sde (version 2.0.9)

rcOU: Ornstein-Uhlenbeck or Vasicek process conditional law

Description

Density, distribution function, quantile function, and random generation for the conditional law $X(t+D_t) | X(t)=x_0$ of the Ornstein-Uhlenbeck process, also known as the Vasicek process.

Usage

dcOU(x, Dt, x0, theta, log = FALSE)
pcOU(x, Dt, x0, theta, lower.tail = TRUE, log.p = FALSE) 
qcOU(p, Dt, x0, theta, lower.tail = TRUE, log.p = FALSE)
rcOU(n=1, Dt, x0, theta)

Arguments

x

vector of quantiles.

p

vector of probabilities.

Dt

lag or time.

x0

the value of the process at time t; see details.

theta

parameter of the Ornstein-Uhlenbeck process; see details.

n

number of random numbers to generate from the conditional distribution.

log, log.p

logical; if TRUE, probabilities $p$ are given as $\log(p)$.

lower.tail

logical; if TRUE (default), probabilities are P[X <= x]<="" code="">; otherwise P[X > x].

Value

• x
  a numeric vector

Details

This function returns quantities related to the conditional law of the process solution of $${\rm d}X_t = (\theta_1 - \theta_2 X_t){\rm d}t + \theta_3 {\rm d}W_t.$$

Constraints: $\theta_2>0, \theta_3>0$.

Please note that the process is stationary only if $\theta_2>0$.

References

Uhlenbeck, G. E., Ornstein, L. S. (1930) On the theory of Brownian motion, Phys. Rev., 36, 823-841.

Vasicek, O. (1977) An Equilibrium Characterization of the Term Structure, Journal of Financial Economics, 5, 177-188.

See Also

rsOU

Examples

rcOU(n=1, Dt=0.1, x0=1, theta=c(0,2,1))