# Brownian motion, Brownian bridge, and geometric Brownian motion simulators

### Description

Brownian motion, Brownian bridge, and geometric Brownian motion simulators.

### Usage

1 2 3 |

### Arguments

`x` |
initial value of the process at time |

`y` |
terminal value of the process at time |

`t0` |
initial time. |

`r` |
the interest rate of the GBM. |

`sigma` |
the volatility of the GBM. |

`T` |
final time. |

`N` |
number of intervals in which to split |

### Details

These functions return an invisible `ts`

object containing
a trajectory of the process calculated on a grid of `N+1`

equidistant points between `t0`

and `T`

; i.e.,
`t[i] = t0 + (T-t0)*i/N`

, `i in 0:N`

. `t0=0`

for the
geometric Brownian motion.

The function `BBridge`

returns a trajectory of the Brownian bridge
starting at `x`

at time `t0`

and
ending at `y`

at time `T`

; i.e.,

*(B(t), t0 <= t <= T | B(t0)=x, B(T)=y)*

The function `BM`

returns
a trajectory of the translated
Brownian motion *(B(t), t >= t0 | B(t0)=x)*;
i.e., *x+B(t-t0)* for `t >= t0`

.
The standard Brownian motion is obtained
choosing `x=0`

and `t0=0`

(the default values).

The function `GBM`

returns a trajectory of the geometric Brownian motion
starting at `x`

at time `t0=0`

; i.e., the process

*S(t) = x * exp((r-sigma^2/2)*t + sigma*B(t))*

### Value

`X` |
an invisible |

### Author(s)

Stefano Maria Iacus

### Examples

1 2 3 |