naginterfaces.library.rand.bb_​init

naginterfaces.library.rand.bb_init(t0, tend, times)

bb_init initializes the Brownian bridge generator bb(). It must be called before any calls to bb().

For full information please refer to the NAG Library document for g05xa

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g05/g05xaf.html

Parameters

t0float

The starting value $t_0$ of the time interval.

tendfloat

The end value $T$ of the time interval.

timesfloat, array-like, shape $\left(\text{ntimes}\right)$

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 bb_make_bridge_order() can be used to create predefined bridge construction orders from a set of input times.

Returns

commdict, communication object

Communication structure.

Raises

NagValueError

(errno $-99$)

An unexpected error occurred during execution of bb_init. Please contact NAG with the following error message: error in $⟨value⟩$, $\text{ASSERT}=⟨value⟩$.

(errno $1$)

On entry, $tend=⟨value⟩$ and $t0=⟨value⟩$.

Constraint: $tend>t0$.

(errno $2$)

On entry, $\text{ntimes}=⟨value⟩$.

Constraint: $\text{ntimes}\ge 1$.

(errno $3$)

On entry, $times[⟨value⟩]=⟨value⟩$, $t0=⟨value⟩$ and $tend=⟨value⟩$.

Constraint: $t0<times[i-1]<tend$ for all $i$.

(errno $4$)

On entry, $times[⟨value⟩]$ and $times[⟨value⟩]$ both equal $⟨value⟩$.

Constraint: all elements of $times$ must be unique.

Notes

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 the G05 Introduction. We briefly recall some notation and definitions.

Fix two times $t_0<T$ and let ${\left(t_i\right)}_{1\le i\le N}$ be any set of time points satisfying $t_0<t_1<t_2<\cdots <t_N<T$. Let ${\left(X_{t_i}\right)}_{1\le i\le N}$ 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 the G05 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 the G05 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 $D=d(N+1)$ or $D=dN$ respectively, depending on whether a free or non-free Wiener process is required. When bb() is called, the $p$th sample path for $1\le p\le P$ is constructed as follows: if a non-free Wiener process is required set $X_T$ equal to the terminal value $w$, otherwise construct $X_T$ as

$$\begin{array}{c}{X}_T=X_{t_0}+C\sqrt{T-t_0}\left[\begin{array}{c}{Z}_1^p\\ ⋮\\ Z_d^p\end{array}\right]\end{array}$$

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 $r=times[j-1]$, find

$$q=\text{max}\{t_0,times[i-1]:1\le i<j,times[i-1]<r\}$$

and

$$s=\text{min}\{T,times[i-1]:1\le i<j,times[i-1]>r\}$$

and construct the point $X_r$ as

$$\begin{array}{c}{X}_r=\frac{X_q(s-r)+X_s(r-q)}{s-q}+C\sqrt{\frac{(s-r)(r-q)}{(s-q)}}\left[\begin{array}{c}{Z}_{jd-ad+1}^p\\ ⋮\\ Z_{jd-ad+d}^p\end{array}\right]\end{array}$$

where $a=0$ or $a=1$ 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 bb_make_bridge_order() can be used to initialize the $times$ array for several predefined bridge construction orders.

References

Glasserman, P, 2004, Monte Carlo Methods in Financial Engineering, Springer