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Monte-Carlo Simulations and Analysis of Stochastic Differential Equations"
In Sim.DiffProc: Simulation of Diffusion Processes
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").
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:
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:
- 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.
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 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.
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"))
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:
$$
\Sigma=
\begin{pmatrix}
1 & 0.3 \
0.3 & 2
\end{pmatrix}
$$
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:
- 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 $\Delta t = 0.001$.
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")
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.
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")
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 $\sigma = 1$.
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"))
Further reading
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.
References
-
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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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
driftanddiffusioncoefficients as R expressions that depend on the state variablexand time variablet. - 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=10and end timeT=1(by default:t0=0,x0=0andT=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 ortype="str"for Stratonovich (by defaulttype="ito"). - The numerical method to be used by
method(by defaultmethod="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 $t$:
- 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
momentwithorder=2andcenter=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
minandmax. - The skewness and the kurtosis of $X_{t}$ at time $t$, using the command
skewnessandkurtosis. - 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:
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:
- The
drift(2d) anddiffusion(2d) coefficients as R expressions that depend on the state variablex,yand time variablet. corrthe 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,x0and end timeT(default:t0=0,x0=c(0,0)andT=1). - The integration step size
Dt(default:Dt=(T-t0)/N). - The choice of process types by the argument
type="ito"for Ito ortype="str"for Stratonovich (defaulttype="ito"). - The numerical method to be used by
method(defaultmethod="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.
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 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.
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"))
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: $$ \Sigma= \begin{pmatrix} 1 & 0.3 \ 0.3 & 2 \end{pmatrix} $$
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:
- The
drift(3d) anddiffusion(3d) coefficients as R expressions that depend on the state variablesx,y,zand time variablet. corrthe 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,x0and end timeT(default:t0=0,x0=c(0,0,0)andT=1). - The integration step size
Dt(default:Dt=(T-t0)/N). - The choice of process types by the argument
type="ito"for Ito ortype="str"for Stratonovich (defaulttype="ito"). - The numerical method to be used by
method(defaultmethod="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 $\Delta t = 0.001$.
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")
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.
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")
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 $\sigma = 1$.
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"))
Further reading
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.
References
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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.
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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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