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 10.99598
## Variance 31.52028
## Median 9.71960
## First quartile 7.01976
## Third quartile 13.57172
## Skewness 1.20481
## Kurtosis 4.47704
## Moment of order 3 213.20873
## Moment of order 4 4448.05997
## Moment of order 5 67075.89171
## Int.conf Inf (95%) 3.54499
## Int.conf Sup (95%) 25.19046
summary(mod2, at = 1)
##
## Monte-Carlo Statistics for X(t) at time t = 1
##
## Mean 10.09247
## Variance 27.14782
## Median 9.22196
## First quartile 6.49679
## Third quartile 12.70541
## Skewness 1.66113
## Kurtosis 7.46548
## Moment of order 3 234.96690
## Moment of order 4 5502.08997
## Moment of order 5 114509.25603
## Int.conf Inf (95%) 3.81760
## Int.conf Sup (95%) 23.35448
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. : 2.368 Min. : 2.276
## 1st Qu.: 7.020 1st Qu.: 6.497
## Median : 9.720 Median : 9.222
## Mean :10.996 Mean :10.092
## 3rd Qu.:13.572 3rd Qu.:12.705
## Max. :34.148 Max. :37.802
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 0.21799 14.83489
## Variance 0.71889 20.87251
## Median 0.20895 14.93429
## First quartile -0.38134 11.63571
## Third quartile 0.75892 18.10022
## Skewness 0.24042 0.03821
## Kurtosis 3.24206 3.04422
## Moment of order 3 0.14654 3.64326
## Moment of order 4 1.67552 1326.24895
## Moment of order 5 1.31262 2100.80674
## Int.conf Inf (95%) -1.38838 6.25775
## Int.conf Sup (95%) 1.95350 23.29807
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. :-2.0829 Min. : 0.2984
## 1st Qu.:-0.3813 1st Qu.:11.6357
## Median : 0.2090 Median :14.9343
## Mean : 0.2180 Mean :14.8349
## 3rd Qu.: 0.7589 3rd Qu.:18.1002
## Max. : 3.4639 Max. :31.6856
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' = 0.2202
##
## x f(x)
## Min. :-2.743438 Min. :0.0000409
## 1st Qu.:-1.026477 1st Qu.:0.0053135
## Median : 0.690484 Median :0.0736112
## Mean : 0.690484 Mean :0.1454632
## 3rd Qu.: 2.407445 3rd Qu.:0.2763057
## Max. : 4.124406 Max. :0.5007931
##
## Marginal density for the conditional law of Y(t)|Y(0) at time t = 10
##
## Data: y (500 obs.); Bandwidth 'bw' = 1.186
##
## y f(y)
## Min. :-3.26083 Min. :0.00000763
## 1st Qu.: 6.36559 1st Qu.:0.00072703
## Median :15.99201 Median :0.00975566
## Mean :15.99201 Mean :0.02594470
## 3rd Qu.:25.61843 3rd Qu.:0.05366683
## Max. :35.24485 Max. :0.09078235
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. :-2.082894 Min. : 0.29841 Min. :0.00000000
## 1st Qu.:-0.696205 1st Qu.: 8.14521 1st Qu.:0.00005552
## Median : 0.690484 Median :15.99201 Median :0.00089418
## Mean : 0.690484 Mean :15.99201 Mean :0.00560450
## 3rd Qu.: 2.077173 3rd Qu.:23.83881 3rd Qu.:0.00666960
## Max. : 3.463862 Max. :31.68561 Max. :0.04477607
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,main="Bivariate Density")
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 -0.78788 0.87272 0.78982
## Variance 0.01014 0.10102 0.01012
## Median -0.79794 0.85339 0.79719
## First quartile -0.85617 0.67141 0.72217
## Third quartile -0.73330 1.06789 0.85607
## Skewness 0.70273 0.37686 -0.48184
## Kurtosis 4.13518 3.47136 3.36557
## Moment of order 3 0.00072 0.01210 -0.00049
## Moment of order 4 0.00043 0.03543 0.00034
## Moment of order 5 0.00009 0.01364 -0.00005
## Int.conf Inf (95%) -0.96890 0.25598 0.58101
## Int.conf Sup (95%) -0.57456 1.58272 0.96022
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' = 0.02381
##
## x f(x)
## Min. :-1.1152504 Min. :0.000388
## 1st Qu.:-0.9119133 1st Qu.:0.075024
## Median :-0.7085761 Median :0.476136
## Mean :-0.7085761 Mean :1.228278
## 3rd Qu.:-0.5052390 3rd Qu.:2.216015
## Max. :-0.3019018 Max. :4.196779
##
## Marginal density for the conditional law of Y(t)|Y(0) at time t = 1
##
## Data: y (500 obs.); Bandwidth 'bw' = 0.07684
##
## y f(y)
## Min. :-0.1396524 Min. :0.0001174
## 1st Qu.: 0.4713330 1st Qu.:0.0273182
## Median : 1.0823185 Median :0.2032677
## Mean : 1.0823185 Mean :0.4087731
## 3rd Qu.: 1.6933039 3rd Qu.:0.7411125
## Max. : 2.3042894 Max. :1.3366929
##
## Marginal density for the conditional law of Z(t)|Z(0) at time t = 1
##
## Data: z (500 obs.); Bandwidth 'bw' = 0.02595
##
## z f(z)
## Min. :0.3019458 Min. :0.000348
## 1st Qu.:0.5014142 1st Qu.:0.047529
## Median :0.7008826 Median :0.561927
## Mean :0.7008826 Mean :1.252100
## 3rd Qu.:0.9003510 3rd Qu.:2.393278
## Max. :1.0998194 Max. :4.295844
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="persp",col="blue")
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 0.02038 -0.00022 -0.04226
## Variance 0.75673 0.98774 0.93418
## Median 0.00419 -0.03371 -0.05310
## First quartile -0.50761 -0.68390 -0.72646
## Third quartile 0.56291 0.69763 0.63591
## Skewness -0.25218 0.07301 0.01379
## Kurtosis 4.04460 2.66176 2.79796
## Moment of order 3 -0.16600 0.07167 0.01245
## Moment of order 4 2.31607 2.59691 2.44178
## Moment of order 5 -2.88379 0.38041 0.00787
## Int.conf Inf (95%) -1.97639 -1.90679 -1.91798
## Int.conf Sup (95%) 1.75167 1.93111 1.78561
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$resx
##
## Call:
## density.default(x = x, na.rm = TRUE)
##
## Data: x (500 obs.); Bandwidth 'bw' = 0.2075
##
## x y
## Min. :-4.7641 Min. :0.0000021
## 1st Qu.:-2.8019 1st Qu.:0.0026348
## Median :-0.8396 Median :0.0396680
## Mean :-0.8396 Mean :0.1272792
## 3rd Qu.: 1.1226 3rd Qu.:0.2325336
## Max. : 3.0849 Max. :0.4738560
MASS::truehist(den$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: \(dX_{t} = \theta X_{t} \circ dW_{t}\).↩