Sim.DiffProc (version 2.5)

PEOU: Parametric Estimation of Ornstein-Uhlenbeck Model (Exact likelihood inference)

Description

Parametric estimation of Ornstein-Uhlenbeck Model.

Usage

PEOU(X, delta, starts = list(r= 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 : $$dX(t) = - r * X(t) * dt + sigma * dW(t)$$ It is ergodic for r > 0. We have also shown its exact conditional and stationary densities. In particular, the conditional density p(t,.|x) is the density of a Gaussian law with mean = x0 * exp(-r*t) and variance = ((sigma^2)/(2*r))*(1-exp(-2*r*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

PEABM Parametric Estimation of Arithmetic Brownian Motion, 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 Ornstein-Uhlenbeck Model.
## t0 = 0 ,T = 10
 data(DATA1)
 res <- PEOU(DATA1,delta=0.01,starts=list(r=2,sigma=1),leve = 0.90)
 res
 OUF(N=1000,M=10,t0=0,T=10,x0=40,r=0.1979284,sigma=3.972637)
 points(seq(0,10,length=length(DATA1)),DATA1,type="l",lwd=3,col="blue")