LaTeX table for object of class MCM.sde

The Monte Carlo results of MCM.sde class can be presented in terms of LaTeX tables.

\[\begin{equation}\label{eq01} \begin{cases} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t\\ dY_t = X_{t} dt \end{cases} \end{equation}\]

R> mu=1;sigma=0.5;theta=2
R> x0=0;y0=0;init=c(x0,y0)
R> f <- expression(1/mu*(theta-x), x)  
R> g <- expression(sqrt(sigma),0)
R> mod2d <- snssde2d(drift=f,diffusion=g,M=500,Dt=0.015,x0=c(x=0,y=0))
R> ## true values of first and second moment at time 10
R> Ex <- function(t) theta+(x0-theta)*exp(-t/mu)
R> Vx <- function(t) 0.5*sigma*mu *(1-exp(-2*(t/mu)))
R> Ey <- function(t) y0+theta*t+(x0-theta)*mu*(1-exp(-t/mu))
R> Vy <- function(t) sigma*mu^3*((t/mu)-2*(1-exp(-t/mu))+0.5*(1-exp(-2*(t/mu))))
R> covxy <- function(t) 0.5*sigma*mu^2 *(1-2*exp(-t/mu)+exp(-2*(t/mu)))
R> tvalue = list(m1=Ex(15),m2=Ey(15),S1=Vx(15),S2=Vy(15),C12=covxy(15))
R> ## function of the statistic(s) of interest.
R> sde.fun2d <- function(data, i){
+   d <- data[i,]
+   return(c(mean(d$x),mean(d$y),var(d$x),var(d$y),cov(d$x,d$y)))
+ }
R> ## Parallel Monte-Carlo of 'OUI' at time 10
R> mcm.mod2d = MCM.sde(mod2d,statistic=sde.fun2d,time=15,R=10,exact=tvalue,parallel="snow",ncpus=2)
R> mcm.mod2d$MC

    Exact Estimate     Bias Std.Error    RMSE    CI( 2.5 % , 97.5 % )
m1   2.00  2.01117  0.01117   0.00485 0.01835   ( 2.00166 , 2.02068 )
m2  28.00 28.05632  0.05632   0.04436 0.14450 ( 27.96938 , 28.14326 )
S1   0.25  0.25157  0.00157   0.00348 0.01057   ( 0.24475 , 0.25839 )
S2   6.75  6.66725 -0.08275   0.08543 0.26932   ( 6.49981 , 6.83469 )
C12  0.25  0.26155  0.01155   0.01401 0.04359   ( 0.23409 , 0.28901 )

In R we create simple LaTeX table for this object using the following code:

R> TEX.sde(object = mcm.mod2d, booktabs = TRUE, align = "r", caption ="LaTeX 
+           table for Monte Carlo results generated by `TEX.sde()` method.")

%%% LaTeX table generated in R 4.0.2 by TEX.sde() method 
%%% Copy and paste the following output in your LaTeX file 

\begin{table}

\caption{\label{tab:unnamed-chunk-2}LaTeX 
          table for Monte Carlo results generated by `TEX.sde()` method.}
\centering
\begin{tabular}[t]{lrrrrrr}
\toprule
  & Exact & Estimate & Bias & Std.Error & RMSE & CI( 2.5 \% , 97.5 \% )\\
\midrule
$m_{1}(t)$ & 2.00 & 2.01117 & 0.01117 & 0.00485 & 0.01835 & ( 2.00166 , 2.02068 )\\
$m_{2}(t)$ & 28.00 & 28.05632 & 0.05632 & 0.04436 & 0.14450 & ( 27.96938 , 28.14326 )\\
$S_{1}(t)$ & 0.25 & 0.25157 & 0.00157 & 0.00348 & 0.01057 & ( 0.24475 , 0.25839 )\\
$S_{2}(t)$ & 6.75 & 6.66725 & -0.08275 & 0.08543 & 0.26932 & ( 6.49981 , 6.83469 )\\
$C_{12}(t)$ & 0.25 & 0.26155 & 0.01155 & 0.01401 & 0.04359 & ( 0.23409 , 0.28901 )\\
\bottomrule
\end{tabular}
\end{table} 

For inclusion in LaTeX documents, and optionally if we use booktabs = TRUE in the previous function, the LaTeX add-on package booktabs must be loaded into the .tex document.

LaTeX mathematic for object of class MEM.sde

we want to automatically generate the LaTeX code appropriate to moment equations obtained from the previous model using TEX.sde() method.

R> mem.oui <- MEM.sde(drift = f, diffusion = g)
R> mem.oui

Itô Sde 2D:
 | dX(t) = 1/mu * (theta - X(t)) * dt + sqrt(sigma) * dW1(t)
 | dY(t) = X(t) * dt + 0 * dW2(t)
 | t in [t0,T].

Moment equations: 
 | dm1(t)  = (theta - m1(t))/mu
 | dm2(t)  = m1(t)
 | dS1(t)  = sigma - 2 * (S1(t)/mu)
 | dS2(t)  = 2 * C12(t)
 | dC12(t) = S1(t) - C12(t)/mu

In R we create LaTeX mathematical expressions for this object using the following code:

R> TEX.sde(object = mem.oui)

%%% LaTeX equation generated in R 4.0.2 by TEX.sde() method
%%% Copy and paste the following output in your LaTeX file

\begin{equation}\label{eq:}
\begin{cases}
\begin{split}
\frac{d}{dt} m_{1}(t) ~&= \frac{\left( \theta - m_{1}(t) \right)}{\mu} \\
\frac{d}{dt} m_{2}(t) ~&= m_{1}(t) \\
\frac{d}{dt} S_{1}(t) ~&= \sigma - 2 \, \left( \frac{S_{1}(t)}{\mu} \right) \\
\frac{d}{dt} S_{2}(t) ~&= 2 \, C_{12}(t) \\
\frac{d}{dt} C_{12}(t) &= S_{1}(t) - \frac{C_{12}(t)}{\mu}
\end{split}
\end{cases}
\end{equation}

that can be typed with LaTeX to produce a system:

\[\begin{equation} \begin{cases} \begin{split} \frac{d}{dt} m_{1}(t) ~&= \frac{\left( \theta - m_{1}(t) \right)}{\mu} \\ \frac{d}{dt} m_{2}(t) ~&= m_{1}(t) \\ \frac{d}{dt} S_{1}(t) ~&= \sigma - 2 \, \left( \frac{S_{1}(t)}{\mu} \right) \\ \frac{d}{dt} S_{2}(t) ~&= 2 \, C_{12}(t) \\ \frac{d}{dt} C_{12}(t) &= S_{1}(t) - \frac{C_{12}(t)}{\mu} \end{split} \end{cases} \end{equation}\]

Note that it is obvious the LaTeX package amsmath must be loaded into the .tex document.

LaTeX mathematic for an R expression of SDEs

In this section, we will convert the R expressions of a SDEs, i.e., drift and diffusion coefficients into their LaTeX mathematical equivalents with the same procedures previous. An example sophisticated that will make this clear.

R> f <- expression((alpha*x *(1 - x / beta)- delta * x^2 * y / (kappa + x^2)),
+                 (gamma * x^2 * y / (kappa + x^2) - mu * y^2)) 
R> g <- expression(sqrt(sigma1)*x*(1-y), abs(sigma2)*y*(1-x))  
R> TEX.sde(object=c(drift = f, diffusion = g))

%%% LaTeX equation generated in R 4.0.2 by TEX.sde() method
%%% Copy and paste the following output in your LaTeX file

\begin{equation}\label{eq:}
\begin{cases}
\begin{split}
dX_{t} &= \left( \alpha \, X_{t} \, \left( 1 - \frac{X_{t}}{\beta} \right) - \frac{\delta \, X_{t}^2 \, Y_{t}}{\left( \kappa + X_{t}^2 \right)} \right) \:dt +  \sqrt{\sigma_{1}} \, X_{t} \, \left( 1 - Y_{t} \right) \:dW_{1,t} \\
dY_{t} &= \left( \frac{\gamma \, X_{t}^2 \, Y_{t}}{\left( \kappa + X_{t}^2 \right)} - \mu \, Y_{t}^2 \right) \:dt +  \left| \sigma_{2}\right|  \, Y_{t} \, \left( 1 - X_{t} \right) \:dW_{2,t}
\end{split}
\end{cases}
\end{equation}

under LaTeX will create this system:

\[\begin{equation*} \begin{cases} \begin{split} dX_{t} &= \left( \alpha \, X_{t} \, \left( 1 - \frac{X_{t}}{\beta} \right) - \frac{\delta \, X_{t}^2 \, Y_{t}}{\left( \kappa + X_{t}^2 \right)} \right) \:dt + \sqrt{\sigma_{1}} \, X_{t} \, \left( 1 - Y_{t} \right) \:dW_{1,t} \\ dY_{t} &= \left( \frac{\gamma \, X_{t}^2 \, Y_{t}}{\left( \kappa + X_{t}^2 \right)} - \mu \, Y_{t}^2 \right) \:dt + \left| \sigma_{2}\right| \, Y_{t} \, \left( 1 - X_{t} \right) \:dW_{2,t} \end{split} \end{cases} \end{equation*}\]