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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_bb_inc_init (g05xc)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_rand_bb_inc_init (g05xc) initializes the Brownian bridge increments generator nag_rand_bb_inc (g05xd). It must be called before any calls to nag_rand_bb_inc (g05xd).


[rcomm, ifail] = g05xc(t0, tend, times, 'ntimes', ntimes)
[rcomm, ifail] = nag_rand_bb_inc_init(t0, tend, times, 'ntimes', ntimes)


Brownian Bridge Algorithm

Details on the Brownian bridge algorithm and the Brownian bridge process (sometimes also called a non-free Wiener process) can be found in Brownian Bridge in the G05 Chapter Introduction. We briefly recall some notation and definitions.
Fix two times t0<T and let ti 1iN  be any set of time points satisfying t0<t1<t2<⋯<tN<T. Let Xti 1iN  denote a d-dimensional Wiener sample path at these time points, and let C be any d by d matrix such that CCT is the desired covariance structure for the Wiener process. Each point Xti of the sample path is constructed according to the Brownian bridge interpolation algorithm (see Glasserman (2004) or Brownian Bridge in the G05 Chapter Introduction). We always start at some fixed point Xt0 = xd . If we set XT =x+ C T-t0 Z  where Z is any d-dimensional standard Normal random variable, then X will behave like a normal (free) Wiener process. However if we fix the terminal value XT = wd , then X will behave like a non-free Wiener process.
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 .  
Such increments can be useful in computing numerical solutions to stochastic differential equations driven by (free or non-free) Wiener processes.


Conceptually, the output of the Wiener increments generator is the same as if nag_rand_bb_init (g05xa) and nag_rand_bb (g05xb) were called first, and the scaled increments then constructed from their output. The implementation adopts a much more efficient approach whereby the scaled increments are computed directly without first constructing the Wiener sample path.
Given the start and end points of the process, the order in which successive interpolation times tj are chosen is called the bridge construction order. The construction order is given by the array times. Further information on construction orders is given in Brownian Bridge Algorithm in the G05 Chapter Introduction. For clarity we consider here the common scenario where the Brownian bridge algorithm is used with quasi-random points. If pseudorandom numbers are used instead, these details can be ignored.
Suppose we require the increments of P Wiener sample paths each of dimension d. The main input to the Brownian bridge increments generator is then an array of quasi-random points Z1,Z2,…,ZP where each point Zp = Z1p,Z2p,,ZDp  has dimension D=dN+1 or D=dN depending on whether a free or non-free Wiener process is required. When nag_rand_bb_inc (g05xd) is called, the pth sample path for 1pP is constructed as follows: if a non-free Wiener process is required set XT equal to the terminal value w, otherwise construct XT as
XT = Xt0 + C T-t0 Z1p Zdp  
where C is the matrix described in Brownian Bridge Algorithm. The array times holds the remaining time points t1 , t2 ,… tN  in the order in which the bridge is to be constructed. For each j=1,…,N set r=timesj, find
q = max t0, timesi : 1i<j , timesi < r  
s = min T, timesi : 1i<j , timesi > r  
and construct the point Xr as
Xr = Xq s-r + Xs r-q s-q + C s-r r-q s-q Zjd-ad+1p Zjd-ad+dp  
where a=0 or a=1 depending on whether a free or non-free Wiener process is required. The function nag_rand_bb_make_bridge_order (g05xe) can be used to initialize the times array for several predefined bridge construction orders. Lastly, the scaled Wiener increments
Xt1 - Xt0 t1 - t0 , Xt2 - Xt1 t2 - t1 ,…, XtN - XtN-1 tN - tN-1 , XT - XtN T - tN  
are computed.


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


Compulsory Input Parameters

1:     t0 – double scalar
The starting value t0 of the time interval.
2:     tend – double scalar
The end value T of the time interval.
Constraint: tend>t0.
3:     timesntimes – double array
The points in the time interval t0,T at which the Wiener process is to be constructed. The order in which points are listed in times determines the bridge construction order. The function nag_rand_bb_make_bridge_order (g05xe) can be used to create predefined bridge construction orders from a set of input times.
  • t0<timesi<tend, for i=1,2,,ntimes;
  • timesi timesj, for i,j=1,2,ntimes and ij.

Optional Input Parameters

1:     ntimes int64int32nag_int scalar
Default: the dimension of the array times.
The length of times, denoted by N in Brownian Bridge Algorithm.
Constraint: ntimes1.

Output Parameters

1:     rcomm12×ntimes+1 – double array
Communication array, used to store information between calls to nag_rand_bb_inc (g05xd). This array must not be directly modified.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
Constraint: tend>t0.
Constraint: ntimes1.
Constraint: t0<timesi<tend for all i.
Constraint: all elements of times must be unique.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Not applicable.

Further Comments

The efficient implementation of a Brownian bridge algorithm requires the use of a workspace array called the working stack. Since previously computed points will be used to interpolate new points, they should be kept close to the hardware processing units so that the data can be accessed quickly. Ideally the whole stack should be held in hardware cache. Different bridge construction orders may require different amounts of working stack. Indeed, a naive bridge algorithm may require a stack of size N 4  or even N 2 , which could be very inefficient when N is large. nag_rand_bb_inc_init (g05xc) performs a detailed analysis of the bridge construction order specified by times. Heuristics are used to find an execution strategy which requires a small working stack, while still constructing the bridge in the order required.


The following example program calls nag_rand_bb_init (g05xa) and nag_rand_bb (g05xb) to generate two sample paths from a two-dimensional free Wiener process. It then calls nag_rand_bb_inc_init (g05xc) and nag_rand_bb_inc (g05xd) with the same input arguments to obtain the scaled increments of the Wiener sample paths. Lastly, the program prints the Wiener sample paths from nag_rand_bb (g05xb), the scaled increments from nag_rand_bb_inc (g05xd), and the cumulative sum of the unscaled increments side by side. Note that the cumulative sum of the unscaled increments is identical to the output of nag_rand_bb (g05xb).
Please see Example in nag_rand_bb_inc (g05xd) for additional examples.
function g05xc_example

fprintf('g05xc 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
[rcommb, ifail] = g05xa( ...
                         t0, tend, times);
[rcommd, ifail] = g05xc( ...
                         t0, tend, times);

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

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

% Call the Brownian bridge generator routine
[zb, bb, ifail] = g05xb( ...
                         npaths, start, term, z, c, rcommb, 'a', a);
[zd, bd, ifail] = g05xd( ...
                         npaths, dif, z, c, rcommd, 'a', a);

% Display the results
unscaled = zeros(1, d);
for n = 1:npaths
  fprintf('Weiner Path %d, %d time steps, %d dimensions\n', n, ntimes+1, d);
  fprintf('     Output of g05xb    Output of g05xd    Sum of g05xd\n');

  cum = start;
  unscaled = bd(1:d, n)*(intime(1)-t0);
  cum = cum + unscaled;
  fprintf('%2d %8.4f %8.4f  %8.4f %8.4f  %8.4f %8.4f\n', ...
          1, bb(1:d,n), bd(1:d,n), cum(1:d));

  jd1 = 1;
  for j = 2:ntimes
    jd1 = jd1 + d;
    unscaled = bd(jd1:j*d, n)*(intime(j)-intime(j-1));
    cum = cum + unscaled;
    fprintf('%2d %8.4f %8.4f  %8.4f %8.4f  %8.4f %8.4f\n', ...
            j, bb(jd1:j*d,n), bd(jd1:j*d,n), cum(1:d));

  j = ntimes+1;
  jd1 = ntimes*d + 1;
  unscaled = bd(jd1:j*d, n)*(tend-intime(j-1));
  cum = cum + unscaled;
  fprintf('%2d %8.4f %8.4f  %8.4f %8.4f  %8.4f %8.4f\n\n', ...
          j, bb(jd1:j*d,n), bd(jd1:j*d,n), cum(1:d));

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

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

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

function [z] = get_z(npaths, d, a, ntimes)
  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)]);

  % Generate the pseudorandom points from N(0,1)
  [state, z, ifail] = g05sk( ...
                             int64(idim*npaths), 0, 1, state);

  z = transpose(reshape(z, npaths, idim));

function [state] = initialize_prng(genid, subid, seed)
  % Initialize the generator to a repeatable sequence
  [state, ifail] = g05kf( ...
                          genid, subid, seed);
g05xc example results

Weiner Path 1, 11 time steps, 2 dimensions
     Output of g05xb    Output of g05xd    Sum of g05xd
 1  -2.2323   1.6656   -2.2323  -0.3344   -2.2323   1.6656
 2  -5.2301   1.2812   -2.9978  -0.3844   -5.2301   1.2812
 3  -0.9025  -1.2421    4.3276  -2.5234   -0.9025  -1.2421
 4  -3.6799  -0.3972   -2.7774   0.8449   -3.6799  -0.3972
 5  -6.5789  -2.0358   -2.8990  -1.6386   -6.5789  -2.0358
 6 -11.2879  -1.1972   -4.7090   0.8385  -11.2879  -1.1972
 7  -8.8959  -1.6751    2.3919  -0.4779   -8.8959  -1.6751
 8  -9.7103  -2.0523   -0.8144  -0.3772   -9.7103  -2.0523
 9  -8.5720  -3.3306    1.1383  -1.2783   -8.5720  -3.3306
10  -9.8245  -3.2035   -1.2524   0.1271   -9.8245  -3.2035
11  -4.9941  -8.3506    4.8304  -5.1471   -4.9941  -8.3506

Weiner Path 2, 11 time steps, 2 dimensions
     Output of g05xb    Output of g05xd    Sum of g05xd
 1  -1.4101   0.0576   -1.4101  -1.9424   -1.4101   0.0576
 2  -3.5738   0.2519   -2.1637   0.1943   -3.5738   0.2519
 3  -5.2528   1.7232   -1.6790   1.4713   -5.2528   1.7232
 4  -0.8540   1.0897    4.3988  -0.6335   -0.8540   1.0897
 5   0.4905  -0.9098    1.3445  -1.9995    0.4905  -0.9098
 6   2.3322   1.3415    1.8417   2.2514    2.3322   1.3415
 7   3.0105  -4.3312    0.6783  -5.6728    3.0105  -4.3312
 8   2.6776  -3.4437   -0.3329   0.8875    2.6776  -3.4437
 9   0.6546  -2.7291   -2.0230   0.7146    0.6546  -2.7291
10  -1.3175  -3.8166   -1.9721  -1.0875   -1.3175  -3.8166
11  -3.0214  -3.5439   -1.7039   0.2727   -3.0214  -3.5439

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