g05xbf uses a Brownian bridge algorithm to construct sample paths for a free or non-free Wiener process. The initialization routine g05xaf must be called prior to the first call to g05xbf.

2 Specification

Fortran Interface

Subroutine g05xbf (

npaths, rcord, d, start, a, term, z, ldz, c, ldc, b, ldb, rcomm, ifail)

Integer, Intent (In)	::	npaths, rcord, d, a, ldz, ldc, ldb
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	start(d), term(d), c(ldc,), rcomm()
Real (Kind=nag_wp), Intent (Inout)	::	z(ldz,), b(ldb,)

C Header Interface

#include <nag.h>

void	g05xbf_ (const Integer npaths, const Integer rcord, const Integer d, const double start[], const Integer a, const double term[], double z[], const Integer ldz, const double c[], const Integer ldc, double b[], const Integer ldb, const double rcomm[], Integer ifail)

The routine may be called by the names g05xbf or nagf_rand_bb.

3 Description

For details on the Brownian bridge algorithm and the bridge construction order see Section 2.6 in the G05 Chapter Introduction and Section 3 in g05xaf. 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.

4 References

Glasserman P (2004) Monte Carlo Methods in Financial Engineering Springer

5 Arguments

Note: the following variable is used in the parameter descriptions:

N = ntimes

, the length of the array times passed to the initialization routine g05xaf.

1: $npaths$ – Integer Input

On entry: the number of Wiener sample paths to create.

Constraint:

npaths \geq 1

2: $rcord$ – Integer Input

On entry: the order in which Normal random numbers are stored in z and in which the generated values are returned in b.

Constraint:

rcord = 1

2

3: $d$ – Integer Input

On entry: the dimension of each Wiener sample path.

Constraint:

d \geq 1

4: $start (d)$ – Real (Kind=nag_wp) array Input

On entry: the starting value of the Wiener process.

5: $a$ – Integer Input

On entry: if

a = 0

, a free Wiener process is created beginning at start and term is ignored.

a = 1

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

Constraint:

a = 0

1

6: $term (d)$ – Real (Kind=nag_wp) array Input

On entry: the terminal value at which the non-free Wiener process should end. If

a = 0

, term is ignored.

7: $z (ldz, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array z must be at least

npaths

rcord = 1

and at least

d \times (N + 1 - a)

rcord = 2

On entry: the Normal random numbers used to construct the sample paths.

rcord = 1

and quasi-random numbers are used, the

d \times (N + 1 - a)

, where

N = nint

rcomm (2)

-dimensional quasi-random points should be stored in successive columns of z.

rcord = 2

and quasi-random numbers are used, the

d \times (N + 1 - a)

, where

N = nint

rcomm (2)

-dimensional quasi-random points should be stored in successive rows of z.

On exit: the Normal random numbers premultiplied by

C

8: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which g05xbf is called.

Constraints:

if $rcord = 1$ , $ldz \geq d \times (N + 1 - a)$ ;
if $rcord = 2$ , $ldz \geq npaths$ .

9: $c (ldc, *)$ – Real (Kind=nag_wp) array Input

Note: the second dimension of the array c must be at least

d

On entry: 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.

10: $ldc$ – Integer Input

On entry: the first dimension of the array c as declared in the (sub)program from which g05xbf is called.

Constraint:

ldc \geq d

11: $b (ldb, *)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array b must be at least

npaths

rcord = 1

and at least

d \times (N + 1)

rcord = 2

On exit: 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 \leq k \leq d

1 \leq i \leq N + 1

and

1 \leq p \leq npaths

rcord = 1

, the point

X_{p, i}^{k}

will be stored at

b (k + (i - 1) \times d, p)

rcord = 2

, the point

X_{p, i}^{k}

will be stored at

b (p, k + (i - 1) \times d)

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

12: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which g05xbf is called.

Constraints:

if $rcord = 1$ , $ldb \geq d \times (N + 1)$ ;
if $rcord = 2$ , $ldb \geq npaths$ .

13: $rcomm (*)$ – Real (Kind=nag_wp) array Communication Array

Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument rcomm in the previous call to g05xaf or g05xbf.

On entry: communication array as returned by the last call to g05xaf or g05xbf. This array must not be directly modified.

14: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, rcomm was not initialized or has been corrupted.

$ifail = 2$: On entry, $npaths = ⟨ value ⟩$ .
Constraint: $npaths \geq 1$ .

$ifail = 3$: On entry, $rcord = ⟨ value ⟩$ was an illegal value.

$ifail = 4$: On entry, $d = ⟨ value ⟩$ .
Constraint: $d \geq 1$ .

$ifail = 5$: On entry, $a = ⟨ value ⟩$ .
Constraint: $a = 0 or 1$ .

$ifail = 6$: On entry, $ldz = ⟨ value ⟩$ and $d \times (ntimes + 1 - a) = ⟨ value ⟩$ .
Constraint: $ldz \geq d \times (ntimes + 1 - a)$ .

On entry, $ldz = ⟨ value ⟩$ and $npaths = ⟨ value ⟩$ .
Constraint: $ldz \geq npaths$ .

$ifail = 7$: On entry, $ldc = ⟨ value ⟩$ .
Constraint: $ldc \geq ⟨ value ⟩$ .

$ifail = 8$: On entry, $ldb = ⟨ value ⟩$ and $d \times (ntimes + 1) = ⟨ value ⟩$ .
Constraint: $ldb \geq d \times (ntimes + 1)$ .

On entry, $ldb = ⟨ value ⟩$ and $npaths = ⟨ value ⟩$ .
Constraint: $ldb \geq npaths$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05xbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05xbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example calls g05xbf, g05xaf and g05xef to generate two sample paths of a three-dimensional non-free Wiener process. The process starts at zero and each sample path terminates at the point