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.
The Brownian bridge algorithm (see Glasserman (2004)) is a popular method for constructing a Wiener process at a set of discrete times, , for . To ease notation we assume that has the index so that . Inherent in the algorithm is the notion of a bridge construction order which specifies the order in which the points of the Wiener process, and , for , are generated. The value of is always assumed known, and the first point to be generated is always the final time . 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.
Supported Bridge Construction Orders
g05xef accepts as input an array of time points 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 which has index . The remaining points are constructed by iteratively bisecting (sub-intervals of) the time indices interval , as Figure 1 illustrates:
The time indices interval is processed in levels , for . Each level contains points where . The number of points at each level depends on the value of . The points for and are computed as follows: define and set
By convention the maximum of the empty set is taken to be to be zero. Figure 1 illustrates the algorithm when is a power of two. When is not a power of two, one must decide how to round the divisions by . For example, if one rounds down to the nearest integer, then one could get the following:
From the series of bisections outlined above, two ways of ordering the time indices are supported. In both cases, levels are always processed from coarsest to finest (i.e., increasing ). Within a level, the time indices can either be processed left to right (i.e., increasing ) or right to left (i.e., decreasing ). For example, when processing left to right, the sequence of time indices could be generated as:
while when processing right to left, the same sequence would be generated as:
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
the Brownian bridge would be constructed in the following orders:
(processing left to right, rounding down)
(processing left to right, rounding up)
(processing right to left, rounding down)
(processing right to left, rounding up)
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 holds the output of g05xef when (i.e., the bridge construction order as specified by bgord only). Let
be the array of all times identified by move, and let be the array with all the elements in removed, i.e.,
Then the output of g05xef when is given by
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.
Glasserman P (2004) Monte Carlo Methods in Financial Engineering Springer
1: – IntegerInput
On entry: the bridge construction order to use.
, , or .
2: – Real (Kind=nag_wp)Input
On entry: , the start value of the time interval on which the Wiener process is to be constructed.
3: – Real (Kind=nag_wp)Input
On entry: , the largest time at which the Wiener process is to be constructed.
4: – IntegerInput
On entry: , the number of time points in the Wiener process, excluding and .
5: – Real (Kind=nag_wp) arrayInput
On entry: the time points, , at which the Wiener process is to be constructed. Note that the final time is not included in this array.
and , for ;
6: – IntegerInput
On entry: the number of elements in the array move.
7: – Integer arrayInput
On entry: the indices of the entries in intime which should be moved to the front of the times array, with setting the th element of times to . Note that ranges from to ntimes. When , move is not referenced.
On exit: the output bridge construction order. This should be passed to g05xaf or g05xcf.
9: – IntegerInput/Output
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, . Constraint: , , or
On entry, . Constraint: .
On entry, and . Constraint: .
On entry, and .
Constraint: the elements in intime must be in increasing order.
On entry, and . Constraint: .
On entry, , and . Constraint: .
On entry, .
Constraint: for all .
On entry, and . Constraint: for all .
On entry, and both equal .
Constraint: all elements in move must be unique.
An unexpected error has been triggered by this routine. Please
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
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.
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.