g05xec 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 g05xacorg05xcc to initialize the Brownian bridge generators with the chosen bridge construction order.
The function may be called by the names: g05xec or nag_rand_bb_make_bridge_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.
3.1Supported Bridge Construction Orders
g05xec 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:
g05xec, 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 g05xec 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 g05xec 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: – Nag_BridgeOrderInput
On entry: the bridge construction order to use.
, , or .
2: – doubleInput
On entry: , the start value of the time interval on which the Wiener process is to be constructed.
3: – doubleInput
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: – const doubleInput
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: – const IntegerInput
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 and may be NULL.
On exit: the output bridge construction order. This should be passed to g05xacorg05xcc.
9: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, and . Constraint: .
On entry, . 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 internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, and .
Constraint: the elements in intime must be in increasing order.
On entry, and . Constraint: .
On entry, , and . Constraint: .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g05xec 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
This example calls g05xec, g05xacandg05xbc 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.