R Under development (unstable) (2025-12-12 r89163 ucrt) -- "Unsuffered Consequences"
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> library(Sim.DiffProc)
> 
> 
> ## 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 495 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             0.3407838 0.0734218
Variance         0.0004332 0.0002749
Median           0.3397345 0.0703638
Mode             0.3395577 0.0659129
First quartile   0.3268930 0.0616801
Third quartile   0.3547007 0.0816789
Minimum          0.2924059 0.0432869
Maximum          0.4071547 0.1487750
Skewness         0.2109819 1.3231472
Kurtosis         2.8566997 5.8199024
Coef-variation   0.0610757 0.2258136
3th-order moment 0.0000019 0.0000060
4th-order moment 0.0000005 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"] (495 obs.);	Bandwidth 'bw' = 0.005399857

       x                  f(x)             
 Min.   :0.2762063   Min.   : 0.001694035  
 1st Qu.:0.3129933   1st Qu.: 0.650832189  
 Median :0.3497803   Median : 4.265654319  
 Mean   :0.3497803   Mean   : 6.782542437  
 3rd Qu.:0.3865673   3rd Qu.:12.237906137  
 Max.   :0.4233543   Max.   :19.666683283  

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

Data: out[, "y"] (495 obs.);	Bandwidth 'bw' = 0.003883478

       y                   f(y)            
 Min.   :0.03163643   Min.   : 0.00256984  
 1st Qu.:0.06383367   1st Qu.: 0.57057738  
 Median :0.09603092   Median : 1.97939294  
 Mean   :0.09603092   Mean   : 7.74941929  
 3rd Qu.:0.12822817   3rd Qu.:13.34977503  
 Max.   :0.16042542   Max.   :29.13295753  
> 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 495 obs.);

       x              y              f(x,y)          
 Min.   :0.28   Min.   :0.0400   Min.   :  0.000000  
 1st Qu.:0.31   1st Qu.:0.0625   1st Qu.:  7.816838  
 Median :0.34   Median :0.0850   Median : 33.905408  
 Mean   :0.34   Mean   :0.0850   Mean   : 89.890149  
 3rd Qu.:0.37   3rd Qu.:0.1075   3rd Qu.:122.429334  
 Max.   :0.40   Max.   :0.1300   Max.   :476.205859  
> 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 
   3.50    0.43    3.92