Sim.DiffProc (version 2.5)

RadialP_1: Radial Process Model(S = 1,Sigma) Or Attractive Model

Description

Simulation the radial process one-dimensional (S = 1).

Usage

RadialP_1(N, t0, Dt, T = 1, R0, K, Sigma, Output = FALSE, 
          Methods = c("Euler", "Milstein", "MilsteinS", 
          "Ito-Taylor", "Heun", "RK3"), ...)

Arguments

N

size of process.

t0

initial time.

Dt

time step of the simulation (discretization).

T

final time.

R0

initial value of the process at time t0 ,(R0 > 0).

K

constant K > 0.

Sigma

constant Sigma > 0.

Output

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

Methods

method of simulation ,see details snssde.

...

Value

• data.frame(time,R(t)) and plot of process R(t).

Details

The attractive models is defined by the system for stochastic differential equation two-dimensional : $$dX(t) = (-K * X(t)/(sqrt(X(t)^2 + Y(t)^2))^(S+1) )* dt + Sigma* dW1(t)$$ $$dY(t) = (-K * Y(t)/(sqrt(X(t)^2 + Y(t)^2))^(S+1) )* dt + Sigma* dW2(t)$$ dW1(t) and dW2(t) are brownian motions independent. Using Ito transform, it is shown that the Radial Process R(t) with R(t)=||(X(t),Y(t))|| is a markovian diffusion, solution of the stochastic differential equation one-dimensional: $$dR(t) = ((0.5 * Sigma^2 * R(t)^(S-1) - K)/ R(t)^S )* dt + Sigma* dW(t)$$ If S = 1 (ie M(S=1,Sigma)) the R(t) is : $$dR(t) = ((0.5*Sigma^2 -K )/R(t) ) * dt + Sigma* dW(t)$$ Where ||.|| is the Euclidean norm and dW(t) is a determined brownian motions. 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

RadialP2D_1, RadialP2D_1PC, RadialP3D_1, tho_M1, fctgeneral, hist_general, Kern_meth.

Examples

## Example 1
 RadialP_1(N=1000, t0=0, Dt=0.001, T = 1, R0=2, K=2, 
           Sigma=0.5, Output = FALSE)
## Example 2
 RadialP_1(N=1000, t0=0, Dt=0.001, T = 1, R0=3, K=3.5, 
           Sigma=0.5, Output = FALSE,Methods="Heun")