naginterfaces.library.rand.bb_​inc

naginterfaces.library.rand.bb_inc(npaths, diff, z, c, comm, rcord=2, a=1)

bb_inc computes scaled increments of sample paths of a free or non-free Wiener process, where the sample paths are constructed by a Brownian bridge algorithm. The initialization function bb_inc_init() must be called prior to the first call to bb_inc.

For full information please refer to the NAG Library document for g05xd

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g05/g05xdf.html

Parameters

npathsint

The number of Wiener sample paths.

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

The difference between the terminal value and starting value of the Wiener process. If $a=0$, $diff$ 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 $d\times (N+1-a)$-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_inc_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 and $diff$ is ignored.

If $a=1$, a non-free Wiener process is created where $diff$ is the difference between the terminal value and the starting value of the process.

Returns

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

The Normal random numbers premultiplied by $c$.

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

The scaled Wiener increments.

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 increment $\frac{(X_{p,i}^k-X_{p,i-1}^k)}{(t_i-t_{i-1})}$ is stored at $B(p,k+(i-1)\times d)$.

Raises

NagValueError

(errno $-99$)

An unexpected error occurred during execution of bb_inc. 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_inc_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.

Fix two times $t_0<T$, let ${\left(t_i\right)}_{1\le i\le N}$ be any set of time points satisfying $t_0<t_1<t_2<\cdots <t_N<T$, and let $X_{t_0}$, ${\left(X_{t_i}\right)}_{1\le i\le N}$, $X_T$ denote a $d$-dimensional Wiener sample path at these time points.

The Brownian bridge increments generator uses the Brownian bridge algorithm to construct sample paths for the (free or non-free) Wiener process $X$, and then uses this to compute the scaled Wiener increments

$$\frac{X_{t_1}-X_{t_0}}{t_1-t_0},\frac{X_{t_2}-X_{t_1}}{t_2-t_1},\dots ,\frac{X_{t_N}-X_{t_{N-1}}}{t_N-t_{N-1}},\frac{X_T-X_{t_N}}{T-t_N}$$

References

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

See also

naginterfaces.library.examples.rand.bb_inc_ex.main()