NAG Toolbox: nag_rand_bb (g05xb)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{npaths}$ – int64int32nag_int scalar

The number of Wiener sample paths to create.

Constraint: $\mathbf{npaths}\ge 1$.

2:     $\mathrm{start}\left(\mathbf{d}\right)$ – double array

The starting value of the Wiener process.

3:     $\mathrm{term}\left(\mathbf{d}\right)$ – double array

The terminal value at which the non-free Wiener process should end. If $\mathbf{a}=0$, term is ignored.

4:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – double array

The first dimension, $\mathit{ldz}$, of the array z must satisfy

• if $\mathbf{rcord}=1$, $\mathit{ldz}\ge \mathbf{d}\times \left(\mathit{n}+1-\mathbf{a}\right)$;
• if $\mathbf{rcord}=2$, $\mathit{ldz}\ge \mathbf{npaths}$.

The second dimension of the array z must be at least $\mathbf{npaths}$ if $\mathbf{rcord}=1$ and at least $\mathbf{d}\times \left(\mathit{n}+1-\mathbf{a}\right)$ if $\mathbf{rcord}=2$.

The Normal random numbers used to construct the sample paths.

If $\mathbf{rcord}=1$ and quasi-random numbers are used, the $\mathbf{d}\times \left(\mathit{n}+1-\mathbf{a}\right)$, where $\mathit{n}=\mathrm{nint}$ $\mathbf{rcomm}\left(2\right)$-dimensional quasi-random points should be stored in successive columns of z.

If $\mathbf{rcord}=2$ and quasi-random numbers are used, the $\mathbf{d}\times \left(\mathit{n}+1-\mathbf{a}\right)$, where $\mathit{n}=\mathrm{nint}$ $\mathbf{rcomm}\left(2\right)$-dimensional quasi-random points should be stored in successive rows of z.

5:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array

The first dimension of the array c must be at least $\mathbf{d}$.

The second dimension of the array c must be at least $\mathbf{d}$.

The lower triangular Cholesky factorization $C$ such that $C{C}^{\mathrm{T}}$ gives the covariance matrix of the Wiener process. Elements of $C$ above the diagonal are not referenced.

6:     $\mathrm{rcomm}\left(*\right)$ – double 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 nag_rand_bb_init (g05xa) or nag_rand_bb (g05xb).

Communication array as returned by the last call to nag_rand_bb_init (g05xa) or nag_rand_bb (g05xb). This array must not be directly modified.

Optional Input Parameters

1:     $\mathrm{rcord}$ – int64int32nag_int scalar

Default: $1$

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

Constraint: $\mathbf{rcord}=1$ or $2$.

2:     $\mathrm{d}$ – int64int32nag_int scalar

Default: the dimension of the arrays term, start and the first dimension of the array c. (An error is raised if these dimensions are not equal.)

The dimension of each Wiener sample path.

Constraint: $\mathbf{d}\ge 1$.

3:     $\mathrm{a}$ – int64int32nag_int scalar

Default: $1$

If $\mathbf{a}=0$, a free Wiener process is created beginning at start and term is ignored.

If $\mathbf{a}=1$, a non-free Wiener process is created beginning at start and ending at term.

Constraint: $\mathbf{a}=0$ or $1$.

Output Parameters

1:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – double array

The first dimension, $\mathit{ldz}$, of the array z will be

• if $\mathbf{rcord}=1$, $\mathit{ldz}=\mathbf{d}\times \left(\mathit{n}+1-\mathbf{a}\right)$;
• if $\mathbf{rcord}=2$, $\mathit{ldz}=\mathbf{npaths}$.

The second dimension of the array z will be $\mathbf{npaths}$ if $\mathbf{rcord}=1$ and at least $\mathbf{d}\times \left(\mathit{n}+1-\mathbf{a}\right)$ if $\mathbf{rcord}=2$.

The Normal random numbers premultiplied by $C$.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array

The first dimension, $\mathit{ldb}$, of the array b will be

• if $\mathbf{rcord}=1$, $\mathit{ldb}=\mathbf{d}\times \left(\mathit{n}+1\right)$;
• if $\mathbf{rcord}=2$, $\mathit{ldb}=\mathbf{npaths}$.

The second dimension of the array b will be $\mathbf{npaths}$ if $\mathbf{rcord}=1$ and at least $\mathbf{d}\times \left(\mathit{n}+1\right)$ if $\mathbf{rcord}=2$.

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 $1\le k\le \mathbf{d}$, $1\le i\le \mathit{n}+1$ and $1\le p\le \mathbf{npaths}$.

If $\mathbf{rcord}=1$, the point $X_{p,i}^k$ will be stored at $\mathbf{b}\left(k+\left(i-1\right)\times \mathbf{d},p\right)$.

If $\mathbf{rcord}=2$, the point $X_{p,i}^k$ will be stored at $\mathbf{b}\left(p,k+\left(i-1\right)\times \mathbf{d}\right)$.

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

3:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry, rcomm was not initialized or has been corrupted.

   $\mathbf{ifail}=2$

Constraint: $\mathbf{npaths}\ge 1$.

   $\mathbf{ifail}=3$

On entry, $\mathbf{rcord}=\_$ was an illegal value.

   $\mathbf{ifail}=4$

Constraint: $\mathbf{d}\ge 1$.

   $\mathbf{ifail}=5$

Constraint: $\mathbf{a}=0\text{ or }1$.

   $\mathbf{ifail}=6$

Constraint: $\mathit{ldz}\ge \mathbf{d}\times \left(\mathbf{ntimes}+1-\mathbf{a}\right)$.

Constraint: $\mathit{ldz}\ge \mathbf{npaths}$.

   $\mathbf{ifail}=7$

ldc is too small.

   $\mathbf{ifail}=8$

Constraint: $\mathit{ldb}\ge \mathbf{d}\times \left(\mathbf{ntimes}+1\right)$.

Constraint: $\mathit{ldb}\ge \mathbf{npaths}$.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g05xb_example

function g05xb_example


fprintf('g05xb example results\n\n');

% Get information required to set up the bridge
[bgord,t0,tend,ntimes,intime,nmove,move] = get_bridge_init_data();

% Make the bridge construction bgord
[times, ifail] = g05xe( ...
                        t0, tend, intime, move, 'bgord', bgord);

% Initialize the Brownian bridge generator
[rcomm, ifail] = g05xa( ...
                        t0, tend, times);

% Get additional information required by the bridge generator
[npaths, d, start, term, c] = get_bridge_gen_data();

% Generate the Z values
[z] = get_z(npaths, d, ntimes);

% Call the Brownian bridge generator routine
[z, b, ifail] = g05xb( ...
                       npaths, start, term, z, c, rcomm);

% Display the results
for i = 1:npaths
  fprintf('Weiner Path %d, %d time steps, %d dimensions\n', i, ntimes+1, d);
  w = transpose(reshape(b(:,i), d, ntimes+1));

  ifail = x04ca('G', ' ', w, '');

  fprintf('\n');
end



function [bgord,t0,tend,ntimes,intime,nmove,move] = get_bridge_init_data()
  % Set the basic parameters for a Wiener process
  t0 = 0;
  n = 10;
  ntimes = int64(n);

  % We want to generate the Wiener process at these time points
  intime = 1:n + t0;
  tend = t0 + n + 1;

  nmove= int64(0);
  move = zeros(nmove, 1, 'int64');
  bgord = int64(3);

function [npaths,d,start,term,c] = get_bridge_gen_data();
  % Set the basic parameters for a non-free Wiener process
  npaths = int64(2);
  d = 3;
  start = zeros(d, 1);
  term  = [1, 0.5, 0];

  % We want the following covariance matrix
  c = [ 6,   1,   -0.2; 
        1,   5,    0.3; 
       -0.2, 0.3,  4];

  % Cholesky factorize the covariance matrix c
  [c, info] = f07fd('l', c);

function [z] = get_z(npaths, d, ntimes)
  % Non-free Wiener process (the default)
  a = int64(1); 
  idim = d*(ntimes+1-a);

  % We now need to generate the input pseudorandom points

  % First initialize the base pseudorandom number generator
  state = initialize_prng(int64(6), int64(0), [int64(1023401)]);

  % Scrambled quasi-random sequences preserve the good discrepancy
  % properties of quasi-random sequences while counteracting the bias
  % some applications experience when using quasi-random sequences.
  % Initialize the scrambled quasi-random generator.
  [iref, state] = initialize_scrambled_qrng(int64(1), int64(2), ...
                                            idim, state);

  % Generate the quasi-random points from N(0,1)
  xmean = zeros(idim, 1);
  std   = ones(idim, 1);
  [z, iref, ifail] = g05yj( ...
                            xmean, std, npaths, iref);
  z = z';

function [state] = initialize_prng(genid, subid, seed)
  % Initialize the generator to a repeatable sequence
  [state, ifail] = g05kf( ...
                          genid, subid, seed);


function [iref, state] = initialize_scrambled_qrng(genid,stype,idim,state)
  iskip = int64(0);
  nsdigits = int64(32);
  [iref, state, ifail] = g05yn( ...
                                genid, stype, int64(idim), iskip, ...
                                nsdigits, state);

g05xb example results

Weiner Path 1, 11 time steps, 3 dimensions
              1          2          3
  1     -1.0602    -2.8701    -0.9415
  2     -3.0575    -1.9502     0.2596
  3     -6.8274    -2.4434     0.4597
  4     -5.2855    -3.4475     0.0795
  5     -8.1784    -5.2296    -0.0921
  6     -4.6874    -5.0220     1.4862
  7     -3.0959    -4.8623    -4.4076
  8     -2.9605    -1.8936    -3.9539
  9     -5.4685    -2.3856    -3.2031
 10      0.1205    -5.0520    -1.0385
 11      1.0000     0.5000     0.0000

Weiner Path 2, 11 time steps, 3 dimensions
              1          2          3
  1      0.6564     3.5142     1.5911
  2     -2.3773     3.1618     3.0316
  3      0.3020     6.8815     2.0875
  4     -0.2169     4.6026     1.1982
  5     -2.0684     4.1503     2.4758
  6     -5.1075     3.7303     2.7563
  7     -3.8497     3.6682     2.4827
  8     -1.8292     4.4153     0.1916
  9     -2.0649     0.6952    -2.1201
 10      0.1962     1.7769    -5.7685
 11      1.0000     0.5000     0.0000