g05xef takes a set of input times and permutes them to specify one of several predefined Brownian bridge construction orders. The permuted times can be passed to g05xaf or g05xcf to initialize the Brownian bridge generators with the chosen bridge construction order.

2 Specification

Fortran Interface

Subroutine g05xef (

bgord, t0, tend, ntimes, intime, nmove, move, times, ifail)

Integer, Intent (In)	::	bgord, ntimes, nmove, move(nmove)
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	t0, tend, intime(ntimes)
Real (Kind=nag_wp), Intent (Out)	::	times(ntimes)

C Header Interface

#include <nag.h>

void	g05xef_ (const Integer bgord, const double t0, const double tend, const Integer ntimes, const double intime[], const Integer nmove, const Integer move[], double times[], Integer ifail)

The routine may be called by the names g05xef or nagf_rand_bb_make_bridge_order.

3 Description

The Brownian bridge algorithm (see Glasserman (2004)) is a popular method for constructing a Wiener process at a set of discrete times,

t_{0} < t_{1} < t_{2} <, \dots, < t_{N} < T

, for

N \geq 1

. To ease notation we assume that

T

has the index

N + 1

so that

T = t_{N + 1}

. Inherent in the algorithm is the notion of a bridge construction order which specifies the order in which the

N + 2

points of the Wiener process,

X_{t_{0}}, X_{T}

and

X_{t_{i}}

, for

i = 1, 2, \dots, N

, are generated. The value of

X_{t_{0}}

is always assumed known, and the first point to be generated is always the final time

X_{T}

. Thereafter, successive points are generated iteratively by an interpolation formula, using points which were computed at previous iterations. In many cases the bridge construction order is not important, since any construction order will yield a correct process. However, in certain cases, for example when using quasi-random variates to construct the sample paths, the bridge construction order can be important.

3.1 Supported Bridge Construction Orders

g05xef accepts as input an array of time points

t_{1}, t_{2}, \dots, t_{N}, T

at which the Wiener process is to be sampled. These time points are then permuted to construct the bridge. In all of the supported construction orders the first construction point is

T

which has index

N + 1

. The remaining points are constructed by iteratively bisecting (sub-intervals of) the time indices interval

[0, N + 1]

, as Figure 1 illustrates:

Figure 1

The time indices interval is processed in levels

L^{i}

, for

i = 1, 2, \dots

. Each level

L^{i}

contains

n_{i}

points

L_{1}^{i}, \dots, L_{n_{i}}^{i}

where

n_{i} \leq 2^{i - 1}

. The number of points at each level depends on the value of

N

. The points

L_{j}^{i}

for

i \geq 1

and

j = 1, 2, \dots n_{i}

are computed as follows: define

L_{0}^{0} = N + 1

and set

\begin{matrix} L_{j}^{i} = J + (K - J) / 2 & where \\ J = \max {L_{k}^{p} : 1 \leq k \leq n_{p}, ​ 0 \leq p < i ​ and ​ L_{k}^{p} < L_{j}^{i}} & ​ and ​ \\ K = \min {L_{k}^{p} : 1 \leq k \leq n_{p}, ​ 0 \leq p < i ​ and ​ L_{k}^{p} > L_{j}^{i}} \end{matrix}

By convention the maximum of the empty set is taken to be to be zero. Figure 1 illustrates the algorithm when

N + 1

is a power of two. When

N + 1

is not a power of two, one must decide how to round the divisions by

2

. For example, if one rounds down to the nearest integer, then one could get the following:

Figure 2

From the series of bisections outlined above, two ways of ordering the time indices

L_{j}^{i}

are supported. In both cases, levels are always processed from coarsest to finest (i.e., increasing

i

). Within a level, the time indices can either be processed left to right (i.e., increasing

j

) or right to left (i.e., decreasing

j

). For example, when processing left to right, the sequence of time indices could be generated as:

\begin{matrix} N + 1 & L_{1}^{1} & L_{1}^{2} & L_{2}^{2} & L_{1}^{3} & L_{2}^{3} & L_{3}^{3} & L_{4}^{3} & \dots \end{matrix}

while when processing right to left, the same sequence would be generated as:

\begin{matrix} N + 1 & L_{1}^{1} & L_{2}^{2} & L_{1}^{2} & L_{4}^{3} & L_{3}^{3} & L_{2}^{3} & L_{1}^{3} & \dots \end{matrix}

g05xef, therefore, offers four bridge construction methods; processing either left to right or right to left, with rounding either up or down. Which method is used is controlled by the bgord argument. For example, on the set of times

\begin{matrix} t_{1} & t_{2} & t_{3} & t_{4} & t_{5} & t_{6} & t_{7} & t_{8} & t_{9} & t_{10} & t_{11} & t_{12} & T \end{matrix}

the Brownian bridge would be constructed in the following orders:

$bgord = 1$ (processing left to right, rounding down)

$\begin{matrix} T & t_{6} & t_{3} & t_{9} & t_{1} & t_{4} & t_{7} & t_{11} & t_{2} & t_{5} & t_{8} & t_{10} & t_{12} \end{matrix}$
$bgord = 2$ (processing left to right, rounding up)

$\begin{matrix} T & t_{7} & t_{4} & t_{10} & t_{2} & t_{6} & t_{9} & t_{12} & t_{1} & t_{3} & t_{5} & t_{8} & t_{11} \end{matrix}$
$bgord = 3$ (processing right to left, rounding down)

$\begin{matrix} T & t_{6} & t_{9} & t_{3} & t_{11} & t_{7} & t_{4} & t_{1} & t_{12} & t_{10} & t_{8} & t_{5} & t_{2} \end{matrix}$
$bgord = 4$ (processing right to left, rounding up)

$\begin{matrix} T & t_{7} & t_{10} & t_{4} & t_{12} & t_{9} & t_{6} & t_{2} & t_{11} & t_{8} & t_{5} & t_{3} & t_{1} \end{matrix} .$

The four construction methods described above can be further modified through the use of the input array move. To see the effect of this argument, suppose that an array

A

holds the output of g05xef when

nmove = 0

(i.e., the bridge construction order as specified by bgord only). Let

B = {t_{j} : j = move (i), i = 1, 2, \dots, nmove}

be the array of all times identified by move, and let

C

be the array

A

with all the elements in

B

removed, i.e.,

C = {A (i) : A (i) \neq B (j), i = 1, 2, \dots, ntimes, j = 1, 2, \dots, nmove} .

Then the output of g05xef when

nmove > 0

is given by

\begin{matrix} B (1) & B (2) & \dots & B (nmove) & C (1) & C (2) & \dots & C (ntimes - nmove) \end{matrix}

When the Brownian bridge is used with quasi-random variates, this functionality can be used to allow specific sections of the bridge to be constructed using the lowest dimensions of the quasi-random points.

4 References

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

5 Arguments

1: $bgord$ – Integer Input

On entry: the bridge construction order to use.

Constraint:

bgord = 1

2

3

4

2: $t0$ – Real (Kind=nag_wp) Input

On entry:

t_{0}

, the start value of the time interval on which the Wiener process is to be constructed.

3: $tend$ – Real (Kind=nag_wp) Input

On entry:

T

, the largest time at which the Wiener process is to be constructed.

4: $ntimes$ – Integer Input

On entry:

N

, the number of time points in the Wiener process, excluding

t_{0}

and

T

Constraint:

ntimes \geq 1

5: $intime (ntimes)$ – Real (Kind=nag_wp) array Input

On entry: the time points,

t_{1}, t_{2}, \dots, t_{N}

, at which the Wiener process is to be constructed. Note that the final time

T

is not included in this array.

Constraints:

$t0 < intime (i)$ and $intime (i) < intime (i + 1)$ , for $i = 1, 2, \dots, ntimes - 1$ ;
$intime (ntimes) < tend$ .

6: $nmove$ – Integer Input

On entry: the number of elements in the array move.

Constraint:

0 \leq nmove \leq ntimes

7: $move (nmove)$ – Integer array Input

On entry: the indices of the entries in intime which should be moved to the front of the times array, with

move (j) = i

setting the

j

th element of times to

t_{i}

. Note that

i

ranges from

1

to ntimes. When

nmove = 0

, move is not referenced.

Constraint:

1 \leq move (j) \leq ntimes

, for

j = 1, 2, \dots, nmove

The elements of move must be unique.

8: $times (ntimes)$ – Real (Kind=nag_wp) array Output

On exit: the output bridge construction order. This should be passed to g05xaf or g05xcf.

9: $ifail$ – 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, $bgord = ⟨ value ⟩$ .
Constraint: $bgord = 1$ , $2$ , $3$ or $4$

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

$ifail = 3$: On entry, $nmove = ⟨ value ⟩$ and $ntimes = ⟨ value ⟩$ .
Constraint: $0 \leq nmove \leq ntimes$ .

$ifail = 4$: On entry, $intime (⟨ value ⟩) = ⟨ value ⟩$ and $intime (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: the elements in intime must be in increasing order.

On entry, $intime (1) = ⟨ value ⟩$ and $t0 = ⟨ value ⟩$ .
Constraint: $intime (1) > t0$ .

On entry, $ntimes = ⟨ value ⟩$ , $intime (ntimes) = ⟨ value ⟩$ and $tend = ⟨ value ⟩$ .
Constraint: $intime (ntimes) < tend$ .

$ifail = 5$: On entry, $move (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $move (i) \geq 1$ for all $i$ .

On entry, $move (⟨ value ⟩) = ⟨ value ⟩$ and $ntimes = ⟨ value ⟩$ .
Constraint: $move (i) \leq ntimes$ for all $i$ .

$ifail = 6$: On entry, $move (⟨ value ⟩)$ and $move (⟨ value ⟩)$ both equal $⟨ value ⟩$ .
Constraint: all elements in move must be unique.

$ifail = - 99$: 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

g05xef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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 g05xef, g05xaf and g05xbf to generate two sample paths of a three-dimensional free Wiener process. The array move is used to ensure that a certain part of the sample path is always constructed using the lowest dimensions of the input quasi-random points. For further details on using quasi-random points with the Brownian bridge algorithm, please see Section 2.6 in the G05 Chapter Introduction.