# BM: Brownian motion, Brownian bridge, and geometric Brownian... In sde: Simulation and Inference for Stochastic Differential Equations

## Description

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

## Usage

 ```1 2 3``` ```BBridge(x=0, y=0, t0=0, T=1, N=100) BM(x=0, t0=0, T=1, N=100) GBM(x=1, r=0, sigma=1, T=1, N=100) ```

## Arguments

 `x` initial value of the process at time `t0`. `y` terminal value of the process at time `T`. `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 `[t0,T]`.

## 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 `ts` object

## Author(s)

Stefano Maria Iacus

## Examples

 ```1 2 3``` ```plot(BM()) plot(BBridge()) plot(GBM()) ```

### 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")
```

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