R Under development (unstable) (2024-03-03 r86036 ucrt) -- "Unsuffered Consequences"
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> library(Sim.DiffProc)
Package 'Sim.DiffProc', version 4.9
browseVignettes('Sim.DiffProc') for more informations.
> 
> 
> ## Example 1: 
> 
> # SDE's 2d
> fx <- expression(5*(-1-y)*x , 5*(-1-x)*y)
> gx <- expression(0.5 , 0.5)
> mod2d <- snssde2d(drift=fx,diffusion=gx,x0=c(2,-2),M=500)
> 
> # boundary
> 
> St <- expression(-1+5*t)
> 
> # random fpt
> 
> out <- fptsde2d(mod2d,boundary=St)
> out
Itô Sde 2D:
	| dX(t) = 5 * (-1 - Y(t)) * X(t) * dt + 0.5 * dW1(t)
	| dY(t) = 5 * (-1 - X(t)) * Y(t) * dt + 0.5 * dW2(t)
	| t in [0,1].
Boundary:
	| S(t) = -1 + 5 * t
F.P.T:
	| T(S(t),X(t)) = inf{t >=  0 : X(t) <=  -1 + 5 * t }
	| 	And 
	| T(S(t),Y(t)) = inf{t >=  0 : Y(t) >=  -1 + 5 * t }
	| Crossing realized 492 among 500.
> summary(out)

Monte-Carlo Statistics for the F.P.T of (X(t),Y(t))
	| T(S(t),X(t)) = inf{t >=  0 : X(t) <=  -1 + 5 * t }
	| 	 And
	| T(S(t),Y(t)) = inf{t >=  0 : Y(t) >=  -1 + 5 * t }
                    T(S,X)    T(S,Y)
Mean             0.3387521 0.0727487
Variance         0.0004008 0.0002565
Median           0.3389694 0.0698733
Mode             0.3398444 0.0684433
First quartile   0.3255841 0.0619255
Third quartile   0.3512447 0.0791862
Minimum          0.2828480 0.0430992
Maximum          0.3932701 0.1522165
Skewness         0.0925895 1.3700591
Kurtosis         2.7322311 6.1671521
Coef-variation   0.0590984 0.2201691
3th-order moment 0.0000007 0.0000056
4th-order moment 0.0000004 0.0000004
5th-order moment 0.0000000 0.0000000
6th-order moment 0.0000000 0.0000000
> 
> # Marginal density 
> 
> denM <- dfptsde2d(out,pdf="M")
> denM

Marginal density for the F.P.T of X(t)
	| T(S,X) = inf{t >= 0 : X(t) <= -1 + 5 * t}

Data: out[, "x"] (492 obs.);	Bandwidth 'bw' = 0.004988977

       x                  f(x)          
 Min.   :0.2678811   Min.   : 0.001932  
 1st Qu.:0.3029701   1st Qu.: 0.648889  
 Median :0.3380591   Median : 5.645862  
 Mean   :0.3380591   Mean   : 7.110790  
 3rd Qu.:0.3731481   3rd Qu.:11.753469  
 Max.   :0.4082370   Max.   :21.405074  

Marginal density for the F.P.T of Y(t)
	| T(S,Y) = inf{t >= 0 : Y(t) >= -1 + 5 * t}

Data: out[, "y"] (492 obs.);	Bandwidth 'bw' = 0.003355852

       y                   f(y)         
 Min.   :0.03303167   Min.   : 0.00419  
 1st Qu.:0.06534477   1st Qu.: 0.39119  
 Median :0.09765787   Median : 2.75703  
 Mean   :0.09765787   Mean   : 7.72162  
 3rd Qu.:0.12997096   3rd Qu.:12.24683  
 Max.   :0.16228406   Max.   :31.65309  
> plot(denM)
> 
> # Joint density
> 
> denJ <- dfptsde2d(out,pdf="J",n=200,lims=c(0.28,0.4,0.04,0.13))
> denJ

Joint density for the F.P.T of (X(t),Y(t))
	| T(S,X) = inf{t >= 0 : X(t) <=  -1 + 5 * t } 
	| 	And 
	| T(S,Y) = inf{t >= 0 : Y(t) >=  -1 + 5 * t }

Data: (x,y) (2 x 492 obs.);

       x              y              f(x,y)        
 Min.   :0.28   Min.   :0.0400   Min.   :  0.0000  
 1st Qu.:0.31   1st Qu.:0.0625   1st Qu.:  7.8959  
 Median :0.34   Median :0.0850   Median : 29.1431  
 Mean   :0.34   Mean   :0.0850   Mean   : 90.8076  
 3rd Qu.:0.37   3rd Qu.:0.1075   3rd Qu.:118.7386  
 Max.   :0.40   Max.   :0.1300   Max.   :609.0383  
> plot(denJ)
> plot(denJ,display="image")
> plot(denJ,display="image",drawpoints=TRUE,cex=0.5,pch=19,col.pt='green')
> plot(denJ,display="contour")
> plot(denJ,display="contour",color.palette=colorRampPalette(c('white','green','blue','red')))
> 
> proc.time()
   user  system elapsed 
   2.39    0.18    2.57