# rvasicek: Simulation and Density of Vasicek Process In rpgm: Fast Simulation of Normal/Exponential Random Variables and Stochastic Differential Equations / Poisson Processes

## Description

The definition of the process used here is:

dX_t = -a(X_t - mu) + sd*dW_t,

where (mu, a, sd) are the three real parameters.

## Usage

 ```1 2 3 4``` ```rvasicek(n, m, x0 = 0, mu = 0, a = 1, sd = 1, T = 1, drop = TRUE) dvasicek(x, mu=0, a=1, sd=1, T=1, log = FALSE) lvasicek(x, mu=0, a=1, sd=1, T=1) evasicek(x, a0=1, T=1) ```

## Arguments

 `n` integer, number of paths. `m` integer, number of steps, the step size will be T/m. `x` double, the vector of the observed values of a Vasicek process. `x0` double, the initial value. `mu` double, the value on which the process is centered and has an attraction when it is away. `a` double, the coefficient of how strong is the mean reversion when the process is away from `mu`. `sd` double, the volatility. `T` double, the final date on which the brownian motion is simulated. `drop` logical, if `n = 1` and `drop = TRUE` then the function returns the single path of the brownian motion as a vector instead of a matrix. `log` logical, if TRUE, returns the log-density, if FALSE, returns the density. `a0` double, starting value of `a` in the estimation algorithm.

## Value

`rvasicek` returns a `(n, m+1)` matrix of n path of the Vasicek process. `dvasicek` returns a vector of size `length(x)-1`. Note that the first value has no density. `lvasicek` returns the log-liklihood associated to `dvasicek` and `evasicek` returns the Maximum Likelihood Estimator of the parameters `(mu, a, sd)`.

## Note

If `mu = 0`, the process coincides with the Ornstein-Uhlenbeck process.

## Author(s)

Nicolas Baradel - PGM Solutions

## References

https://en.wikipedia.org/wiki/Vasicek_model

## Examples

 ```1 2``` ```x <- rvasicek(5, 10) dvasicek(x[1L, ]) ```

### Example output

```  0.6621352 1.0899338 1.2677005 1.1900134 0.7833885 0.4952588 1.3187989
 1.0308698 1.3115995 0.7758035
```

rpgm documentation built on March 18, 2018, 2:24 p.m.