naginterfaces.library.rand.bb

naginterfaces.library.rand.bb(npaths, start, term, z, c, comm, rcord=2, a=1)

bb uses a Brownian bridge algorithm to construct sample paths for a free or non-free Wiener process. The initialization function bb_init() must be called prior to the first call to bb.

For full information please refer to the NAG Library document for g05xb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g05/g05xbf.html

Parameters

npathsint

The number of Wiener sample paths to create.

startfloat, array-like, shape $\left(d\right)$

The starting value of the Wiener process.

termfloat, array-like, shape $\left(d\right)$

The terminal value at which the non-free Wiener process should end. If $a=0$, $term$ is ignored.

zfloat, array-like, shape $(:,:)$

Note: the required extent for this argument in dimension 1 is determined as follows: if $rcord=1$: $d\times (\text{N}+1-a)$; if $rcord=2$: $npaths$; otherwise: $0$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $rcord=1$: $npaths$; if $rcord=2$: $d\times (\text{N}+1-a)$; otherwise: $0$.

The Normal random numbers used to construct the sample paths.

If quasi-random numbers are used, the $d\times (\text{N}+1-a)$-dimensional quasi-random points should be stored in successive rows of $Z$.

cfloat, array-like, shape $(d,d)$

The lower triangular Cholesky factorization $C$ such that $C{C}^T$ gives the covariance matrix of the Wiener process. Elements of $C$ above the diagonal are not referenced.

commdict, communication object

Communication structure.

This argument must have been initialized by a prior call to bb_init().

rcordint, optional

The order in which Normal random numbers are stored in $z$ and in which the generated values are returned in $b$.

aint, optional

If $a=0$, a free Wiener process is created beginning at $start$ and $term$ is ignored.

If $a=1$, a non-free Wiener process is created beginning at $start$ and ending at $term$.

Returns

zfloat, ndarray, shape $(:,:)$

The Normal random numbers premultiplied by $C$.

bfloat, ndarray, shape $(:,:)$

The values of the Wiener sample paths.

Let $X_{p,i}^k$ denote the $k$th dimension of the $i$th point of the $p$th sample path where $1\le k\le d$, $1\le i\le \text{N}+1$ and $1\le p\le npaths$.

The point $X_{p,i}^k$ is stored at $B(p,k+(i-1)\times d)$.

The starting value $start$ is never stored, whereas the terminal value is always stored.

Raises

NagValueError

(errno $-99$)

An unexpected error occurred during execution of bb. Please contact NAG with the following error message: error in $⟨value⟩$, $\text{ASSERT}=⟨value⟩$.

(errno $1$)

On entry, $comm$[‘rcomm’] was not initialized or has been corrupted.

(errno $2$)

On entry, $npaths=⟨value⟩$.

Constraint: $npaths\ge 1$.

(errno $3$)

On entry, the value of $rcord$ is invalid.

(errno $4$)

On entry, $d=⟨value⟩$.

Constraint: $d\ge 1$.

(errno $5$)

On entry, $a=⟨value⟩$.

Constraint: $a=0\text{or}1$.

(errno $6$)

On entry, $\text{ldz}=⟨value⟩$ and $d\times (\text{ntimes}+1-a)=⟨value⟩$.

Constraint: $\text{ldz}\ge d\times (\text{ntimes}+1-a)$.

Notes

For details on the Brownian bridge algorithm and the bridge construction order see the G05 Introduction and Notes for bb_init. Recall that the terms Wiener process (or free Wiener process) and Brownian motion are often used interchangeably, while a non-free Wiener process (also known as a Brownian bridge process) refers to a process which is forced to terminate at a given point.

References

Glasserman, P, 2004, Monte Carlo Methods in Financial Engineering, Springer