# rcOU: Ornstein-Uhlenbeck or Vasicek process conditional law In sde: Simulation and Inference for Stochastic Differential Equations

## Description

Density, distribution function, quantile function, and random generation for the conditional law X(t+Dt) | X(t)=x0 of the Ornstein-Uhlenbeck process, also known as the Vasicek process.

## Usage

 ```1 2 3 4``` ```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]`; otherwise `P[X > x]`.

## Details

This function returns quantities related to the conditional law of the process solution of

dX_t = (theta[1] - theta[2]*Xt)*dt + theta[3]*dWt.

Constraints: theta[2]>0, theta[3]>0.

Please note that the process is stationary only if theta[2]>0.

## Value

 `x` a numeric vector

## Author(s)

Stefano Maria Iacus

## 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.

`rsOU`

## Examples

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

### Example output

```Loading required package: MASS

Attaching package: 'fda'

The following object is masked from 'package:graphics':

matplot

Attaching package: 'zoo'

The following objects are masked from 'package:base':

as.Date, as.Date.numeric

sde 2.0.15
Companion package to the book
'Simulation and Inference for Stochastic Differential Equations With R Examples'
Iacus, Springer NY, (2008)
To check the errata corrige of the book, type vignette("sde.errata")
[1] 1.026021
```

sde documentation built on May 31, 2017, 3:58 a.m.