Details on the Brownian bridge algorithm and the Brownian bridge process (sometimes also called a non-free Wiener process) can be found in Section 2.6 in the G05 Chapter Introduction. We briefly recall some notation and definitions.

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 \times 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 Section 2.6 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.

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

\frac{X_{t_{1}} - X_{t_{0}}}{t_{1} - t_{0}}, \frac{X_{t_{2}} - X_{t_{1}}}{t_{2} - t_{1}}, \dots, \frac{X_{t_{N}} - X_{t_{N - 1}}}{t_{N} - t_{N - 1}}, \frac{X_{T} - X_{t_{N}}}{T - t_{N}} .

Such increments can be useful in computing numerical solutions to stochastic differential equations driven by (free or non-free) Wiener processes.

3.2 Implementation

Conceptually, the output of the Wiener increments generator is the same as if g05xac and g05xbc 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

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 Section 2.6.2 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

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

depending on whether a free or non-free Wiener process is required. When g05xdc 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 Section 3.1. 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 - 1]

, find

q = max {t_{0}, times [i - 1] : 1 \leq i < j, times [i - 1] < r}

and

s = min {T, times [i - 1] : 1 \leq i < j, times [i - 1] > 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

depending on whether a free or non-free Wiener process is required. The function g05xec can be used to initialize the times array for several predefined bridge construction orders. Lastly, the scaled Wiener increments

\frac{X_{t_{1}} - X_{t_{0}}}{t_{1} - t_{0}}, \frac{X_{t_{2}} - X_{t_{1}}}{t_{2} - t_{1}},\dots, \frac{X_{t_{N}} - X_{t_{N - 1}}}{t_{N} - t_{N - 1}}, \frac{X_{T} - X_{t_{N}}}{T - t_{N}}

are computed.

4 References

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

5 Arguments

1: $t0$ – double Input

On entry: the starting value

t_{0}

of the time interval.

2: $tend$ – double Input

On entry: the end value

T

of the time interval.

Constraint:

tend > t0

3: $times [ntimes]$ – const double Input

On entry: the points in the time interval

(t_{0}, 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 g05xec can be used to create predefined bridge construction orders from a set of input times.

Constraints:

$t0 < times [i - 1] < tend$ , for $i = 1, 2, \dots, ntimes$ ;
$times [i - 1] \neq times [j - 1]$ , for $i, j = 1, 2, \dots ntimes$ and $i \neq j$ .

4: $ntimes$ – Integer Input

On entry: the length of times, denoted by

N

in Section 3.1.

Constraint:

ntimes \geq 1

5: $rcomm [12 \times (ntimes + 1)]$ – double Communication Array

On exit: communication array, used to store information between calls to g05xdc. This array MUST NOT be directly modified.

6: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $ntimes = ⟨ value ⟩$ .
Constraint: $ntimes \geq 1$ .
NE_INTERNAL_ERROR: 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.
NE_NO_LICENCE: 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.
NE_REAL: On entry, $tend = ⟨ value ⟩$ and $t0 = ⟨ value ⟩$ .
Constraint: $tend > t0$ .
NE_REAL_ARRAY: On entry, $times [i - 1] = times [j - 1] = ⟨ value ⟩$ , for $i = ⟨ value ⟩$ and $j = ⟨ value ⟩$ .
Constraint: all elements of times must be unique.

On entry, $times [⟨ value ⟩] = ⟨ value ⟩$ , $t0 = ⟨ value ⟩$ and $tend = ⟨ value ⟩$ .
Constraint: $t0 < times [i - 1] < tend$ for all $i$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

g05xcc is not threaded in any implementation.

9 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. g05xcc 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.

10 Example

The following example program calls g05xac and g05xbc to generate two sample paths from a two-dimensional free Wiener process. It then calls g05xcc and g05xdc with the same input arguments to obtain the scaled increments of the Wiener sample paths. Lastly, the program prints the Wiener sample paths from g05xbc, the scaled increments from g05xdc, 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 g05xbc.

Please see g05xdc for additional examples.