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Re: Cumulative probability that random walk variable

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104879] Re: [mg104834] Cumulative probability that random walk variable
  • From: Leonid Shifrin <lshifr at gmail.com>
  • Date: Thu, 12 Nov 2009 06:05:31 -0500 (EST)
  • References: <200911110928.EAA29352@smc.vnet.net>

Hi Kelly,

I am not sure if the notion of "continuous random walk" makes
sense without refinement (for me at least). I have a feeling that before you
start using Mathematica to solve your problem, you may want to better
formulate it and see what is the question you really want to ask.

The probability you ask for is exactly 1 if you assume (as you presumably
do) that x[t] are independent random variables for different t, and that <t>
is a continuous variable.

The simplest way to see this is to note that the probability you
want is

1 - P{all x[t] are smaller than 2} ==
1-Product_{all values of t}(P{x[t]<2})

The product above contains (uncountably) infinitely many terms,
(one for each value of t), and all of them except for the very first one (at
t=0)  are strictly less  than 1 (at least for the normally-distributed x[t],
where for your numbers, each term in the product  is 1/2 (1 +
Erf[Sqrt[2]/Sqrt[t]])), therefore the product is zero, and the probability
in question is 1.

"Physically", this happens because, by making x[t] independent
normally-distributed random variables, you allow for arbitrarily large jumps
in your  trajectories, even for infinitesimally close to each other points
in time,   so a generic trajectory x[t] will be a very messy function of
<t>, jumping at every point, something like a Dirichlet function (but with
unbounded jumps). Therefore, the probability that it will not jump over 2
(or any other finite number for that matter) is just zero.

I can see two ways of making the problem more meaningful. One is to
discretize the time and fix the number of jumps (steps). In this case, the
above-mentioned product will be a (possibly very small)  finite number.

Another is to somehow restrict your random trajectories to be more "smooth"
and well-behaved functions. This however will mean that variables x[t] will
no longer be independently distributed.
This may eventually lead you to the functional (path) integrals,
which can be modeled and computed numerically and may provide you with the
answers. But doing so it sufficiently more complicated and will still
require a precise formulation of the random system you wish to model, and
the properties you want to test. You may find that in this approach you will
be asking different sorts of questions. If you wish to go this way, you may
find it of interest to look at how this is done for certain simple
one-dimensional quantum-mechanical or statistical systems. Keywords will
probably include Quantum Mechanics, Statistical Mechanics, Path Integrals,
Metropolis algorithm, Monte-Carlo simulations.

Hope this helps. My apologies if I misunderstood the question.

Regards,
Leonid




On Wed, Nov 11, 2009 at 12:28 PM, Kelly Jones
<kelly.terry.jones at gmail.com>wrote:

> How can I use Mathematica to solve this problem?
>
> Let x[t] be a normally-distributed random variable with mean 0 and
> standard deviation Sqrt[t].
>
> In other words, x[0] is 0, x[1] follows the standard normal
> distribution, x[2] follows the normal distribution with mean 0 and
> standard deviation Sqrt[2], etc.
>
> It's easy to compute the probability that x[5] > 2 (for example).
>
> How do I compute the probability that x[t] > 2 for 0 <= t <= 5.
>
> In other words, the probablity that x[t] surpassed 2 at some point
> between t=0 and t=5, even though x[5] may be less than 2 itself. Notes:
>
>  % My goal: predicting whether a continuous random walk will exceed a
>  given value in a given period of time.
>
>  % I realize that saying things like "x[5] may be less than 2" is
>  sloppy, since x[5] is a distribution, not a value. Hopefully, my
>  meaning is clear.
>
>  % I tried doing this by adding/integrating probabilities like this
>  (psuedo-code):
>
> P(x[t] > 2 for 0 <= t <= 5) = Integral[P(x[t] > 2),{t,0,5}]
>
> but this overcounts if x[t] > 2 for multiple values of t.
>
> --
> We're just a Bunch Of Regular Guys, a collective group that's trying
> to understand and assimilate technology. We feel that resistance to
> new ideas and technology is unwise and ultimately futile.
>
>



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