qu.:-0.866 1st Qu.: 0.6518 1st Qu.:0.732
Median :-0.802 Median : 0.8534 Median :0.801
Mean :-0.795 Mean : 0.8784 Mean :0.795
3rd Qu.:-0.732 3rd Qu.: 1.0809 3rd Qu.:0.868
Max. :-0.335 Max. : 2.8702 Max. :1.136
R> cov(out)
x y z
x 0.0098277 -0.018378 -0.0042096
y -0.0183785 0.106949 0.0197329
z -0.0042096 0.019733 0.0102475
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' = 0.01414
x f(x)
Min. :-1.11598 Min. :0.0000
1st Qu.:-0.91001 1st Qu.:0.0434
Median :-0.70404 Median :0.4680
Mean :-0.70404 Mean :1.2126
3rd Qu.:-0.49807 3rd Qu.:2.3853
Max. :-0.29209 Max. :4.0569
Marginal density of Y(t-t0)|Y(t0)=-2 at time t = 1
Data: y (10000 obs.); Bandwidth 'bw' = 0.04567
y f(y)
Min. :-0.20935 Min. :0.00001
1st Qu.: 0.59479 1st Qu.:0.00099
Median : 1.39893 Median :0.06041
Mean : 1.39893 Mean :0.31059
3rd Qu.: 2.20307 3rd Qu.:0.55081
Max. : 3.00721 Max. :1.27133
Marginal density of Z(t-t0)|Z(t0)=-2 at time t = 1
Data: z (10000 obs.); Bandwidth 'bw' = 0.01444
z f(z)
Min. :0.25470 Min. :0.0000
1st Qu.:0.48593 1st Qu.:0.0121
Median :0.71715 Median :0.2991
Mean :0.71715 Mean :1.0802
3rd Qu.:0.94837 3rd Qu.:2.0534
Max. :1.17959 Max. :3.9860
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 -0.00675 0.01140 -0.00362
Variance 0.82947 1.02639 1.01124
Median 0.00356 0.01360 -0.00226
Mode 0.02129 0.11475 0.10404
First quartile -0.57297 -0.67813 -0.68853
Third quartile 0.57052 0.69228 0.67940
Minimum -5.37654 -3.82762 -3.53156
Maximum 4.80013 4.04002 4.19328
Skewness -0.04712 0.00645 -0.00476
Kurtosis 4.25285 2.92843 2.98956
Coef-variation -134.86881 88.86827 -277.70256
3th-order moment -0.03560 0.00670 -0.00484
4th-order moment 2.92603 3.08501 3.05716
5th-order moment -0.77761 0.04420 0.02058
6th-order moment 24.55746 15.26236 15.43598
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$resx
Call:
density.default(x = x, na.rm = TRUE)
Data: x (10000 obs.); Bandwidth 'bw' = 0.1217
x y
Min. :-5.742 Min. :0.00000
1st Qu.:-3.015 1st Qu.:0.00070
Median :-0.288 Median :0.00728
Mean :-0.288 Mean :0.09159
3rd Qu.: 2.438 3rd Qu.:0.11802
Max. : 5.165 Max. :0.48233
R> MASS::truehist(den$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.TEX.sde()
- Converting Sim.DiffProc Objects to LaTeX.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 (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)↩
The equivalently of \(X_{t}^{\text{mod1}}\) the following Stratonovich SDE: \(dX_{t} = \theta X_{t} \circ dW_{t}\).↩