--- 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