Sim.DiffProc (version 2.1)

BMRWF: Creating Flow of Brownian Motion (by a Random Walk)

Description

Simulation flow of the brownian motion model by a Random Walk.

Usage

BMRWF(N, M, t0, T, C, output = FALSE)

Arguments

N

size of process.

M

number of trajectories.

t0

initial time.

T

final time.

C

constant positive (if C = 1 it is standard brownian motion).

output

if output = TRUE write a output to an Excel 2007.

Value

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

Details

One characterization of the Brownian motion says that it can be seen as the limit of a random walk in the following sense. Given a sequence of independent and identically distributed random variables X1, X2, . . . , Xn, taking only two values +1 and -1 with equal probability and considering the partial sum, Sn = X1+ X2+ . . . + Xn. then, as n --> lnf,P(Sn/sqrt(N) < x) = P( W(t) < x). Where [x] is the integer part of the real number x. Please note that this result is a refinement of the central limit theorem that, in our case, asserts that Sn/sqrt(n) ~~> N(0,1).

See Also

BMN simulation brownian motion by the normal distribution, BMRW simulation brownian motion by a random walk, BB Simulation of brownian bridge model, GBM simulation geometric brownian motion Model.

Examples

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BMRWF(N=1000,M=5,t0=0,T=1,C=1)
BMRWF(N=1000,M=5,t0=0,T=1,C=10)