A user guide and a brief description of the C-program 
for generating integrated Brownian motion + drift


1- How to compile the program:

The C-program is currently under the name ÔIntbrownGen.cÕ and can be 
compiled under both unix and linux with the command:
                                      
      g++ -o intbrown IntbrownGen.c

To get the output printed in a .txt file, the user may use the command

      ./intbrown m c k myfile.txt

where m is a parameter that controls the precision of the approximation 
of Brownian motion on a grid, [-c, c] is the interval on which two-sided 
Brownian motion and its integrals are considered, k is an integer that 
should be at least be equal to 2, and it is the same parameter considered 
in the nonparametric shape constrained (k-monotone) density estimation problem.   

2- A brief description of the program:

For chosen parameters m, c and k, the output is of the form of matrix 
of n=   rows and k columns. The rows correspond to the points of the 
grid obtained by subdividing the interval [-c, c] to sub-intervals of 
length  , whereas the j-th column store the value of the 
j-th fold integral of two-sided Brownian motion +   for j=0, ..., k-1.
The approximation of Brownian motion and its j-th fold integral  is 
based on the construction of the Haar functions on [0,1] (see Rogers 
and Williams (1994))  and their integrals,  which can be calculated 
in closed form (see Balabdaoui (2004), appendix). The output can be 
fed to the iterative (2k-1) spline algorithm (coded in S) to find an 
approximation to the envelopes (invelopes) of the (k-1)-fold integral 
of Brownian motion + the corresponding drift when k is odd (even) on [-c, c].
Beside the main function, IntbrownGen.c consists essentially of the 
following sub-functions:

- Schauder and InSchauder: return the value of the Schauder function 
  and its integral (of any chosen degree).
- IntBrownFunc: returns the j-th integral of Brownian motion on [0,1] 
  (approximated on a grid of mesh size   ) , for j=0, ..., k-1.
- IntBrown0C: uses InBrownFunc to extend the  j-th integral of approximated 
   Brownian motion on the interval [0,c] (c > 1).
- IntBrownCCdrift: uses IntBrown0C to extend the above approximation to 
  [-c, c], and adds the drift   for j=0, ..., k-1.