NAG FL Interface
g05xdf (bb_​inc)

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1 Purpose

g05xdf 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 routine g05xcf must be called prior to the first call to g05xdf.

2 Specification

Fortran Interface
Subroutine g05xdf ( npaths, rcord, d, a, diff, 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) :: diff(d), c(ldc,*), rcomm(*)
Real (Kind=nag_wp), Intent (Inout) :: z(ldz,*), b(ldb,*)
C Header Interface
#include <nag.h>
void  g05xdf_ (const Integer *npaths, const Integer *rcord, const Integer *d, const Integer *a, const double diff[], 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 g05xdf or nagf_rand_bb_inc.

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 g05xcf. 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 t0<T, let (ti) 1iN be any set of time points satisfying t0<t1<t2<<tN<T, and let Xt0, (Xti) 1iN , XT 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
Xt1 - Xt0 t1 - t0 , Xt2 - Xt1 t2 - t1 ,, XtN - XtN-1 tN - tN-1 , XT - XtN T - tN  

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 g05xcf.
1: npaths Integer Input
On entry: the number of Wiener sample paths.
Constraint: npaths1.
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 or 2.
3: d Integer Input
On entry: the dimension of each Wiener sample path.
Constraint: d1.
4: a Integer Input
On entry: 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.
Constraint: a=0 or 1.
5: diff(d) Real (Kind=nag_wp) array Input
On entry: the difference between the terminal value and starting value of the Wiener process. If a=0, diff is ignored.
6: z(ldz,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array z must be at least npaths if rcord=1 and at least d×(N+1-a) if rcord=2.
On entry: the Normal random numbers used to construct the sample paths.
If quasi-random numbers are used, the d×(N+1-a)-dimensional quasi-random points should be stored in successive rows of Z.
On exit: the Normal random numbers premultiplied by c.
7: ldz Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which g05xdf is called.
  • if rcord=1, ldzd×(N+1-a);
  • if rcord=2, ldznpaths.
8: 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 CCT gives the covariance matrix of the Wiener process. Elements of C above the diagonal are not referenced.
9: ldc Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which g05xdf is called.
Constraint: ldcd.
10: b(ldb,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array b must be at least npaths if rcord=1 and at least d×(N+1) if rcord=2.
On exit: the scaled Wiener increments.
Let Xp,ik denote the kth dimension of the ith point of the pth sample path where 1kd, 1iN+1 and 1pnpaths. The increment (Xp,ik-Xp,i-1k)(ti-ti-1) is stored at B(p,k+(i-1)×d).
11: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which g05xdf is called.
  • if rcord=1, ldbd×(N+1);
  • if rcord=2, ldbnpaths.
12: 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 g05xcf or g05xdf.
On entry: communication array as returned by the last call to g05xcf or g05xdf. This array must not be directly modified.
13: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 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 or 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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, rcomm was not initialized or has been corrupted.
On entry, npaths=value.
Constraint: npaths1.
On entry, rcord=value was an illegal value.
On entry, d=value.
Constraint: d1.
On entry, a=value.
Constraint: a=0​ or ​1.
On entry, ldz=value and d×(ntimes+1-a)=value.
Constraint: ldzd×(ntimes+1-a).
On entry, ldz=value and npaths=value.
Constraint: ldznpaths.
On entry, ldc=value.
Constraint: ldcvalue.
On entry, ldb=value and d×(ntimes+1)=value.
Constraint: ldbd×(ntimes+1).
On entry, ldb=value and npaths=value.
Constraint: ldbnpaths.
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.
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.
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.
g05xdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05xdf 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


10 Example

The scaled Wiener increments produced by this routine can be used to compute numerical solutions to stochastic differential equations (SDEs) driven by (free or non-free) Wiener processes. Consider an SDE of the form
d Yt = f(t,Yt) dt + σ(t,Yt) dXt  
on the interval [t0,T] where (Xt) t0 tT is a (free or non-free) Wiener process and f and σ are suitable functions. A numerical solution to this SDE can be obtained by the Euler–Maruyama method. For any discretization t0 < t1 < t2 << tN+1=T of [t0,T] , set
Y ti+1 = Y ti + f (ti,Yti) (ti+1-ti) + σ (ti,Yti) (Xti+1-Xti)  
for i=1,,N so that YtN+1 is an approximation to YT. The scaled Wiener increments produced by g05xdf can be used in the Euler–Maruyama scheme outlined above by writing
Yti+1 = Yti + (ti+1-ti) (f(ti,Yti)+σ(ti,Yti)( Xti+1 - Xti ti+1 - ti )) .  
The following example program uses this method to solve the SDE for geometric Brownian motion
d St = rSt dt + σSt dXt  
where X is a Wiener process, and compares the results against the analytic solution
ST = S0 exp ((r-σ2/2)T+σXT) .  
Quasi-random variates are used to construct the Wiener increments.

10.1 Program Text

Program Text (g05xdfe.f90)

10.2 Program Data


10.3 Program Results

Program Results (g05xdfe.r)