Sim.DiffProc (version 2.5)

RadialP2D_1PC: Two-Dimensional Attractive Model in Polar Coordinates Model(S = 1,Sigma)

Description

Simulation 2-dimensional attractive model (S = 1) in polar coordinates.

Usage

RadialP2D_1PC(N, R0, t0, T, ThetaMax, K, sigma, output = FALSE)

Arguments

N

size of process.

R0

initial valueR0 > 0 at time t0.

t0

initial time.

T

final time.

ThetaMax

polar coordinates, example ThetaMax = 2*pi.

K

constant K > 0.

sigma

constant sigma > 0.

output

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

Value

• data.frame(time,R(t),theta(t)) and plot of process 2-D in polar coordinates.

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. R(t)=sqrt(X(t)^2 + Y(t)^2) it is distance between X(t) and Y(t), then X(t)=R(t)*cos(theta(t)) and Y(t)=R(t)*sin(theta(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

snssde2D, PredCorr2D, RadialP2D_2PC, RadialP3D_1, tho_M1, fctgeneral, hist_general, Kern_meth.

Examples

RadialP2D_1PC(N=1000, R0=3, t0=0, T=1, ThetaMax=4*pi, K=2, sigma=1, 
               output = FALSE)