library(Sim.DiffProc) library(knitr) knitr::opts_chunk$set(comment="",prompt=TRUE, fig.show='hold', warning=FALSE, message=FALSE) options(prompt="R> ",scipen=16,digits=5,warning=FALSE, message=FALSE, width = 70)
snssde1d()
Assume that we want to describe the following SDE:
Ito form^[The equivalently of $X_{t}^{\text{mod1}}$ the following Stratonovich SDE: $dX_{t} = \theta X_{t} \circ dW_{t}$.]:
\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=1000
(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"
).set.seed(1234, kind = "L'Ecuyer-CMRG") theta = 0.5 f <- expression( (0.5*theta^2*x) ) g <- expression( theta*x ) mod1 <- snssde1d(drift=f,diffusion=g,x0=10,M=1000,type="ito") # Using Ito mod2 <- snssde1d(drift=f,diffusion=g,x0=10,M=1000,type="str") # Using Stratonovich mod1 mod2
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 summary of the results of mod1
and mod2
at time $t=1$ of class snssde1d()
is given by:
summary(mod1, at = 1) summary(mod2, at = 1)
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 $t = 1$.
x1 <- rsde1d(object = mod1, at = 1) # X(t=1) | X(0)=x0 (Ito SDE) x2 <- rsde1d(object = mod2, at = 1) # X(t=1) | X(0)=x0 (Stratonovich SDE) head(data.frame(x1,x2),n=5)
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:
mu1 = log(10); sigma1= sqrt(theta^2) # log mean and log variance for mod1 mu2 = log(10)-0.5*theta^2 ; sigma2 = sqrt(theta^2) # log mean and log variance for mod2 AppdensI <- dsde1d(mod1, at = 1) AppdensS <- dsde1d(mod2, at = 1) plot(AppdensI , dens = function(x) dlnorm(x,meanlog=mu1,sdlog = sigma1)) plot(AppdensS , dens = function(x) dlnorm(x,meanlog=mu2,sdlog = sigma2))
```r|X_{0}$ at time $t-s=1$ with log-normal curves. mod1: Ito and mod2: Stratonovich ', fig.env='figure*',fig.width=10,fig.height=10} knitr::include_graphics(c("Figures/fig007.png","Figures/fig008.png"))
In Figure 2, we present the flow of trajectories, the mean path (red lines) of solution of \eqref{eq:05} and \eqref{eq:06}, 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 ## Ito plot(mod1,ylab=expression(X^mod1)) lines(time(mod1),apply(mod1$X,1,mean),col=2,lwd=2) lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[1,],col=4,lwd=2) lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[2,],col=4,lwd=2) legend("topleft",c("mean path",paste("bound of", 95,"% confidence")),inset = .01,col=c(2,4),lwd=2,cex=0.8) ## Stratonovich plot(mod2,ylab=expression(X^mod2)) lines(time(mod2),apply(mod2$X,1,mean),col=2,lwd=2) lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[1,],col=4,lwd=2) lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[2,],col=4,lwd=2) legend("topleft",c("mean path",paste("bound of",95,"% confidence")),col=c(2,4),inset =.01,lwd=2,cex=0.8)
knitr::include_graphics(c("Figures/fig07.png","Figures/fig08.png"))
snssde2d()
The following $2$-dimensional SDE's with a vector of drift and matrix of diffusion coefficients:
Ito 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}
where $(W_{1,t}, W_{2,t})$ are a two independent standard Wiener process if corr = NULL
. 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
.corr
the correlation structure of two standard Wiener process $(W_{1,t},W_{2,t})$; must be a real symmetric positive-definite square matrix of dimension $2$ (default: corr=NULL
).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"
).The Ornstein-Uhlenbeck (OU) process has a long history in physics. Introduced in essence by Langevin in his famous 1908 paper on Brownian motion, the process received a more thorough mathematical examination several decades later by Uhlenbeck and Ornstein (1930). The OU process is understood here to be the univariate continuous Markov process $X_t$. In mathematical terms, the equation is written as an Ito equation: \begin{equation}\label{eq016} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t,\quad X_{0}=x_{0} \end{equation} In these equations, $\mu$ and $\sigma$ are positive constants called, respectively, the relaxation time and the diffusion constant. The time integral of the OU process $X_t$ (or indeed of any process $X_t$) is defined to be the process $Y_t$ that satisfies: \begin{equation}\label{eq017} Y_{t} = Y_{0}+\int X_{t} dt \Leftrightarrow dY_t = X_{t} dt ,\quad Y_{0}=y_{0} \end{equation} $Y_t$ is not itself a Markov process; however, $X_t$ and $Y_t$ together comprise a bivariate continuous Markov process. We wish to find the solutions $X_t$ and $Y_t$ to the coupled time-evolution equations: \begin{equation}\label{eq018} \begin{cases} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t\ dY_t = X_{t} dt \end{cases} \end{equation}
We simulate a flow of $1000$ trajectories of $(X_{t},Y_{t})$, with integration step size $\Delta t = 0.01$, and using second Milstein method.
set.seed(1234, kind = "L'Ecuyer-CMRG") x0=5;y0=0 mu=3;sigma=0.5 fx <- expression(-(x/mu),x) gx <- expression(sqrt(sigma),0) mod2d <- snssde2d(drift=fx,diffusion=gx,Dt=0.01,M=1000,x0=c(x0,y0),method="smilstein") mod2d
The summary of the results of mod2d
at time $t=10$ of class snssde2d()
is given by:
summary(mod2d, at = 10)
For plotting in time (or in plane) using the command plot
(plot2d
), the results of the simulation are shown in Figure 3.
## in time plot(mod2d) ## in plane (O,X,Y) plot2d(mod2d,type="n") points2d(mod2d,col=rgb(0,100,0,50,maxColorValue=255), pch=16)
knitr::include_graphics(c("Figures/fig09.png","Figures/fig009.png"))
Hence we can just make use of the rsde2d()
function to build our random number for $(X_{t},Y_{t})$ at time $t = 10$.
out <- rsde2d(object = mod2d, at = 10) head(out,n=3)
The density of $X_t$ and $Y_t$ at time $t=10$ are reported using dsde2d()
function, see e.g. Figure 4: the marginal density of $X_t$ and $Y_t$ at time $t=10$. For 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
, see:
## the marginal density denM <- dsde2d(mod2d,pdf="M",at =10) plot(denM, main="Marginal Density") ## the Joint density denJ <- dsde2d(mod2d, pdf="J", n=100,at =10) plot(denJ,display="contour",main="Bivariate Transition Density at time t=10")
knitr::include_graphics(c("Figures/fig1001.png","Figures/fig1002.png"))
A $3$D plot of the transition density at $t=10$ obtained with:
plot(denJ,display="persp",main="Bivariate Transition Density at time t=10")
knitr::include_graphics(c("Figures/fig1003.png"))
We approximate the bivariate transition density over the set transition horizons $t\in [1,10]$ by $\Delta t = 0.005$ using the code:
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)) }
The Van der Pol (1922) equation is an ordinary differential equation that can be derived from the Rayleigh differential equation by differentiating and setting $\dot{x}=y$, see Naess and Hegstad (1994); Leung (1995) and for more complex dynamics in Van-der-Pol equation see Jing et al. (2006). It is an equation describing self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given by: \begin{equation}\label{eq:12} \ddot{X}-\mu (1-X^{2}) \dot{X} + X = 0 \end{equation} where $x$ is the position coordinate (which is a function of the time $t$), and $\mu$ is a scalar parameter indicating the nonlinearity and the strength of the damping, to simulate the deterministic equation see Grayling (2014) for more details. Consider stochastic perturbations of the Van-der-Pol equation, and random excitation force of such systems by White noise $\xi_{t}$, with delta-type correlation function $\text{E}(\xi_{t}\xi_{t+h})=2\sigma \delta (h)$ \begin{equation}\label{eq:13} \ddot{X}-\mu (1-X^{2}) \dot{X} + X = \xi_{t}, \end{equation} where $\mu > 0$ . It's solution cannot be obtained in terms of elementary functions, even in the phase plane. The White noise $\xi_{t}$ is formally derivative of the Wiener process $W_{t}$. The representation of a system of two first order equations follows the same idea as in the deterministic case by letting $\dot{x}=y$, from physical equation we get the above system: \begin{equation}\label{eq:14} \begin{cases} \dot{X} = Y \ \dot{Y} = \mu \left(1-X^{2}\right) Y - X + \xi_{t} \end{cases} \end{equation} The system can mathematically explain by a Stratonovitch equations: \begin{equation}\label{eq:15} \begin{cases} dX_{t} = Y_{t} dt \ dY_{t} = \left(\mu (1-X^{2}{t}) Y{t} - X_{t}\right) dt + 2 \sigma \circ dW_{2,t} \end{cases} \end{equation}
Implemente in R as follows, with integration step size $\Delta t = 0.01$ and using stochastic Runge-Kutta methods 1-stage.
set.seed(1234, kind = "L'Ecuyer-CMRG") mu = 4; sigma=0.1 fx <- expression( y , (mu*( 1-x^2 )* y - x)) gx <- expression( 0 ,2*sigma) mod2d <- snssde2d(drift=fx,diffusion=gx,N=10000,Dt=0.01,type="str",method="rk1")
For plotting (back in time) using the command plot
, and plot2d
in plane the results of the simulation are shown in Figure 6.
plot(mod2d,ylim=c(-8,8)) ## back in time plot2d(mod2d) ## in plane (O,X,Y)
knitr::include_graphics(c("Figures/fig1004.png","Figures/fig1005.png"))
Consider a system of stochastic differential equations:
\begin{equation}\label{eq:115} \begin{cases} dX_{t} = \mu X_{t} dt + X_{t}\sqrt{Y_{t}} dB_{1,t}\ dY_{t} = \nu (\theta-Y_{t}) dt + \sigma \sqrt{Y_{t}} dB_{2,t} \end{cases} \end{equation}
Conditions to ensure positiveness of the volatility process are that $2\nu \theta > \sigma^2$, and the two Brownian motions $(B_{1,t},B_{2,t})$ are correlated. $\Sigma$ to describe the correlation structure, for example: $$ \Sigma= \begin{pmatrix} 1 & 0.3 \ 0.3 & 2 \end{pmatrix} $$
set.seed(1234, kind = "L'Ecuyer-CMRG") mu = 1.2; sigma=0.1;nu=2;theta=0.5 fx <- expression( mu*x ,nu*(theta-y)) gx <- expression( x*sqrt(y) ,sigma*sqrt(y)) Sigma <- matrix(c(1,0.3,0.3,2),nrow=2,ncol=2) # correlation matrix HM <- snssde2d(drift=fx,diffusion=gx,Dt=0.001,x0=c(100,1),corr=Sigma,M=1000) HM
Hence we can just make use of the rsde2d()
function to build our random number for $(X_{t},Y_{t})$ at time $t = 1$.
out <- rsde2d(object = HM, at = 1) head(out,n=3)
The density of $X_t$ and $Y_t$ at time $t=1$ are reported using dsde2d()
function. See:
denJ <- dsde2d(HM,pdf="J",at =1,lims=c(-100,900,0.4,0.75)) plot(denJ,display="contour",main="Bivariate Transition Density at time t=10") plot(denJ,display="persp",main="Bivariate Transition Density at time t=10")
snssde3d()
The following $3$-dimensional SDE's with a vector of drift and matrix of diffusion coefficients:
Ito 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}
$(W_{1,t},W_{2,t},W_{3,t})$ are three independents standard Wiener process if corr = NULL
. 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
.corr
the correlation structure of three standard Wiener process $(W_{1,t},W_{2,t},W_{2,t})$; must be a real symmetric positive-definite square matrix of dimension $3$ (default: corr=NULL
).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"
).Assume that we want to describe the following SDE's (3D) in Ito form: \begin{equation}\label{eq0166} \begin{cases} dX_t = 4 (-1-X_{t}) Y_{t} dt + 0.2 dW_{1,t}\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t}\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t} \end{cases} \end{equation} with $(W_{1,t},W_{2,t},W_{3,t})$ are three indpendant standard Wiener process.
We simulate a flow of $1000$ trajectories, with integration step size $\Delta t = 0.001$.
set.seed(1234, kind = "L'Ecuyer-CMRG") fx <- expression(4*(-1-x)*y , 4*(1-y)*x , 4*(1-z)*y) gx <- rep(expression(0.2),3) mod3d <- snssde3d(x0=c(x=2,y=-2,z=-2),drift=fx,diffusion=gx,M=1000) mod3d
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
:
s = 1 mean(mod3d, at = s) moment(mod3d, at = s , center = TRUE , order = 2) ## variance Median(mod3d, at = s) Mode(mod3d, at = s) quantile(mod3d , at = s) kurtosis(mod3d , at = s) skewness(mod3d , at = s) cv(mod3d , at = s ) min(mod3d , at = s) max(mod3d , at = s) moment(mod3d, at = s , center= TRUE , order = 4) moment(mod3d, at = s , center= FALSE , order = 4)
The summary of the results of mod3d
at time $t=1$ of class snssde3d()
is given by:
summary(mod3d, at = t)
For plotting (back in time) using the command plot
, and plot3D
in space the results of the simulation are shown in Figure 7.
plot(mod3d,union = TRUE) ## back in time plot3D(mod3d,display="persp") ## in space (O,X,Y,Z)
knitr::include_graphics(c("Figures/fig10.png","Figures/fig11.png"))
Hence we can just make use of the rsde3d()
function to build our random number for $(X_{t},Y_{t},Z_{t})$ at time $t = 1$.
out <- rsde3d(object = mod3d, at = 1) head(out,n=3)
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 8.
den <- dsde3d(mod3d,pdf="M",at =1) 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.)
denJ <- dsde3d(mod3d,pdf="J") plot(denJ,display="rgl")
If we assume that $U_w( x , y , z , t )$, $V_w( x , y , z , t )$ and $S_w( x , y , z , t )$ are neglected and the dispersion coefficient $D( x , y , z )$ is constant. A system becomes (see Boukhetala,1996):
\begin{eqnarray}\label{eq19}
% \nonumber to remove numbering (before each equation)
\begin{cases}
dX_t = \left(\frac{-K X_{t}}{\sqrt{X^{2}{t} + Y^{2}{t} + Z^{2}{t}}}\right) dt + \sigma dW{1,t} \nonumber\
dY_t = \left(\frac{-K Y_{t}}{\sqrt{X^{2}{t} + Y^{2}{t} + Z^{2}{t}}}\right) dt + \sigma dW{2,t} \
dZ_t = \left(\frac{-K Z_{t}}{\sqrt{X^{2}{t} + Y^{2}{t} + Z^{2}{t}}}\right) dt + \sigma dW{3,t} \nonumber
\end{cases}
\end{eqnarray}
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
.
set.seed(1234, kind = "L'Ecuyer-CMRG") K = 4; s = 1; sigma = 0.2 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)) ) gx <- rep(expression(sigma),3) mod3d <- snssde3d(drift=fx,diffusion=gx,N=10000,x0=c(x=1,y=1,z=1))
The results of simulation (3D) are shown in Figure 9:
plot3D(mod3d,display="persp",col="blue")
Next is an example of one-dimensional SDE driven by three correlated Wiener process ($B_{1,t}$,$B_{2,t}$,$B_{3,t}$), as follows:
\begin{equation}\label{eq20}
dX_{t} = B_{1,t} dt + B_{2,t} dB_{3,t}
\end{equation}
with:
$$
\Sigma=
\begin{pmatrix}
1 & 0.2 &0.5\
0.2 & 1 & -0.7 \
0.5 &-0.7&1
\end{pmatrix}
$$
To simulate the solution of the process $X_t$, we make a transformation to a system of three equations as follows:
\begin{eqnarray}\label{eq21}
\begin{cases}
% \nonumber to remove numbering (before each equation)
dX_t = Y_{t} dt + Z_{t} dB_{3,t} \nonumber\
dY_t = dB_{1,t} \
dZ_t = dB_{2,t} \nonumber
\end{cases}
\end{eqnarray}
run by calling the function snssde3d()
to produce a simulation of the solution, with $\mu = 1$ and $\sigma = 1$.
set.seed(1234, kind = "L'Ecuyer-CMRG") fx <- expression(y,0,0) gx <- expression(z,1,1) Sigma <-matrix(c(1,0.2,0.5,0.2,1,-0.7,0.5,-0.7,1),nrow=3,ncol=3) modtra <- snssde3d(drift=fx,diffusion=gx,M=1000,corr=Sigma) modtra
The histogram and kernel density of $X_t$ at time $t=1$ are reported using rsde3d()
function, and we calculate emprical variance-covariance matrix $C(s,t)=\text{Cov}(X_{s},X_{t})$, see e.g. Figure 10.
X <- rsde3d(modtra,at=1)$x MASS::truehist(X,xlab = expression(X[t==1]));box() lines(density(X),col="red",lwd=2) legend("topleft",c("Distribution histogram","Kernel Density"),inset =.01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"), lwd=2,cex=0.8) ## Cov-Matrix color.palette=colorRampPalette(c('white','green','blue','red')) filled.contour(time(modtra), time(modtra), cov(t(modtra$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 = ""))
knitr::include_graphics(c("Figures/fig1007.png","Figures/fig1006.png"))
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 (2020). "Performing Parallel Monte Carlo and Moment Equations Methods for Itô and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc". Journal of Statistical Software, 96(2), 1--82. https://doi.org/10.18637/jss.v096.i02
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