Assume that we want to describe the following SDE:
Ito form1:
\[\begin{equation}\label{eq:05} dX_{t} = \frac{1}{2}\theta^{2} X_{t} dt + \theta X_{t} dW_{t},\qquad X_{0}=x_{0} > 0 \end{equation}\] 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:
drift
and diffusion
coefficients as R expressions that depend on the state variable x
and time variable t
.N=1000
(by default: N=1000
).M=500
(by default: M=1
).t0=0
, x0=10
and end time T=1
(by default: t0=0
, x0=0
and T=1
).Dt=0.001
(by default: Dt=(T-t0)/N
).type="ito"
for Ito or type="str"
for Stratonovich (by default type="ito"
).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 \(t\):
mean
.median
.quantile
.skewness
and kurtosis
.moment
.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 11.40353
## Variance 34.13210
## Median 10.25887
## First quartile 7.25841
## Third quartile 14.08331
## Skewness 2.46228
## Kurtosis 18.19171
## Moment of order 3 491.00057
## Moment of order 4 21193.33999
## Moment of order 5 976221.18991
## Int.conf Inf (95%) 4.12423
## Int.conf Sup (95%) 23.59981
summary(mod2, at = 1)
##
## Monte-Carlo Statistics for X(t) at time t = 1
##
## Mean 10.10946
## Variance 30.84606
## Median 8.73994
## First quartile 6.40033
## Third quartile 12.43298
## Skewness 1.96516
## Kurtosis 9.70802
## Moment of order 3 336.66411
## Moment of order 4 9236.98164
## Moment of order 5 254177.41724
## Int.conf Inf (95%) 3.62197
## Int.conf Sup (95%) 24.37419
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 (Itô SDE)
x2 <- rsde1d(object = mod2, at = 1) # X(t=1) | X(0)=x0 (Stratonovich SDE)
summary(data.frame(x1,x2))
## x1 x2
## Min. : 2.245 Min. : 1.403
## 1st Qu.: 7.258 1st Qu.: 6.400
## Median :10.259 Median : 8.740
## Mean :11.404 Mean :10.109
## 3rd Qu.:14.083 3rd Qu.:12.433
## Max. :64.986 Max. :48.100
The function dsde1d()
can be used to show the kernel density estimation for \(X_{t}|X_{0}\) at time \(t=1\) 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))
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)
The following \(2\)-dimensional SDE’s with a vector of drift and a diagonal 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: \[\begin{equation}\label{eq:10} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) \circ dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) \circ dW_{2,t} \end{cases} \end{equation}\]\(W_{1,t}\) and \(W_{2,t}\) is a two independent standard Wiener process. To simulate \(2d\) models using snssde2d()
function we need to specify:
drift
(2d) and diffusion
(2d) coefficients as R expressions that depend on the state variable x
, y
and time variable t
.N
(default: N=1000
).M
(default: M=1
).t0
, x0
and end time T
(default: t0=0
, x0=c(0,0)
and T=1
).Dt
(default: Dt=(T-t0)/N
).type="ito"
for Ito or type="str"
for Stratonovich (default type="ito"
).method
(default method="euler"
).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.
x=5;y=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=500,x0=c(x,y),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 0.15038 14.03061
## Variance 0.76853 26.45717
## Median 0.12574 13.97368
## First quartile -0.43664 10.27947
## Third quartile 0.72965 17.67226
## Skewness 0.04405 0.01798
## Kurtosis 2.85390 2.94868
## Moment of order 3 0.02968 2.44685
## Moment of order 4 1.68563 2064.02471
## Moment of order 5 0.12083 -1315.88866
## Int.conf Inf (95%) -1.56840 4.59106
## Int.conf Sup (95%) 1.84745 24.27626
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: \[ \begin{cases} \text{E}(X_{t}) =x_{0} e^{-t/\mu} &\text{and}\quad\text{Var}(X_{t})=\frac{\sigma \mu}{2} \left (1-e^{-2t/\mu}\right )\\ \text{E}(Y_{t}) = y_{0}+x_{0}\mu \left (1-e^{-t/\mu}\right ) &\text{and}\quad\text{Var}(Y_{t})=\sigma\mu^{3}\left (\frac{t}{\mu}-2\left (1-e^{-t/\mu}\right )+\frac{1}{2}\left (1-e^{-2t/\mu}\right )\right ) \end{cases} \]
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)
summary(out)
## x y
## Min. :-2.4036 Min. :-2.885
## 1st Qu.:-0.4366 1st Qu.:10.279
## Median : 0.1257 Median :13.974
## Mean : 0.1504 Mean :14.031
## 3rd Qu.: 0.7296 3rd Qu.:17.672
## Max. : 2.9342 Max. :29.632
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 4: the marginal density of \(X_t\) and \(Y_t\) at time \(t=10\).
denM <- dsde2d(mod2d,pdf="M",at =10)
denM
##
## Marginal density for the conditional law of X(t)|X(0) at time t = 10
##
## Data: x (500 obs.); Bandwidth 'bw' = 0.226
##
## x f(x)
## Min. :-3.081710 Min. :0.0000398
## 1st Qu.:-1.408215 1st Qu.:0.0077504
## Median : 0.265281 Median :0.0836797
## Mean : 0.265281 Mean :0.1492413
## 3rd Qu.: 1.938777 3rd Qu.:0.2912292
## Max. : 3.612272 Max. :0.4404050
##
## Marginal density for the conditional law of Y(t)|Y(0) at time t = 10
##
## Data: y (500 obs.); Bandwidth 'bw' = 1.336
##
## y f(y)
## Min. :-6.89196 Min. :0.00000785
## 1st Qu.: 3.24084 1st Qu.:0.00139667
## Median :13.37364 Median :0.01218128
## Mean :13.37364 Mean :0.02464810
## 3rd Qu.:23.50644 3rd Qu.:0.04939563
## Max. :33.63924 Max. :0.07406468
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
.
denJ <- dsde2d(mod2d,pdf="J",at =10)
denJ
##
## Joint density for the conditional law of X(t),Y(t)|X(0),Y(0) at time t = 10
##
## Data: (x,y) (2 x 500 obs.);
##
## x y f(x,y)
## Min. :-2.403643 Min. :-2.884757 Min. :0.00000000
## 1st Qu.:-1.069181 1st Qu.: 5.244442 1st Qu.:0.00011450
## Median : 0.265281 Median :13.373641 Median :0.00145132
## Mean : 0.265281 Mean :13.373641 Mean :0.00561250
## 3rd Qu.: 1.599743 3rd Qu.:21.502839 3rd Qu.:0.00739674
## Max. : 2.934206 Max. :29.632038 Max. :0.03511061
plot(denJ,display="contour",main="Bivariate Density")
plot(denJ,display="image",drawpoints=TRUE,col.pt="green",cex=0.25,pch=19,main="Bivariate Density")
A \(3\)D plot of the density obtained with:
plot(denJ,display="rgl",main="Bivariate Density")
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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 , (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")
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
The following \(3\)-dimensional SDE’s with a vector of drift and a diagonal 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: \[\begin{equation}\label{eq18} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) \circ dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) \circ dW_{2,t}\\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) \circ dW_{3,t} \end{cases} \end{equation}\]\(W_{1,t}\), \(W_{2,t}\) and \(W_{3,t}\) is a 3 independent standard Wiener process. To simulate this system using snssde3d()
function we need to specify:
drift
(3d) and diffusion
(3d) coefficients as R expressions that depend on the state variables x
, y
, z
and time variable t
.N
(default: N=1000
).M
(default: M=1
).t0
, x0
and end time T
(default: t0=0
, x0=c(0,0,0)
and T=1
).Dt
(default: Dt=(T-t0)/N
).type="ito"
for Ito or type="str"
for Stratonovich (default type="ito"
).method
(default method="euler"
).We simulate a flow of 500 trajectories, with integration step size \(\Delta t = 0.001\).
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,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 -0.79363 0.89824 0.79638
## Variance 0.01017 0.10923 0.00896
## Median -0.80195 0.87184 0.80025
## First quartile -0.86826 0.66836 0.73532
## Third quartile -0.73147 1.12565 0.86637
## Skewness 0.26669 0.24028 -0.34912
## Kurtosis 2.74278 3.18534 2.83584
## Moment of order 3 0.00027 0.00867 -0.00030
## Moment of order 4 0.00028 0.03801 0.00023
## Moment of order 5 0.00002 0.00945 -0.00002
## Int.conf Inf (95%) -0.97932 0.27582 0.59075
## Int.conf Sup (95%) -0.58567 1.59040 0.96242
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 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,at =1)
den
##
## Marginal density for the conditional law of X(t)|X(0) at time t = 1
##
## Data: x (500 obs.); Bandwidth 'bw' = 0.02619
##
## x f(x)
## Min. :-1.1148777 Min. :0.000402
## 1st Qu.:-0.9431618 1st Qu.:0.199276
## Median :-0.7714459 Median :1.067292
## Mean :-0.7714459 Mean :1.454461
## 3rd Qu.:-0.5997300 3rd Qu.:2.717240
## Max. :-0.4280140 Max. :3.923853
##
## Marginal density for the conditional law of Y(t)|Y(0) at time t = 1
##
## Data: y (500 obs.); Bandwidth 'bw' = 0.08583
##
## y f(y)
## Min. :-0.3775533 Min. :0.0001050
## 1st Qu.: 0.3028136 1st Qu.:0.0142478
## Median : 0.9831806 Median :0.1460117
## Mean : 0.9831806 Mean :0.3670882
## 3rd Qu.: 1.6635476 3rd Qu.:0.7306455
## Max. : 2.3439145 Max. :1.2486437
##
## Marginal density for the conditional law of Z(t)|Z(0) at time t = 1
##
## Data: z (500 obs.); Bandwidth 'bw' = 0.02458
##
## z f(z)
## Min. :0.4333812 Min. :0.000405
## 1st Qu.:0.5966982 1st Qu.:0.188503
## Median :0.7600151 Median :0.955669
## Mean :0.7600151 Mean :1.529261
## 3rd Qu.:0.9233321 3rd Qu.:2.865271
## Max. :1.0866490 Max. :3.911343
plot(den, main="Marginal Density")
For Joint density for \((X_t,Y_t,Z_t)\) see package sm or ks.
out <- rsde3d(mod3d,at =1)
library(sm)
sm.density(out,display="rgl")
##
library(ks)
fhat <- kde(x=out)
plot(fhat, drawpoints=TRUE)
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)) , (-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))
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:
plot3D(mod3d,display="rgl",col="blue")
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run by calling the function snssde3d()
to produce a simulation of the solution, with \(\mu = 1\) and \(\sigma = 1\).
fx <- expression(y,0,0)
gx <- expression(z,1,1)
modtra <- snssde3d(drift=fx,diffusion=gx,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 0.03918 0.02304 -0.08398
## Variance 0.86747 1.01719 1.01050
## Median 0.00060 0.07058 -0.12287
## First quartile -0.57320 -0.69130 -0.74918
## Third quartile 0.58203 0.69016 0.58294
## Skewness 0.32959 -0.05206 0.17379
## Kurtosis 4.15533 2.97478 3.31916
## Moment of order 3 0.26629 -0.05341 0.17653
## Moment of order 4 3.12690 3.07794 3.38920
## Moment of order 5 3.79900 -0.22436 1.66325
## Int.conf Inf (95%) -1.70857 -2.00373 -2.05616
## Int.conf Sup (95%) 2.04740 1.90640 2.05156
the following code produces the result in Figure 9.
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 histogram and kernel density of \(X_t\) at time \(t=1\) are reported using dsde3d()
function, see e.g. Figure 10.
den <- dsde3d(modtra,at=1)
den$resx
##
## Call:
## density.default(x = x, na.rm = TRUE)
##
## Data: x (500 obs.); Bandwidth 'bw' = 0.2239
##
## x y
## Min. :-3.5243 Min. :0.0000404
## 1st Qu.:-1.4024 1st Qu.:0.0035571
## Median : 0.7194 Median :0.0359903
## Mean : 0.7194 Mean :0.1177059
## 3rd Qu.: 2.8413 3rd Qu.:0.2143911
## Max. : 4.9631 Max. :0.4566161
MASS::truehist(den$ech$x,xlab = expression(X[t==1]));box()
lines(den$resx,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)
The equivalently of \(X_{t}^{\text{mod1}}\) the following Stratonovich SDE: \(dX_{t} = \theta X_{t} \circ dW_{t}\).↩