Sim.DiffProc (version 2.5)

PEABM: Parametric Estimation of Arithmetic Brownian Motion(Exact likelihood inference)

Description

Parametric estimation of Arithmetic Brownian Motion

Usage

PEABM(X, delta, starts = list(theta= 1, sigma= 1), leve = 0.95)

Arguments

X

a numeric vector of the observed time-series values.

delta

the fraction of the sampling period between successive observations.

starts

named list. Initial values for optimizer.

leve

the confidence level required.

Value

• coef
  Coefficients extracted from the model.
• AIC
  A numeric value with the corresponding AIC.
• vcov
  A matrix of the estimated covariances between the parameter estimates in the linear or non-linear predictor of the model.
• confint
  A matrix (or vector) with columns giving lower and upper confidence limits for each parameter. These will be labelled as (1-level)/2 and 1 - (1-level)/2.

Details

This process solves the stochastic differential equation : $$dX(t) = theta * dt + sigma * dW(t)$$ The conditional density p(t,.|x) is the density of a Gaussian law with mean = x0 + theta * t and variance = sigma^2 * t. R has the [dqpr]norm functions to evaluate the density, the quantiles, and the cumulative distribution or generate pseudo random numbers from the normal distribution.

See Also

PEOU Parametric Estimation of Ornstein-Uhlenbeck Model, PEOUexp Explicit Estimators of Ornstein-Uhlenbeck Model, PEOUG Parametric Estimation of Hull-White/Vasicek Models, PEBS Parametric Estimation of model Black-Scholes.

Examples

## Parametric estimation of Arithmetic Brownian Motion.
## t0 = 0 ,T = 100
 data(DATA3)
 res <- PEABM(DATA3,delta=0.1,starts=list(theta=1,sigma=1),leve = 0.95)
 res
 ABMF(N=1000,M=10,t0=0,T=100,x0=DATA3[1],theta=res$coef[1],sigma=res$coef[2])
 points(seq(0,100,length=length(DATA3)),DATA3,type="l",lwd=3,col="blue")