Sim.DiffProc (version 2.5)

PEOUexp: Parametric Estimation of Ornstein-Uhlenbeck Model (Explicit Estimators)

Description

Explicit estimators of Ornstein-Uhlenbeck Model.

Usage

PEOUexp(X, delta)

Arguments

X

a numeric vector of the observed time-series values.

delta

the fraction of the sampling period between successive observations.

Value

• r
  Estimator of speed of reversion.
• sigma
  Estimator of volatility.

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)), the maximum likelihood estimator of r is available in explicit form and takes the form : $$r = -(1/dt) * log(sum(X(t)*X(t-1)) / sum( X(t-1)^2))$$ which is defined only if sum(X(t)*X(t-1)) > 0, this estimator is consistent and asymptotically Gaussian. The maximum likelihood estimator of : $$sigma^2 = (2*r)/(N*(1-exp(-2*dt*r))) * sum(X(t)-X(t-1)**exp(-dt*r))^2$$

See Also

PEABM Parametric Estimation of Arithmetic Brownian Motion, PEOU Parametric Estimation of Ornstein-Uhlenbeck Model, PEOUG Parametric Estimation of Hull-White/Vasicek Models, PEBS Parametric Estimation of model Black-Scholes.

Examples

## t0 = 0 ,T = 10
 data(DATA1)
 res <- PEOUexp(DATA1,delt=0.01)
 res
 OUF(N=1000,M=10,t0=0,T=10,x0=DATA1[1],r=res$r,sigma=res$sigma)
 points(seq(0,10,length=length(DATA1)),DATA1,type="l",lwd=3,col="blue")