---
title: "Monte-Carlo Simulations and Analysis of Stochastic Differential Equations"
author: 
- A.C. Guidoum^[Department of Mathematics and Computer Science, Faculty of Sciences and Technology, University of Tamanghasset, Algeria, E-mail  (acguidoum@univ-tam.dz)] and K. Boukhetala^[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)]
date: "`r Sys.Date()`"
output: 
  knitr:::html_vignette:
    toc: yes
vignette: >
  %\VignetteIndexEntry{Monte-Carlo Simulation and Analysis of Stochastic Differential Equations}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, echo = F, message = F, results = 'hide',screenshot.force=FALSE}
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:

- The `drift` and `diffusion` coefficients as R expressions that depend on the state variable `x` and time variable `t`.
- The number of simulation steps `N=1000` (by default: `N=1000`).
- The number of the solution trajectories to be simulated by `M=1000` (by default: `M=1`).
- The initial conditions `t0=0`, `x0=10` and end time `T=1` (by default: `t0=0`, `x0=0` and `T=1`).
- The integration step size `Dt=0.001` (by default: `Dt=(T-t0)/N`).
- The choice of process types by the argument `type="ito"` for Ito or `type="str"` for Stratonovich (by default `type="ito"`).
- The numerical method to be used by `method` (by default `method="euler"`).

```{r}
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 $t$:

* The expected value $\text{E}(X_{t})$ at time $t$, using the command `mean`.
* The variance $\text{Var}(X_{t})$ at time $t$, using the command `moment` with `order=2` and `center=TRUE`.
* The median $\text{Med}(X_{t})$ at time $t$, using the command `Median`.
* The mode $\text{Mod}(X_{t})$ at time $t$, using the command `Mode`.
* The quartile of $X_{t}$ at time $t$, using the command `quantile`.
* The maximum and minimum of $X_{t}$ at time $t$, using the command  `min` and `max`.
* The skewness and the kurtosis of $X_{t}$ at time $t$, using the command `skewness` and `kurtosis`.
* The coefficient of variation (relative variability) of $X_{t}$ at time $t$, using the command `cv`.
* The central moments up to order $p$ of $X_{t}$ at time $t$, using the command `moment`.
* The empirical $\alpha \%$ confidence interval of expected value $\text{E}(X_{t})$ at time $t$ (from the $2.5th$ to the $97.5th$ percentile), using the command `bconfint`.
* The result summaries of the results of Monte-Carlo simulation at time $t$, using the command `summary`.

The summary of the results of `mod1` and `mod2` at time $t=1$ of class `snssde1d()` is given by:
```{r}
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 $t = 1$.
```{r}
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:
```{r 01,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
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 001, echo=FALSE, fig.cap='Approximate transitional density for $X_{t}|X_{0}$ 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 02,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
## 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)
```

```{r 100, echo=FALSE, fig.cap='mod1: Ito and mod2: Stratonovich  ', fig.env='figure*',fig.width=7,fig.height=7}
knitr::include_graphics(c("Figures/fig07.png","Figures/fig08.png"))
```

[Return to snssde1d()](#snssde1d)

# `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:

- The `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`).
- The number of simulation steps `N` (default: `N=1000`).
- The number of the solution trajectories to be simulated by `M` (default: `M=1`).
- The initial conditions `t0`, `x0` and end time `T` (default: `t0=0`, `x0=c(0,0)` and `T=1`).
- The integration step size `Dt` (default: `Dt=(T-t0)/N`).
- The choice of process types by the argument `type="ito"` for Ito or `type="str"` for Stratonovich (default `type="ito"`).
- The numerical method to be used by `method` (default `method="euler"`).

## Ornstein-Uhlenbeck process and its integral

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.

```{r}
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:
```{r,eval=FALSE, include=TRUE}
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.

```{r 03,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
## 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)
```

```{r 102, echo=FALSE, fig.cap=' Ornstein-Uhlenbeck process and its integral ', fig.env='figure*',fig.width=10,fig.height=10}
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 $t = 10$.
```{r}
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:


```{r 04,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
## 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")
```

```{r 1002, echo=FALSE, fig.cap='Marginal and Joint density at time t=10 ', fig.env='figure*',fig.width=10,fig.height=10}
knitr::include_graphics(c("Figures/fig1001.png","Figures/fig1002.png"))
```

A $3$D plot of the transition density at $t=10$ obtained with: 

```{r 07,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
plot(denJ,display="persp",main="Bivariate Transition Density at time t=10")
```

```{r 10002, echo=FALSE, fig.cap='Marginal and Joint density at time t=10 ', fig.env='figure*',fig.width=10,fig.height=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:

```{r ,eval=FALSE, include=TRUE}
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))
}
```

[Return to snssde2d()](#snssde2d)

## The stochastic Van-der-Pol equation

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.

```{r}
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.

```{r 9,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
plot(mod2d,ylim=c(-8,8))   ## back in time
plot2d(mod2d)              ## in plane (O,X,Y)
```

```{r 100002, echo=FALSE, fig.cap='The stochastic Van-der-Pol equation', fig.env='figure*',fig.width=10,fig.height=10}
knitr::include_graphics(c("Figures/fig1004.png","Figures/fig1005.png"))
```

[Return to snssde2d()](#snssde2d)

## The Heston Model

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:
$$
\Sigma=
\begin{pmatrix}
1 & 0.3 \\
0.3 & 2
\end{pmatrix}
$$

```{r}
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 $t = 1$.
```{r}
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:

```{r,eval=FALSE, include=TRUE}
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")
```

[Return to snssde2d()](#snssde2d)

# `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:

- The `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`).
- The number of simulation steps `N` (default: `N=1000`).
- The number of the solution trajectories to be simulated by `M` (default: `M=1`).
- The initial conditions `t0`, `x0` and end time `T` (default: `t0=0`, `x0=c(0,0,0)` and `T=1`).
- The integration step size `Dt` (default: `Dt=(T-t0)/N`).
- The choice of process types by the argument `type="ito"` for Ito or `type="str"` for Stratonovich (default `type="ito"`).
- The numerical method to be used by `method` (default `method="euler"`).

## Basic example

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 $\Delta t = 0.001$.

```{r}
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`:

```{r,eval=FALSE, include=TRUE}
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:
```{r,eval=FALSE, include=TRUE}
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.

```{r 10,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
plot(mod3d,union = TRUE)         ## back in time
plot3D(mod3d,display="persp")    ## in space (O,X,Y,Z)
```

```{r 103, echo=FALSE, fig.cap=' Flow of $1000$ trajectories of $(X_t ,Y_t ,Z_t)$ ', fig.env='figure*',fig.width=7,fig.height=7}
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 $t = 1$.
```{r}
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.


```{r 11,fig.env='figure*', fig.cap=' Marginal density of $X_t$, $Y_t$ and $Z_t$ at time $t=1$ ',fig.width=3.5,fig.height=3.5}
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](https://cran.r-project.org/package=sm) or  [ks](https://cran.r-project.org/package=ks).)

```{r 111,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE,fig.width=5,fig.height=5}
denJ <- dsde3d(mod3d,pdf="J")
plot(denJ,display="rgl")
```

[Return to snssde3d()](#snssde3d)

## Attractive model for 3D diffusion processes

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`.

```{r}
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:

```{r 12,fig.env='figure*',fig.width=3.5,fig.height=3.5, fig.cap=' Attractive model for 3D diffusion processes '}
plot3D(mod3d,display="persp",col="blue")
```

[Return to snssde3d()](#snssde3d)

## Transformation of an SDE one-dimensional

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 $\sigma = 1$.

```{r}
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.

```{r 14, fig.cap='  ', fig.env='figure*',eval=FALSE, include=TRUE}
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 = ""))
```

```{r 1000002, echo=FALSE, fig.cap='The histogram and kernel density of $X_t$ at time $t=1$. Emprical variance-covariance matrix', fig.env='figure*',fig.width=10,fig.height=10}
knitr::include_graphics(c("Figures/fig1007.png","Figures/fig1006.png"))
```


[Return to snssde3d()](#snssde3d)

# Further reading

1. [`snssdekd()` & `dsdekd()` & `rsdekd()`- Monte-Carlo Simulation and Analysis of Stochastic Differential Equations](snssde.html).
2. [`bridgesdekd()` & `dsdekd()` & `rsdekd()` - Constructs and Analysis of Bridges Stochastic Differential Equations](bridgesde.html).
3. [`fptsdekd()` & `dfptsdekd()` - Monte-Carlo Simulation and Kernel Density Estimation of First passage time](fptsde.html).
4. [`MCM.sde()` & `MEM.sde()` - Parallel Monte-Carlo and Moment Equations for SDEs](mcmsde.html).
5. [`TEX.sde()` - Converting Sim.DiffProc Objects to LaTeX](sdetotex.html).
6. [`fitsde()` - Parametric Estimation of 1-D Stochastic Differential Equation](fitsde.html).




# References

1. 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.

2. 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

.
