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MathGroup Archive 2005

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Re: Solving a weakly singular integral equation - Take 2.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg54625] Re: Solving a weakly singular integral equation - Take 2.
  • From: Zaeem Burq <Z.Burq at ms.unimelb.edu.au>
  • Date: Thu, 24 Feb 2005 03:21:29 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Thanks again Paul.

> Where does this function come from? For positive a and t, c only makes
> sense for 0 < t < 1/E.

Let c[t] = (1 + a) Sqrt[2 t Log[Log[1/t]]].

1. This function comes from probability theory. It appears in the Law of
Iterated Logarithm for Brownian motion W[t] which says that the integral

\int_{0}^{t} c(t)p(c(t)/\sqrt{t}) t^{-3/2} dt

converges (diverges) if a > 0 (< 0) respectively. See, for example, Ito,
McKean, Diffusion processes and their sample paths - pg 33.

Related theorems tell us that the function 

d[t_]:= (1+a) Sqrt[2t Log[1/t]] 

exhibits similar behaviour near zero (i.e., the integral converges if a 
> 0), but may be easier for Mathematica to handle.

2. Define T = inf {t : W[t] \geq c[t] }.

This random time T is the first time that a trajectory of Brownian motion 
hits c[t]. If a > 0, then the random variable T has a density f given by 
the integral equation I've been trying to solve.

3. Also, I am no expert in integral equations, but by for a Volterra type
2 eqn.

f[t] = g[t] + \int_{0}^{t} K[s,t] f[s] ds, for t \in [0,S]

to have a unique bounded solution, isn't it enough that the inhomogeneous
function g be integrable on [0,S], and the kernel K be integrable in the
triangle 0 \leq t \leq S, and 0 \leq s \leq t? (We'll let S = 1/e here.)
For example, see Jerri, Introduction to integral equations with
applications, Theorem 1 on page 137.

Integrability of the inhomogeneous function is guaranteed by a > 0 in the 
Law of Iterated Logarithm above.

For the integrability of the kernel, see

http://www.ms.unimelb.edu.au/~zab/paul.pdf

4. That business about p killing the singularity rubbish rubbish. But it
seems that this is besides the point.

5. I am very new to both Mathematica and to programming. I hope you'll
forgive my fumbles.

Best, Zaeem.

________________________________

Zaeem Burq
PhD Stochastic Processes,
Dept. of Mathematics and Statistics, 
Unimelb.

Room 201, Richard Berry Building
University of Melbourne,
Parkville, VIC 3052.

ph: 8344 4248.

http://www.ms.unimelb.edu.au/~zab















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