Sim.DiffProc (version 2.5)

RadialP3D_1: Three-Dimensional Attractive Model Model(S = 1,Sigma)

Description

Simulation 3-dimensional attractive model (S = 1).

Usage

RadialP3D_1(N, t0, Dt, T = 1, X0, Y0, Z0, v, K, Sigma, 
            Output = FALSE)

Arguments

N

size of process.

t0

initial time.

Dt

time step of the simulation (discretization).

T

final time.

X0

initial value of the process X(t) at time t0.

Y0

initial value of the process Y(t) at time t0.

Z0

initial value of the process Z(t) at time t0.

v

threshold. 0 < v < sqrt(X0^2 + Y0 ^2 + Z0^2)

K

constant K > 0.

Sigma

constant Sigma > 0.

Output

if Output = TRUE write a Output to an Excel (.csv).

Value

• data.frame(time,X(t),Y(t),Z(t)) and plot of process 3-D.

Details

The attractive models is defined by the system for stochastic differential equation three-dimensional : $$dX(t) = (-K * X(t)/(sqrt(X(t)^2 + Y(t)^2 + Z(t)^2))^(S+1) )* dt + Sigma* dW1(t)$$ $$dY(t) = (-K * Y(t)/(sqrt(X(t)^2 + Y(t)^2 + Z(t)^2))^(S+1) )* dt + Sigma* dW2(t)$$ $$dZ(t) = (-K * Z(t)/(sqrt(X(t)^2 + Y(t)^2 + Z(t)^2))^(S+1) )* dt + Sigma* dW3(t)$$ dW1(t), dW2(t) and dW3(t) are brownian motions independent. If S = 1 (ie M(S=1,Sigma)) the system SDE is : $$dX(t) = (-K * X(t)/(X(t)^2 + Y(t)^2 + Z(t)^2) )* dt + Sigma* dW1(t)$$ $$dY(t) = (-K * Y(t)/(X(t)^2 + Y(t)^2 + Z(t)^2) )* dt + Sigma* dW2(t)$$ $$dZ(t) = (-K * Z(t)/(X(t)^2 + Y(t)^2 + Z(t)^2) )* dt + Sigma* dW3(t)$$ For more detail consulted References.

References

1. K.Boukhetala, Estimation of the first passage time distribution for a simulated diffusion process, Maghreb Math.Rev, Vol.7, No 1, Jun 1998, pp. 1-25.
2. K.Boukhetala, Simulation study of a dispersion about an attractive centre. In proceedings of 11th Symposium Computational Statistics, edited by R.Dutter and W.Grossman, Wien , Austria, 1994, pp. 128-130.
3. K.Boukhetala,Modelling and simulation of a dispersion pollutant with attractive centre, Edited by Computational Mechanics Publications, Southampton ,U.K and Computational Mechanics Inc, Boston, USA, pp. 245-252.
4. K.Boukhetala, Kernel density of the exit time in a simulated diffusion, les Annales Maghrebines De L ingenieur, Vol , 12, N Hors Serie. Novembre 1998, Tome II, pp 587-589.

See Also

RadialP3D_2.

Examples

RadialP3D_1(N=1000, t0=0, Dt=0.001, T = 1, X0=1, Y0=0.5, Z0=0.5, 
 v=0.2, K=3, Sigma=0.2, Output = FALSE)