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> x0=5;y0=0
R> mu=3;sigma=0.5
R> fx <- expression(-(x/mu),x)  
R> gx <- expression(sqrt(sigma),0)
R> mod2d <- snssde2d(drift=fx,diffusion=gx,Dt=0.01,M=1000,x0=c(x0,y0),method="smilstein")
R> mod2d

Itô Sde 2D:
    | dX(t) = -(X(t)/mu) * dt + sqrt(sigma) * dW1(t)
    | dY(t) = X(t) * dt + 0 * dW2(t)
Method:
    | Second-order Milstein scheme
Summary:
    | Size of process   | N  = 1001.
    | Number of simulation  | M  = 1000.
    | Initial values    | (x0,y0) = (5,0).
    | Time of process   | t in [0,10].
    | Discretization    | Dt = 0.01.

The following statistical measures (S3 method) for class snssde2d() can be approximated for the \((X_{t},Y_{t})\) process at any time \(t\), for example at=5:

R> s = 5
R> mean(mod2d, at = s)

[1]  0.92679 12.21988

R> moment(mod2d, at = s , center = TRUE , order = 2) ## variance

[1] 0.68889 6.66566

R> Median(mod2d, at = s)

[1]  0.93172 12.17180

R> Mode(mod2d, at = s)

[1]  1.0114 12.2733

R> quantile(mod2d , at = s)

$x
      0%      25%      50%      75%     100% 
-1.76975  0.32673  0.93172  1.47083  3.93978 

$y
     0%     25%     50%     75%    100% 
 1.1376 10.4418 12.1718 13.9680 20.7411 

R> kurtosis(mod2d , at = s)

[1] 3.0766 3.2350

R> skewness(mod2d , at = s)

[1] 0.040533 0.055974

R> cv(mod2d , at = s )

[1] 0.89601 0.21138

R> min(mod2d , at = s)

[1] -1.7698  1.1376

R> max(mod2d , at = s)

[1]  3.9398 20.7411

R> moment(mod2d, at = s , center= TRUE , order = 4)

[1]   1.463 144.020

R> moment(mod2d, at = s , center= FALSE , order = 4)

[1]     5.837 28461.365

The summary of the results of mod2d at time \(t=5\) of class snssde2d() is given by:

R> summary(mod2d, at = s)

    Monte-Carlo Statistics for (X(t),Y(t)) at time t = 5
                        X          Y
Mean              0.92679   12.21988
Variance          0.68958    6.67233
Median            0.93172   12.17180
Mode              1.01144   12.27334
First quartile    0.32673   10.44181
Third quartile    1.47083   13.96801
Minimum          -1.76975    1.13758
Maximum           3.93978   20.74114
Skewness          0.04053    0.05597
Kurtosis          3.07656    3.23495
Coef-variation    0.89601    0.21138
3th-order moment  0.02321    0.96472
4th-order moment  1.46296  144.02011
5th-order moment  0.09333  -55.09065
6th-order moment  5.28267 6095.24983

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

R> plot(mod2d)

Ornstein-Uhlenbeck process and its integral

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: \[ \begin{cases} \text{E}(X_{t}) =x_{0} e^{-t/\mu} &\text{and}\quad\text{Var}(X_{t})=\frac{\sigma \mu}{2} \left (1-e^{-2t/\mu}\right )\\ \text{E}(Y_{t}) = y_{0}+x_{0}\mu \left (1-e^{-t/\mu}\right ) &\text{and}\quad\text{Var}(Y_{t})=\sigma\mu^{3}\left (\frac{t}{\mu}-2\left (1-e^{-t/\mu}\right )+\frac{1}{2}\left (1-e^{-2t/\mu}\right )\right ) \end{cases} \]

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) 
R> head(out,n=10)

          x       y
1  -0.40797 20.2867
2  -0.19133 12.1643
3  -0.75371  9.4854
4   0.30984 13.3197
5  -0.87097 13.6577
6   1.29402 16.1173
7   0.63369 13.9212
8   1.37924 18.2539
9  -0.44067 20.0510
10 -1.38257  3.5238

R> summary(out)

       x                y        
 Min.   :-3.102   Min.   :-2.02  
 1st Qu.:-0.488   1st Qu.:11.03  
 Median : 0.168   Median :13.91  
 Mean   : 0.151   Mean   :14.27  
 3rd Qu.: 0.765   3rd Qu.:17.75  
 Max.   : 2.860   Max.   :33.57  

R> cov(out)

       x       y
x 0.8335  2.2577
y 2.2577 25.4355

Figure 4, show simulation results for moments of system :

R> mx <- apply(mod2d$X,1,mean)
R> my <- apply(mod2d$Y,1,mean)
R> Sx <- apply(mod2d$X,1,var)
R> Sy <- apply(mod2d$Y,1,var)
R> Cxy <- sapply(1:1001,function(i) cov(mod2d$X[i,],mod2d$Y[i,]))
R> out_b <- data.frame(mx,my,Sx,Sy,Cxy)
R> matplot(time(mod2d), out_b, type = "l", xlab = "time", ylab = "",col=2:6,lwd=2,lty=2:6,las=1)
R> legend("topleft",c(expression(hat(E)(X[t]),hat(E)(Y[t]),hat(Var)(X[t]),hat(Var)(Y[t]),hat(Cov)(X[t],Y[t]))),inset = .05,col=2:6,lty=2:6,lwd=2,cex=0.9)

For each SDE type and for each numerical scheme, the density of \(X_t\) and \(Y_t\) at time \(t=10\) are reported using dsde2d() function, see e.g. Figure 5: the marginal density of \(X_t\) and \(Y_t\) at time \(t=10\).

R> denM <- dsde2d(mod2d,pdf="M",at =10)
R> denM

 Marginal density of X(t-t0)|X(t0)=5 at time t = 10

Data: x (1000 obs.);    Bandwidth 'bw' = 0.2064

       x                f(x)        
 Min.   :-3.7208   Min.   :0.00002  
 1st Qu.:-1.9207   1st Qu.:0.00727  
 Median :-0.1207   Median :0.06398  
 Mean   :-0.1207   Mean   :0.13875  
 3rd Qu.: 1.6794   3rd Qu.:0.27940  
 Max.   : 3.4794   Max.   :0.42392  

 Marginal density of Y(t-t0)|Y(t0)=0 at time t = 10

Data: y (1000 obs.);    Bandwidth 'bw' = 1.133

       y               f(y)         
 Min.   :-5.422   Min.   :0.000004  
 1st Qu.: 5.177   1st Qu.:0.000781  
 Median :15.776   Median :0.008865  
 Mean   :15.776   Mean   :0.023565  
 3rd Qu.:26.375   3rd Qu.:0.046913  
 Max.   :36.973   Max.   :0.084134  

R> plot(denM, main="Marginal Density")

Created using dsde2d() 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.

R> denJ <- dsde2d(mod2d, pdf="J", n=100,at =10)
R> denJ

 Joint density of (X(t-t0),Y(t-t0)|X(t0)=5,Y(t0)=0) at time t = 10

Data: (x,y) (2 x 1000 obs.);

       x                  y              f(x,y)        
 Min.   :-3.10162   Min.   :-2.022   Min.   :0.000000  
 1st Qu.:-1.61115   1st Qu.: 6.877   1st Qu.:0.000028  
 Median :-0.12068   Median :15.776   Median :0.000690  
 Mean   :-0.12068   Mean   :15.776   Mean   :0.004603  
 3rd Qu.: 1.36980   3rd Qu.:24.675   3rd Qu.:0.004779  
 Max.   : 2.86027   Max.   :33.574   Max.   :0.037396  

R> plot(denJ,display="contour",main="Bivariate Transition Density at time t=10")
R> plot(denJ,display="image",main="Bivariate Transition Density at time t=10")

A \(3\)D plot of the transition density at \(t=10\) obtained with:

R> plot(denJ,main="Bivariate Transition Density at time t=10")

We approximate the bivariate transition density over the set transition horizons \(t\in [1,10]\) by \(\Delta t = 0.005\) using the code:

R> 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()

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> mu = 4; sigma=0.1
R> fx <- expression( y ,  (mu*( 1-x^2 )* y - x)) 
R> gx <- expression( 0 ,2*sigma)
R> mod2d <- snssde2d(drift=fx,diffusion=gx,N=10000,Dt=0.01,type="str",method="rk1")
R> 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 with order 1
Summary:
    | Size of process   | N  = 10001.
    | Number of simulation  | M  = 1.
    | Initial values    | (x0,y0) = (0,0).
    | Time of process   | t in [0,100].
    | Discretization    | Dt = 0.01.

For plotting (back in time) using the command plot, and plot2d in plane the results of the simulation are shown in Figure 8.

R> plot2d(mod2d) ## in plane (O,X,Y)
R> plot(mod2d)   ## back in time

Return to snssde2d()

Basic example

Assume that we want to describe the following SDE’s (3D) in It? form: \[\begin{equation}\label{eq0166} \begin{cases} dX_t = 4 (-1-X_{t}) Y_{t} dt + 0.2 dW_{1,t}\\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t}\\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t} \end{cases} \end{equation}\] We simulate a flow of \(10000\) trajectories, with integration step size \(\Delta t = 0.001\).

R> fx <- expression(4*(-1-x)*y , 4*(1-y)*x , 4*(1-z)*y) 
R> gx <- rep(expression(0.2),3)
R> mod3d <- snssde3d(x0=c(x=2,y=-2,z=-2),drift=fx,diffusion=gx,M=1000)
R> mod3d

Itô 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 with order 0.5
Summary:
    | Size of process   | N  = 1001.
    | Number of simulation  | M  = 1000.
    | Initial values    | (x0,y0,z0) = (2,-2,-2).
    | Time of process   | t in [0,1].
    | Discretization    | Dt = 0.001.

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> s = 1
R> mean(mod3d, at = s)

[1] -0.79771  0.87700  0.79586

R> moment(mod3d, at = s , center = TRUE , order = 2) ## variance

[1] 0.0098837 0.1097268 0.0099780

R> Median(mod3d, at = s)

[1] -0.79993  0.83984  0.80182

R> Mode(mod3d, at = s)

[1] -0.78944  0.76454  0.80311

R> quantile(mod3d , at = s)

$x
      0%      25%      50%      75%     100% 
-1.08274 -0.86567 -0.79993 -0.73760 -0.35725 

$y
      0%      25%      50%      75%     100% 
0.057898 0.644261 0.839842 1.073442 2.203195 

$z
     0%     25%     50%     75%    100% 
0.38141 0.73503 0.80182 0.86654 1.07719 

R> kurtosis(mod3d , at = s)

[1] 3.3793 3.5112 3.7198

R> skewness(mod3d , at = s)

[1]  0.32230  0.54207 -0.52200

R> cv(mod3d , at = s )

[1] -0.12469  0.37790  0.12557

R> min(mod3d , at = s)

[1] -1.082739  0.057898  0.381408

R> max(mod3d , at = s)

[1] -0.35725  2.20320  1.07719

R> moment(mod3d, at = s , center= TRUE , order = 4)

[1] 0.00033078 0.04236003 0.00037109

R> moment(mod3d, at = s , center= FALSE , order = 4)

[1] 0.44198 1.20952 0.43782

The summary of the results of mod3d at time \(t=1\) of class snssde3d() is given by:

R> summary(mod3d, at = s)

  Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 1
                        X       Y        Z
Mean             -0.79771 0.87700  0.79586
Variance          0.00989 0.10984  0.00999
Median           -0.79993 0.83984  0.80182
Mode             -0.78944 0.76454  0.80311
First quartile   -0.86567 0.64426  0.73503
Third quartile   -0.73760 1.07344  0.86654
Minimum          -1.08274 0.05790  0.38141
Maximum          -0.35725 2.20320  1.07719
Skewness          0.32230 0.54207 -0.52200
Kurtosis          3.37935 3.51125  3.71979
Coef-variation   -0.12469 0.37790  0.12557
3th-order moment  0.00032 0.01973 -0.00052
4th-order moment  0.00033 0.04236  0.00037
5th-order moment  0.00004 0.02313 -0.00006
6th-order moment  0.00002 0.03233  0.00003

For plotting (back in time) using the command plot, and plot3D in space the results of the simulation are shown in Figure 9.

R> plot(mod3d,union = TRUE)         ## back in time
R> plot3D(mod3d,display="persp")    ## in space (O,X,Y,Z)

3D SDEs

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 = s) 
R> head(out,n=10)

          x       y       z
1  -0.56659 0.40010 0.58340
2  -0.83126 0.54093 0.65433
3  -0.96898 1.30182 0.82791
4  -0.85467 0.91820 0.82323
5  -0.74002 0.85950 0.72057
6  -0.72922 0.84603 0.88307
7  -0.68068 0.56023 0.57196
8  -0.77963 0.62726 0.75782
9  -0.61304 0.54800 0.72629
10 -0.86653 1.26019 0.94123

R> summary(out)

       x                y                z        
 Min.   :-1.083   Min.   :0.0579   Min.   :0.381  
 1st Qu.:-0.866   1st Qu.:0.6443   1st Qu.:0.735  
 Median :-0.800   Median :0.8398   Median :0.802  
 Mean   :-0.798   Mean   :0.8770   Mean   :0.796  
 3rd Qu.:-0.738   3rd Qu.:1.0734   3rd Qu.:0.867  
 Max.   :-0.357   Max.   :2.2032   Max.   :1.077  

R> cov(out)

           x         y          z
x  0.0098936 -0.018792 -0.0043937
y -0.0187917  0.109837  0.0200126
z -0.0043937  0.020013  0.0099880

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 10.

R> den <- dsde3d(mod3d,pdf="M",at =1)
R> den

 Marginal density of X(t-t0)|X(t0)=2 at time t = 1

Data: x (1000 obs.);    Bandwidth 'bw' = 0.02161

       x                 f(x)       
 Min.   :-1.14756   Min.   :0.0002  
 1st Qu.:-0.93378   1st Qu.:0.0279  
 Median :-0.72000   Median :0.4314  
 Mean   :-0.72000   Mean   :1.1683  
 3rd Qu.:-0.50621   3rd Qu.:2.1636  
 Max.   :-0.29243   Max.   :4.0682  

 Marginal density of Y(t-t0)|Y(t0)=-2 at time t = 1

Data: y (1000 obs.);    Bandwidth 'bw' = 0.07241

       y                 f(y)        
 Min.   :-0.15932   Min.   :0.00006  
 1st Qu.: 0.48561   1st Qu.:0.02870  
 Median : 1.13055   Median :0.17131  
 Mean   : 1.13055   Mean   :0.38726  
 3rd Qu.: 1.77548   3rd Qu.:0.75625  
 Max.   : 2.42042   Max.   :1.23074  

 Marginal density of Z(t-t0)|Z(t0)=-2 at time t = 1

Data: z (1000 obs.);    Bandwidth 'bw' = 0.02219

       z                f(z)       
 Min.   :0.31485   Min.   :0.0002  
 1st Qu.:0.52207   1st Qu.:0.0567  
 Median :0.72930   Median :0.4489  
 Mean   :0.72930   Mean   :1.2052  
 3rd Qu.:0.93653   3rd Qu.:2.3340  
 Max.   :1.14375   Max.   :4.2142  

R> 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.)

R> denJ <- dsde3d(mod3d,pdf="J")
R> plot(denJ,display="rgl")

Return to 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> K = 4; s = 1; sigma = 0.2
R> 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)) ) 
R> gx <- rep(expression(sigma),3)
R> mod3d <- snssde3d(drift=fx,diffusion=gx,N=10000,x0=c(x=1,y=1,z=1))
R> mod3d

Itô 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 with order 0.5
Summary:
    | Size of process   | N  = 10001.
    | Number of simulation  | M  = 1.
    | Initial values    | (x0,y0,z0) = (1,1,1).
    | Time of process   | t in [0,1].
    | Discretization    | Dt = 0.0001.

The results of simulation are shown:

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

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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: \[\begin{equation}\label{eq20} dX_{t} = \mu W_{1,t} dt + \sigma W_{2,t} \circ dW_{3,t} \end{equation}\] To simulate the solution of the process \(X_t\), we make a transformation to a system of three equations as follows: \[\begin{eqnarray}\label{eq21} \begin{cases} % \nonumber to remove numbering (before each equation) dX_t = \mu Y_{t} dt + \sigma Z_{t} \circ dW_{3,t} \nonumber\\ dY_t = dW_{1,t} \\ dZ_t = dW_{2,t} \nonumber \end{cases} \end{eqnarray}\] run by calling the function snssde3d() to produce a simulation of the solution, with \(\mu = 1\) and \(\sigma = 1\).

R> fx <- expression(y,0,0) 
R> gx <- expression(z,1,1)
R> modtra <- snssde3d(drift=fx,diffusion=gx,M=1000,type="str")
R> modtra

Stratonovich Sde 3D:
    | dX(t) = Y(t) * dt + Z(t) o dW1(t)
    | dY(t) = 0 * dt + 1 o dW2(t)
    | dZ(t) = 0 * dt + 1 o dW3(t)
Method:
    | Euler scheme with order 0.5
Summary:
    | Size of process   | N  = 1001.
    | Number of simulation  | M  = 1000.
    | Initial values    | (x0,y0,z0) = (0,0,0).
    | Time of process   | t in [0,1].
    | Discretization    | Dt = 0.001.

R> summary(modtra)

  Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 1
                        X          Y         Z
Mean              0.02050   -0.00735   0.00410
Variance          0.76290    1.02520   0.92697
Median           -0.00513    0.00347  -0.00932
Mode              0.08659   -0.10735  -0.07669
First quartile   -0.50274   -0.69009  -0.63180
Third quartile    0.52399    0.69699   0.66827
Minimum          -2.80265   -3.26565  -3.90726
Maximum           5.99953    3.72179   3.17907
Skewness          0.46773   -0.03733  -0.12839
Kurtosis          5.74922    3.15054   3.41486
Coef-variation   42.61668 -137.73620 234.60715
3th-order moment  0.31167   -0.03875  -0.11459
4th-order moment  3.34617    3.31130   2.93428
5th-order moment  8.66966    0.11897  -1.59067
6th-order moment 56.80855   18.73499  18.05038

the following code produces the result in Figure 12.

R> plot(modtra$X,plot.type="single",ylab=expression(X[t]))
R> lines(time(modtra),apply(modtra$X,1,mean),col=2,lwd=2)
R> legend("topleft",c("mean path"),col=2,lwd=2,cex=0.8)

Simulation of \(X_t\)

The histogram and kernel density of \(X_t\) at time \(t=1\) are reported using dsde3d() function, see e.g. Figure 13.

R> den <- dsde3d(modtra,pdf="Marginal",at=1)
R> den$resx

Call:
    density.default(x = x, na.rm = TRUE)

Data: x (1000 obs.);    Bandwidth 'bw' = 0.1732

       x                y          
 Min.   :-3.322   Min.   :0.00000  
 1st Qu.:-0.862   1st Qu.:0.00058  
 Median : 1.598   Median :0.01768  
 Mean   : 1.598   Mean   :0.10151  
 3rd Qu.: 4.059   3rd Qu.:0.14044  
 Max.   : 6.519   Max.   :0.50564  

R> MASS::truehist(den$ech$x,xlab = expression(X[t==1]));box()
R> lines(den$resx,col="red",lwd=2)
R> 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)

Figure 14 and 15, show approximation results for \(m_{1}(t)= \text{E}(X_{t})\), \(S_{1}(t)=\text{V}(X_{t})\) and \(C(s,t)=\text{Cov}(X_{s},X_{t})\):

R> m1  <- apply(modtra$X,1,mean) ## m1(t)
R> S1  <- apply(modtra$X,1,var)  ## s1(t)
R> out_a <- data.frame(m1,S1)
R> matplot(time(modtra), out_a, type = "l", xlab = "time", ylab = "", col=2:3,lwd=2,lty=2:3,las=1)
R> legend("topleft",c(expression(m[1](t),S[1](t))),inset = .09,col=2:3,lty=2:3,lwd=2,cex=0.9)

R> color.palette=colorRampPalette(c('white','green','blue','red'))
R> 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 = ""))

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