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.
> 
> 
> ### 2-dim Ito bridge sde
> set.seed(1234)
> 
> fx <- expression(4*(-1-x) , x)
> gx <- expression(0.2 , 0)
> res <- bridgesde2d(drift=fx,diffusion=gx,Dt=0.005,M=500)
> res
Itô Bridge Sde 2D:
	| dX(t) = 4 * (-1 - X(t)) * dt + 0.2 * dW1(t)
	| dY(t) = X(t) * dt + 0 * dW2(t)
Method:
	| Euler scheme with order 0.5
Summary:
	| Size of process	| N = 1001.
	| Crossing realized	| C = 500 among 500.
	| Initial values	| x0 = (0,0).
	| Ending values		| y = (0,0).
	| Time of process	| t in [0,5].
	| Discretization	| Dt = 0.005.
> summary(res) ## Monte-Carlo statistics at time T/2=2.5

Monte-Carlo Statistics for (X(t),Y(t)) at time t = 2.5
	| Crossing realized 500 among 500
                          X          Y
Mean             -1.0030573 -2.2099784
Variance          0.0045892  0.0034646
Median           -1.0031157 -2.2114024
Mode             -1.0008840 -2.2128791
First quartile   -1.0518003 -2.2487562
Third quartile   -0.9587026 -2.1712329
Minimum          -1.2027993 -2.3974294
Maximum          -0.7814393 -2.0481089
Skewness          0.0711521  0.0360794
Kurtosis          2.8781525  2.9124049
Coef-variation   -0.0675372 -0.0266342
3th-order moment  0.0000221  0.0000074
4th-order moment  0.0000606  0.0000350
5th-order moment  0.0000015 -0.0000001
6th-order moment  0.0000013  0.0000006
> summary(res,at=1) ## Monte-Carlo statistics at time 1

Monte-Carlo Statistics for (X(t),Y(t)) at time t = 1
	| Crossing realized 500 among 500
                          X          Y
Mean             -0.9782661 -0.7554881
Variance          0.0055552  0.0015861
Median           -0.9780195 -0.7558516
Mode             -0.9564967 -0.7555558
First quartile   -1.0325486 -0.7813315
Third quartile   -0.9258241 -0.7306599
Minimum          -1.2036952 -0.8777720
Maximum          -0.7686252 -0.6506899
Skewness         -0.0168452 -0.0099052
Kurtosis          2.7520245  2.9190313
Coef-variation   -0.0761893 -0.0527150
3th-order moment -0.0000070 -0.0000006
4th-order moment  0.0000849  0.0000073
5th-order moment -0.0000003  0.0000000
6th-order moment  0.0000021  0.0000001
> summary(res,at=4) ## Monte-Carlo statistics at time 4

Monte-Carlo Statistics for (X(t),Y(t)) at time t = 4
	| Crossing realized 500 among 500
                          X          Y
Mean             -0.9857418 -0.7527662
Variance          0.0045730  0.0015467
Median           -0.9880801 -0.7512646
Mode             -0.9863887 -0.7470079
First quartile   -1.0290469 -0.7779033
Third quartile   -0.9414837 -0.7273119
Minimum          -1.1878318 -0.8756237
Maximum          -0.7633155 -0.6418674
Skewness          0.0311155 -0.1076027
Kurtosis          2.9955695  3.0991994
Coef-variation   -0.0686023 -0.0522441
3th-order moment  0.0000096 -0.0000065
4th-order moment  0.0000626  0.0000074
5th-order moment  0.0000010 -0.0000001
6th-order moment  0.0000014  0.0000001
> ##
> plot(res,type="n")
> lines(time(res),apply(res$X,1,mean),col=3,lwd=2)
> lines(time(res),apply(res$Y,1,mean),col=4,lwd=2)
> legend("topright",c(expression(E(X[t])),expression(E(Y[t]))),lty=1,inset = .7,col=c(3,4))
> ##
> plot2d(res)
> 
> 
> ##
> 
> fx <- expression(4*(-1-x) , x)
> gx <- expression(0.2 , 0)
> Sigma= matrix(c(1,0.7,0.7,1),nrow=2)
> res <- bridgesde2d(drift=fx,diffusion=gx,Dt=0.005,M=500,corr=Sigma)
> res
Itô Bridge Sde 2D:
	| dX(t) = 4 * (-1 - X(t)) * dt + 0.2 * dB1(t)
	| dY(t) = X(t) * dt + 0 * dB2(t)
	| Correlation structure:                    
             1.0 0.7
             0.7 1.0
Method:
	| Euler scheme with order 0.5
Summary:
	| Size of process	| N = 1001.
	| Crossing realized	| C = 500 among 500.
	| Initial values	| x0 = (0,0).
	| Ending values		| y = (0,0).
	| Time of process	| t in [0,5].
	| Discretization	| Dt = 0.005.
> 
> ## dX(t) = 4*(-1-X(t))*Y(t) dt + 0.2 * dW1(t) ; x01 = 0 and y01 = 0
> ## dY(t) = 4*(1-Y(t)) *X(t) dt + 0.2 * dW2(t) ; x02 = -1 and y02 = -2
> ## dZ(t) = 4*(1-Z(t)) *Y(t) dt + 0.2 * dW3(t) ; x03 = 0.5 and y03 = 0.5       
> ## W1(t), W2(t) and W3(t) are three correlated Brownian motions with Sigma
> set.seed(1234)
> 
> fx <- expression(4*(-1-x)*y, 4*(1-y)*x, 4*(1-z)*y)
> gx <- rep(expression(0.2),3)
> # correlation matrix
> Sigma <-matrix(c(1,0.3,0.5,0.3,1,0.2,0.5,0.2,1),nrow=3,ncol=3) 
> 
> res <- bridgesde3d(x0=c(0,-1,0.5),y=c(0,-2,0.5),drift=fx,diffusion=gx,corr=Sigma,M=200)
> res
Itô Bridge Sde 3D:
	| dX(t) = 4 * (-1 - X(t)) * Y(t) * dt + 0.2 * dB1(t)
	| dY(t) = 4 * (1 - Y(t)) * X(t) * dt + 0.2 * dB2(t)
	| dZ(t) = 4 * (1 - Z(t)) * Y(t) * dt + 0.2 * dB3(t)
	| Correlation structure:                        
             1.0 0.3 0.5
             0.3 1.0 0.2
             0.5 0.2 1.0
Method:
	| Euler scheme with order 0.5
Summary:
	| Size of process	| N = 1001.
	| Crossing realized	| C = 199 among 200.
	| Initial values	| x0 = (0,-1,0.5).
	| Ending values		| y  = (0,-2,0.5).
	| Time of process	| t in [0,1].
	| Discretization	| Dt = 0.001.
> summary(res) ## Monte-Carlo statistics at time T/2=0.5

Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 0.5
	| Crossing realized 199 among 200
                          X         Y          Z
Mean              0.6764265 0.5060498  0.1278097
Variance          0.0117415 0.0074919  0.0294141
Median            0.6798065 0.5126240  0.1348365
Mode              0.6784697 0.5181950  0.1659863
First quartile    0.6182005 0.4487512  0.0215379
Third quartile    0.7489718 0.5609637  0.2432146
Minimum           0.3053625 0.2798871 -0.6098128
Maximum           0.9583460 0.7946176  0.5085510
Skewness         -0.4370167 0.0463749 -0.4678248
Kurtosis          3.7429361 3.2233451  3.9084817
Coef-variation    0.1601923 0.1710418  1.3418808
3th-order moment -0.0005560 0.0000301 -0.0023600
4th-order moment  0.0005160 0.0001809  0.0033816
5th-order moment -0.0000666 0.0000079 -0.0012388
6th-order moment  0.0000393 0.0000075  0.0010249
> summary(res,at=0.25) ## Monte-Carlo statistics at time 0.25

Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 0.25
	| Crossing realized 199 among 200
                         X          Y          Z
Mean             0.9903957 -0.1654947  0.0045809
Variance         0.0098876  0.0081776  0.0317897
Median           0.9879226 -0.1628644 -0.0081038
Mode             0.9814483 -0.1511598 -0.0318484
First quartile   0.9293625 -0.2243503 -0.1044067
Third quartile   1.0471662 -0.1019929  0.1260630
Minimum          0.7210630 -0.3896680 -0.7189567
Maximum          1.3080265  0.0526330  0.5068231
Skewness         0.1856698 -0.1021267 -0.0951232
Kurtosis         3.4350811  2.6662111  3.8314379
Coef-variation   0.1004008 -0.5464238 38.9215147
3th-order moment 0.0001825 -0.0000755 -0.0005392
4th-order moment 0.0003358  0.0001783  0.0038720
5th-order moment 0.0000174 -0.0000033 -0.0007369
6th-order moment 0.0000180  0.0000056  0.0010499
> summary(res,at=0.75) ## Monte-Carlo statistics at time 0.75

Monte-Carlo Statistics for (X(t),Y(t),Z(t)) at time t = 0.75
	| Crossing realized 199 among 200
                         X          Y          Z
Mean             2.0010387  0.1231822 -0.5046211
Variance         0.0097342  0.0041729  0.0501655
Median           1.9955165  0.1289686 -0.4870370
Mode             1.9808758  0.1356478 -0.4096230
First quartile   1.9323563  0.0806200 -0.6596325
Third quartile   2.0711211  0.1653711 -0.3442381
Minimum          1.7766990 -0.0713646 -1.1414503
Maximum          2.2423345  0.2892217 -0.0188320
Skewness         0.0497121 -0.1117937 -0.2686653
Kurtosis         2.4964785  3.0281126  2.5153648
Coef-variation   0.0493055  0.5244122 -0.4438511
3th-order moment 0.0000477 -0.0000301 -0.0030187
4th-order moment 0.0002366  0.0000527  0.0063301
5th-order moment 0.0000031 -0.0000007 -0.0010159
6th-order moment 0.0000081  0.0000010  0.0012145
> ##
> plot(res,type="n")
> lines(time(res),apply(res$X,1,mean),col=3,lwd=2)
> lines(time(res),apply(res$Y,1,mean),col=4,lwd=2)
> lines(time(res),apply(res$Z,1,mean),col=5,lwd=2)
> legend("topleft",c(expression(E(X[t])),expression(E(Y[t])),
+        expression(E(Z[t]))),lty=1,inset = .01,col=c(3,4,5))
> ##
> plot3D(res,display = "persp",main="3-dim bridge sde")
> 
> proc.time()
   user  system elapsed 
   3.23    0.56    3.79