g05xa:: Random Number Generators (NAG Toolbox)

Fix two times

t_{0} < T

and let

{(t_{i})}_{1 \leq i \leq N}

be any set of time points satisfying

t_{0} < t_{1} < t_{2} <\dots< t_{N} < T

. Let

{(X_{t_{i}})}_{1 \leq i \leq N}

denote a

d

-dimensional Wiener sample path at these time points, and let

C

be any

d

d

matrix such that

C C^{T}

is the desired covariance structure for the Wiener process. Each point

X_{t_{i}}

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

X_{t_{0}} = x \in ℝ^{d}

. If we set

X_{T} = x + C \sqrt{T - t_{0}} 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

X_{T} = w \in ℝ^{d}

, then

X

will behave like a non-free Wiener process.

Implementation

Given the start and end points of the process, the order in which successive interpolation times

t_{j}

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

P

Wiener sample paths each of dimension

d

. The main input to the Brownian bridge algorithm is then an array of quasi-random points

Z^{1}, Z^{2},\dots, Z^{P}

where each point

Z^{p} = (Z_{1}^{p}, Z_{2}^{p}, \dots, Z_{D}^{p})

has dimension

D = d (N + 1)

D = d N

respectively, depending on whether a free or non-free Wiener process is required. When nag_rand_bb (g05xb) is called, the

p

th sample path for

1 \leq p \leq P

is constructed as follows: if a non-free Wiener process is required set

X_{T}

equal to the terminal value

w

, otherwise construct

X_{T}

X_{T} = X_{t_{0}} + C \sqrt{T - t_{0}} [\begin{matrix} Z_{1}^{p} \\ ⋮ \\ Z_{d}^{p} \end{matrix}]

where

C

is the matrix described in Brownian Bridge Algorithm. The array times holds the remaining time points

t_{1}, t_{2},\dots t_{N}

in the order in which the bridge is to be constructed. For each

j = 1,\dots, N

set

r = times (j)

, find

q = max \{t_{0}, times (i) : 1 \leq i < j, times (i) < r\}

and

s = min \{T, times (i) : 1 \leq i < j, times (i) > r\}

and construct the point

X_{r}

X_{r} = \frac{X_{q} (s - r) + X_{s} (r - q)}{s - q} + C \sqrt{\frac{(s - r) (r - q)}{(s - q)}} [\begin{matrix} Z_{j d - a d + 1}^{p} \\ ⋮ \\ Z_{j d - a d + d}^{p} \end{matrix}]

where

a = 0

a = 1

respectively depending on whether a free or non-free Wiener process is required. Note that in our discussion

j

is indexed from

1

, and so

X_{r}

is interpolated between the nearest (in time) Wiener points which have already been constructed. The function nag_rand_bb_make_bridge_order (g05xe) can be used to initialize the times array for several predefined bridge construction orders.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

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

\frac{N}{4}

or even

\frac{N}{2}

, which could be very inefficient when

N

is large. nag_rand_bb_init (g05xa) 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.

Example

function g05xa_example


fprintf('g05xa 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,a,term,c] = get_bridge_gen_data();

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

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

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

  % As a=0, term need not be initialized

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

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

function [z] = get_z(npaths, d, a, ntimes)
  idim = d*(ntimes+1-a);

  % 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 = reshape(z, idim, npaths);

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

g05xa example results

Weiner Path 1, 11 time steps, 3 dimensions
              1          2          3
  1      1.6020     0.5611     1.6975
  2      1.2767     0.3972    -1.7199
  3     -0.1895    -0.8812    -5.1908
  4     -2.8083    -4.4484    -6.7697
  5     -5.6251    -6.0375    -3.2551
  6     -6.5404    -6.2009    -5.5638
  7     -4.6398    -4.9675    -7.4454
  8     -5.3501    -4.8563    -9.9002
  9     -7.1683    -7.2638    -9.7825
 10     -1.9440    -7.0725   -10.7577
 11     -4.9941    -3.5442   -10.1561

Weiner Path 2, 11 time steps, 3 dimensions
              1          2          3
  1      2.6097     6.2430     0.0316
  2      3.5442     4.2836     2.5742
  3      1.3068     6.1511     4.5362
  4      2.7487     8.6021     2.6880
  5      3.4584     6.1778    -0.6274
  6      0.5965     8.3014     0.5933
  7     -3.2701     5.4787     1.0727
  8     -4.7527     7.0988     0.9120
  9     -4.9375     7.9486     0.7657
 10     -7.1302     7.3180     0.2706
 11     -0.6289     9.8866    -2.2762

NAG Toolbox: nag_rand_bb_init (g05xa)

▸▿ Contents

Purpose

Syntax

Description

Brownian Bridge Algorithm

Implementation

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example