# snssde1d()

Assume that we want to describe the following SDE:

Ito form1:

$$\label{eq:05} dX_{t} = \frac{1}{2}\theta^{2} X_{t} dt + \theta X_{t} dW_{t},\qquad X_{0}=x_{0} > 0$$ Stratonovich form: $$\label{eq:06} dX_{t} = \frac{1}{2}\theta^{2} X_{t} dt +\theta X_{t} \circ dW_{t},\qquad X_{0}=x_{0} > 0$$

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=500 (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").
theta = 0.5
f <- expression( (0.5*theta^2*x) )
g <- expression( theta*x )
mod1 <- snssde1d(drift=f,diffusion=g,x0=10,M=500,type="ito") # Using Ito
mod2 <- snssde1d(drift=f,diffusion=g,x0=10,M=500,type="str") # Using Stratonovich
mod1
## Ito Sde 1D:
##  | dX(t) = (0.5 * theta^2 * X(t)) * dt + theta * X(t) * dW(t)
## Method:
##  | Euler scheme of order 0.5
## Summary:
##  | Size of process   | N  = 1000.
##  | Number of simulation  | M  = 500.
##  | Initial value     | x0 = 10.
##  | Time of process   | t in [0,1].
##  | Discretization    | Dt = 0.001.
mod2
## Stratonovich Sde 1D:
##  | dX(t) = (0.5 * theta^2 * X(t)) * dt + theta * X(t) o dW(t)
## Method:
##  | Euler scheme of order 0.5
## Summary:
##  | Size of process   | N  = 1000.
##  | Number of simulation  | M  = 500.
##  | Initial value     | x0 = 10.
##  | Time of process   | t in [0,1].
##  | Discretization    | Dt = 0.001.

Using Monte-Carlo simulations, the following statistical measures (S3 method) for class snssde1d() can be approximated for the $$X_{t}$$ process at any time $$t$$:

• The expected value $$\text{E}(X_{t})$$ at time $$t$$, using the command mean.
• The variance $$\text{Var}(X_{t})$$ at time $$t$$.
• The median $$\text{Med}(X_{t})$$ at time $$t$$, using the command median.
• The quartile of $$X_{t}$$ at time $$t$$, using the command quantile.
• The skewness and the kurtosis of $$X_{t}$$ at time $$t$$, using the command skewness and kurtosis.
• The central moments up to order $$p$$ of $$X_{t}$$ at time $$t$$, using the command moment.
• The empirical $$\alpha \%$$ confidence interval of $$X_{t}$$ at time $$t$$ (from the $$2.5th$$ to the $$97.5th$$ percentile), using the command bconfint.

The summary of the results of mod1 and mod2 at time $$t=1$$ of class snssde1d() is given by:

summary(mod1, at = 1)
##
##  Monte-Carlo Statistics for X(t) at time t = 1
##
## Mean                   11.16936
## Variance               39.37456
## Median                  9.90441
## First quartile          6.98896
## Third quartile         13.61834
## Skewness                2.28032
## Kurtosis               13.13066
## Moment of order 3     563.40394
## Moment of order 4   20357.20362
## Moment of order 5  789848.03151
## Int.conf Inf (95%)      3.82463
## Int.conf Sup (95%)     27.19899
summary(mod2, at = 1)
##
##  Monte-Carlo Statistics for X(t) at time t = 1
##
## Mean                   9.87291
## Variance              27.68282
## Median                 8.60328
## First quartile         6.03144
## Third quartile        12.21241
## Skewness               1.48501
## Kurtosis               6.15234
## Moment of order 3    216.29389
## Moment of order 4   4714.77643
## Moment of order 5  87075.26325
## Int.conf Inf (95%)     3.23756
## Int.conf Sup (95%)    23.18922

For each SDE type and for each numerical scheme the histograms of $$X_t$$ at time $$t=1$$ are reported using rsde1d() function, see e.g. Figure 1: the empirical distribution of $$X_t$$ ($$X_{t}^{\text{mod1}}$$ and $$X_{t}^{\text{mod2}}$$) is clearly asymmetric, and in fact the conditional distribution of $$X_t | X_0$$ is log-normal.

x1 <- rsde1d(mod1,at=1) # X(t) at time t = 1 (Ito SDE)
x2 <- rsde1d(mod2,at=1) # X(t) at time t = 1 (Stratonovich SDE)
require(MASS)
truehist(x1,xlab = expression(X[t==1]^mod1));box()
legend("topright",c("Distribution histogram","Exact Log-normal density"),inset =.01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"),lwd=2,cex=0.8)
truehist(x2,xlab = expression(X[t==1]^mod2));box()
legend("topright",c("Distribution histogram","Exact Log-normal density"),inset = .01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"),lwd=2,cex=0.8)

In Figure 2, we present the flow of trajectories, the mean path (red lines) of solution of and , with their empirical $$95\%$$ confidence bands, that is to say from the $$2.5th$$ to the $$97.5th$$ percentile for each observation at time $$t$$ (blue lines):

plot(mod1,plot.type="single",ylab=expression(X^mod1))
lines(time(mod1),mean(mod1),col=2,lwd=2)
lines(time(mod1),bconfint(mod1,level=0.95)[,1],col=4,lwd=2)
lines(time(mod1),bconfint(mod1,level=0.95)[,2],col=4,lwd=2)
legend("topleft",c("mean path",paste("bound of",95,"% confidence")),col=c(2,4),lwd=2,cex=0.8)
plot(mod2,plot.type="single",ylab=expression(X^mod2))
lines(time(mod2),mean(mod2),col=2,lwd=2)
lines(time(mod2),bconfint(mod2,level=0.95)[,1],col=4,lwd=2)
lines(time(mod2),bconfint(mod2,level=0.95)[,2],col=4,lwd=2)
legend("topleft",c("mean path",paste("bound of",95,"% confidence")),col=c(2,4),lwd=2,cex=0.8)

# snssde2d()

The following $$2$$-dimensional SDE’s with a vector of drift and a diagonal matrix of diffusion coefficients:

Ito form: $$\label{eq:09} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) dW_{2,t} \end{cases}$$ Stratonovich form: $$\label{eq:10} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) \circ dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) \circ dW_{2,t} \end{cases}$$

$$W_{1,t}$$ and $$W_{2,t}$$ is a two independent standard Wiener process. To simulate $$2d$$ models using snssde2d() function we need to specify:

• The driftx, drifty and diffx, diffy, respectively, the drift and diffusion coefficients of $$X_t$$ and $$Y_t$$, as R expressions that depend on the state variables x, y and time variable t.
• 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, y0 and end time T (default: t0=0, x0=0, y0=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: $$\label{eq016} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t,\quad X_{0}=x_{0}$$ 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: $$\label{eq017} Y_{t} = Y_{0}+\int X_{t} dt \Leftrightarrow dY_t = X_{t} dt ,\quad Y_{0}=y_{0}$$ $$Y_t$$ 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: $$\label{eq018} \begin{cases} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t\\ dY_t = X_{t} dt \end{cases}$$

We simulate a flow of $$500$$ trajectories of $$(X_{t},Y_{t})$$, with integration step size $$\Delta t = 0.01$$, and using second Milstein method.

x0=5;y0=0
mu=3;sigma=0.5
fx <- expression(-(x/mu)) ; gx <- expression(sqrt(sigma))
fy <- expression(x) ; gy <- expression(0)
mod2d <- snssde2d(driftx=fx,diffx=gx,drifty=fy,diffy=gy,Dt=0.01,M=500,x0=x0,y0=y0,method="smilstein")
mod2d
## Ito Sde 2D:
##  | dX(t) = -(X(t)/mu) * dt + sqrt(sigma) * dW1(t)
##  | dY(t) = X(t) * dt + 0 * dW2(t)
## Method:
##  | Second Milstein scheme of order 1.5
## Summary:
##  | Size of process   | N  = 1000.
##  | Number of simulation  | M  = 500.
##  | Initial values    | (x0,y0) = (5,0).
##  | Time of process   | t in [0,10].
##  | Discretization    | Dt = 0.01.
summary(mod2d)
##
##  Monte-Carlo Statistics for (X(t),Y(t)) at time t = 10
##                           X          Y
## Mean                0.16818   14.13934
## Variance            0.84339   24.77511
## Median              0.16854   14.21884
## First quartile     -0.44738   10.66608
## Third quartile      0.79839   17.33331
## Skewness            0.00165    0.14108
## Kurtosis            3.13997    3.00287
## Moment of order 3   0.00128   17.39713
## Moment of order 4   2.23345 1843.18116
## Moment of order 5   0.11872 4194.18183
## Int.conf Inf (95%) -1.67419    5.01142
## Int.conf Sup (95%)  1.82445   24.19501

For plotting (back in time) using the command plot, the results of the simulation are shown in Figure 3.

plot(mod2d)

Take note of the well known result, which can be derived from either this equations. That for any $$t > 0$$ the OU process $$X_t$$ and its integral $$Y_t$$ will be the normal distribution with mean and variance given by: $\begin{cases} \text{E}(X_{t}) =x_{0} e^{-t/\mu} &\text{and}\quad\text{Var}(X_{t})=\frac{\sigma \mu}{2} \left (1-e^{-2t/\mu}\right )\\ \text{E}(Y_{t}) = y_{0}+x_{0}\mu \left (1-e^{-t/\mu}\right ) &\text{and}\quad\text{Var}(Y_{t})=\sigma\mu^{3}\left (\frac{t}{\mu}-2\left (1-e^{-t/\mu}\right )+\frac{1}{2}\left (1-e^{-2t/\mu}\right )\right ) \end{cases}$

For each SDE type and for each numerical scheme the histograms of $$X_t$$ and $$Y_t$$ at time $$t=10$$ are reported using rsde2d() function, see e.g. Figure 4: the empirical distribution of $$X_t$$ and $$Y_t$$.

out <- rsde2d(mod2d,at =10)
summary(out)
##        x                 y
##  Min.   :-2.7560   Min.   :-0.5359
##  1st Qu.:-0.4474   1st Qu.:10.6661
##  Median : 0.1685   Median :14.2188
##  Mean   : 0.1682   Mean   :14.1393
##  3rd Qu.: 0.7984   3rd Qu.:17.3333
##  Max.   : 3.0054   Max.   :30.8989
cov(out) ## Variance-Covariance matrice
##           x         y
## x 0.8433851  2.282733
## y 2.2827333 24.775109
E_x <- function(t) x0*exp(-t/mu)
V_x <- function(t) 0.5*sigma*mu *(1-exp(-2*(t/mu)))
E_y <- function(t) y0+x0*mu*(1-exp(-t/mu))
V_y <- function(t) sigma*mu^3*((t/mu)-2*(1-exp(-t/mu))+0.5*(1-exp(-2*(t/mu))))
truehist(out$x,xlab = expression(X[t==10]));box() curve(dnorm(x,mean=E_x(10) ,sd =sqrt(V_x(10))),col="red",lwd=2,add=TRUE) legend("topright",c("Distribution histogram","Exact Normal density"),inset = .01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"),lwd=2,cex=0.8) truehist(out$y,xlab = expression(Y[t==10]));box()
legend("topright",c("Distribution histogram","Exact Normal density"),inset = .01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"),lwd=2,cex=0.8)

Created using ggplot2 package plotted in (x, y)-space with dim = 2. A contour plot of density obtained from a realization of system $$(X_{t},Y_{t})$$ at time t=10.

library(ggplot2)
m <- ggplot(out, aes(x = x, y = y))
m + stat_density_2d(aes(fill = ..level..), geom = "polygon")

A $$3$$D plot of the density obtained with sm, from a realization of system $$(X_{t},Y_{t})$$ at time t=10.

library(sm)
sm.density(out,display="persp")

## 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: $$\label{eq:12} \ddot{X}-\mu (1-X^{2}) \dot{X} + X = 0$$ 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 $$\text{E}(\xi_{t}\xi_{t+h})=2\sigma \delta (h)$$ $$\label{eq:13} \ddot{X}-\mu (1-X^{2}) \dot{X} + X = \xi_{t},$$ 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: $$\label{eq:14} \begin{cases} \dot{X} = Y \\ \dot{Y} = \mu \left(1-X^{2}\right) Y - X + \xi_{t} \end{cases}$$ The system can mathematically explain by a Stratonovitch equations: $$\label{eq:15} \begin{cases} dX_{t} = Y_{t} dt \\ dY_{t} = \left(\mu (1-X^{2}_{t}) Y_{t} - X_{t}\right) dt + 2 \sigma \circ dW_{2,t} \end{cases}$$

Implemente in R as follows, with integration step size $$\Delta t = 0.01$$ and using stochastic Runge-Kutta methods 1-stage.

mu = 4; sigma=0.1
fx <- expression( y ) ; gx <- expression( 0 )
fy <- expression( (mu*( 1-x^2 )* y - x) ) ; gy <- expression( 2*sigma)
mod2d <- snssde2d(driftx=fx,diffx=gx,drifty=fy,diffy=gy,N=10000,
Dt=0.01,type="str",method="rk1")
mod2d
## Stratonovich Sde 2D:
##  | dX(t) = Y(t) * dt + 0 o dW1(t)
##  | dY(t) = (mu * (1 - X(t)^2) * Y(t) - X(t)) * dt + 2 * sigma o dW2(t)
## Method:
##  | Runge-Kutta method of order 1
## Summary:
##  | Size of process   | N  = 10000.
##  | Number of simulation  | M  = 1.
##  | Initial values    | (x0,y0) = (0,0).
##  | Time of process   | t in [0,100].
##  | Discretization    | Dt = 0.01.
plot2d(mod2d) ## in plane (O,X,Y)
plot(mod2d)   ## back in time

# snssde3d()

The following $$3$$-dimensional SDE’s with a vector of drift and a diagonal matrix of diffusion coefficients:

Ito form: $$\label{eq17} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) dW_{2,t}\\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) dW_{3,t} \end{cases}$$ Stratonovich form: $$\label{eq18} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) \circ dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) \circ dW_{2,t}\\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) \circ dW_{3,t} \end{cases}$$

$$W_{1,t}$$, $$W_{2,t}$$ and $$W_{3,t}$$ is a 3 independent standard Wiener process. To simulate this system using snssde3d() function we need to specify:

• The driftx, drifty, driftz and diffx, diffy, diffz respectively, the drift and diffusion coefficients of $$X_t$$, $$Y_t$$ and $$Z_t$$, as R expressions that depend on the state variables x, y , z and time variable t.
• 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, y0 , z0 and end time T (default: t0=0, x0=0, y0=0, z0=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: $$\label{eq0166} \begin{cases} dX_t = 4 (-1-X_{t}) Y_{t} dt + 0.2 dW_{1,t}\\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t}\\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t} \end{cases}$$

We simulate a flow of 500 trajectories, with integration step size $$\Delta t = 0.001$$.

fx <- expression(4*(-1-x)*y) ; gx <- expression(0.2)
fy <- expression(4*(1-y)*x) ; gy <- expression(0.2)
fz <- expression(4*(1-z)*y) ; gz <- expression(0.2)
mod3d <- snssde3d(x0=2,y0=-2,z0=-2,driftx=fx,diffx=gx,drifty=fy,diffy=gy,
driftz=fz,diffz=gz,N=1000,M=500)
mod3d
## Ito Sde 3D:
##  | dX(t) = 4 * (-1 - X(t)) * Y(t) * dt + 0.2 * dW1(t)
##  | dY(t) = 4 * (1 - Y(t)) * X(t) * dt + 0.2 * dW2(t)
##  | dZ(t) = 4 * (1 - Z(t)) * Y(t) * dt + 0.2 * dW3(t)
## Method:
##  | Euler scheme of order 0.5
## Summary:
##  | Size of process   | N  = 1000.
##  | Number of simulation  | M  = 500.
##  | Initial values    | (x0,y0,z0) = (2,-2,-2).
##  | Time of process   | t in [0,1].
##  | Discretization    | Dt = 0.001.
summary(mod3d)
##
##   Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 1
##                           X       Y        Z
## Mean               -0.79705 0.86360  0.78893
## Variance            0.00962 0.09202  0.01044
## Median             -0.79622 0.84565  0.80279
## First quartile     -0.86482 0.65893  0.72438
## Third quartile     -0.73336 1.06545  0.85533
## Skewness            0.21842 0.22124 -0.68277
## Kurtosis            2.79617 2.87055  4.37500
## Moment of order 3   0.00021 0.00618 -0.00073
## Moment of order 4   0.00026 0.02430  0.00048
## Moment of order 5   0.00002 0.00320 -0.00013
## Int.conf Inf (95%) -0.97424 0.30089  0.58027
## Int.conf Sup (95%) -0.60419 1.48325  0.95674
plot(mod3d,union = TRUE)         ## back in time
plot3D(mod3d,display="persp")    ## in space (O,X,Y,Z)

For each SDE type and for each numerical scheme the histograms of $$X_t$$, $$Y_t$$ and $$Z_t$$ at time $$t=1$$ are reported using rsde3d() function, see e.g. Figure 9: the empirical distribution of $$X_t$$, $$Y_t$$ and $$Z_t$$.

out <- rsde3d(mod3d,at =1)
summary(out)
##        x                 y                  z
##  Min.   :-1.0679   Min.   :-0.04721   Min.   :0.2476
##  1st Qu.:-0.8648   1st Qu.: 0.65893   1st Qu.:0.7244
##  Median :-0.7962   Median : 0.84565   Median :0.8028
##  Mean   :-0.7970   Mean   : 0.86360   Mean   :0.7889
##  3rd Qu.:-0.7334   3rd Qu.: 1.06545   3rd Qu.:0.8553
##  Max.   :-0.4926   Max.   : 1.72731   Max.   :1.0288
cov(out) ## Variance-Covariance matrice
##              x           y            z
## x  0.009617262 -0.01616923 -0.004372636
## y -0.016169228  0.09201527  0.019273766
## z -0.004372636  0.01927377  0.010436016
truehist(out$x,xlab = expression(X[t==1]));box() lines(density(out$x),col="red",lwd=2)
legend("topright",c("Distribution histogram","Kernel Density"),inset =.01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"),lwd=2,cex=0.8)
truehist(out$y,xlab = expression(Y[t==1]));box() lines(density(out$y),col="red",lwd=2)
legend("topright",c("Distribution histogram","Kernel Density"),inset =.01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"),lwd=2,cex=0.8)
truehist(out$z,xlab = expression(Z[t==1]));box() lines(density(out$z),col="red",lwd=2)
legend("topright",c("Distribution histogram","Kernel Density"),inset =.01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"),lwd=2,cex=0.8)

## 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.

K = 4; s = 1; sigma = 0.2
fx <- expression( (-K*x/sqrt(x^2+y^2+z^2)) ) ; gx <- expression(sigma)
fy <- expression( (-K*y/sqrt(x^2+y^2+z^2)) ) ; gy <- expression(sigma)
fz <- expression( (-K*z/sqrt(x^2+y^2+z^2)) ) ; gz <- expression(sigma)
mod3d <- snssde3d(driftx=fx,diffx=gx,drifty=fy,diffy=gy,driftz=fz,diffz=gz,
N=10000,x0=1,y0=1,z0=1)
mod3d
## Ito Sde 3D:
##  | dX(t) = (-K * X(t)/sqrt(X(t)^2 + Y(t)^2 + Z(t)^2)) * dt + sigma * dW1(t)
##  | dY(t) = (-K * Y(t)/sqrt(X(t)^2 + Y(t)^2 + Z(t)^2)) * dt + sigma * dW2(t)
##  | dZ(t) = (-K * Z(t)/sqrt(X(t)^2 + Y(t)^2 + Z(t)^2)) * dt + sigma * dW3(t)
## Method:
##  | Euler scheme of order 0.5
## Summary:
##  | Size of process   | N  = 10000.
##  | Number of simulation  | M  = 1.
##  | Initial values    | (x0,y0,z0) = (1,1,1).
##  | Time of process   | t in [0,1].
##  | Discretization    | Dt = 1e-04.

The results of simulation are shown in Figure 10.

plot3D(mod3d,display="persp",col="blue")

## Transformation of an SDE one-dimensional

Next is an example of one-dimensional SDE driven by three independent Brownian motions ($$W_{1,t}$$,$$W_{2,t}$$,$$W_{3,t}$$), as follows: $$\label{eq20} dX_{t} = \mu W_{1,t} dt + \sigma W_{2,t} dW_{3,t}$$ 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 = \mu Y_{t} dt + \sigma Z_{t} dW_{3,t} \nonumber\\ dY_t = dW_{1,t} \\ dZ_t = dW_{2,t} \nonumber \end{cases} \end{eqnarray}

run by calling the function snssde3d() to produce a simulation of the solution, with $$\mu = 1$$ and $$\sigma = 1$$.

fx <- expression(y) ; gx <- expression(z)
fy <- expression(0) ; gy <- expression(1)
fz <- expression(0) ; gz <- expression(1)
modtra <- snssde3d(driftx=fx,diffx=gx,drifty=fy,diffy=gy,driftz=fz,diffz=gz,M=500)
modtra
## Ito Sde 3D:
##  | dX(t) = Y(t) * dt + Z(t) * dW1(t)
##  | dY(t) = 0 * dt + 1 * dW2(t)
##  | dZ(t) = 0 * dt + 1 * dW3(t)
## Method:
##  | Euler scheme of order 0.5
## Summary:
##  | Size of process   | N  = 1000.
##  | Number of simulation  | M  = 500.
##  | Initial values    | (x0,y0,z0) = (0,0,0).
##  | Time of process   | t in [0,1].
##  | Discretization    | Dt = 0.001.
summary(modtra)
##
##   Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 1
##                           X        Y        Z
## Mean               -0.05248 -0.02474  0.00375
## Variance            0.75657  0.94239  0.97019
## Median             -0.04265 -0.07412  0.04518
## First quartile     -0.56478 -0.74060 -0.69757
## Third quartile      0.52440  0.65477  0.67345
## Skewness           -0.21980  0.18885 -0.12951
## Kurtosis            4.05917  2.94485  2.90492
## Moment of order 3  -0.14464  0.17277 -0.12377
## Moment of order 4   2.32348  2.61532  2.73434
## Moment of order 5  -1.57945  1.12516 -1.04158
## Int.conf Inf (95%) -1.83152 -1.78882 -1.95137
## Int.conf Sup (95%)  1.52386  1.99609  1.82894

the following code produces the result in Figure 11.

plot(modtra$X,plot.type="single",ylab="X") lines(time(modtra),mean(modtra)$X,col=2,lwd=2)
lines(time(modtra),bconfint(modtra,level=0.95)$X[,1],col=4,lwd=2) lines(time(modtra),bconfint(modtra,level=0.95)$X[,2],col=4,lwd=2)
legend("topleft",c("mean path",paste("bound of",95,"% confidence")),col=c(2,4),lwd=2,cex=0.8)

The histograms of $$X_t$$ at time $$t=1$$ are reported using rsde3d() function, see e.g. Figure 12: the empirical distribution of $$X_t$$.

x <- rsde3d(modtra,at=1)\$x
summary(x)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
## -3.46200 -0.56480 -0.04265 -0.05248  0.52440  3.16300
truehist(x,xlab = expression(X[t==1]));box()
lines(density(x),col="red",lwd=2)
legend("topright",c("Distribution histogram","Kernel Density"),inset =.01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"),lwd=2,cex=0.8)

1. The equivalently of $$X_{t}^{\text{mod1}}$$ the following Stratonovich SDE: $$dX_{t} = \theta X_{t} \circ dW_{t}$$.