Old Dog, New Trick

I've been using integration by parts lately to solve some unique problems. Unfortunately, it seems many people don't seem to think there is anything really interesting about it. I would like to show here a simple example of some of its more complicated things I've done with it recently.

First to start off with a quick definition of the integration by parts transformation.



This transformation is kinda useful for numerical integration as well. If you look at the right hand side, you'll see that a always appears integrated. For this reason, we can use this interpretation of the integral whenever the integral of a is better behaved than a itself.  Take this integral as an example:



This equation has no analytic solution and becomes very difficult to analyse numerically around 0. In fact a simple attempt to numerically integrate it won't give good results.

The oscillations are due to that problematic sine term. The amazing thing is how much better behaved the integral of sine(1/x) is than the original expression. The integral has an analytic solution in terms of  the Cosine Integral function: http://mathworld.wolfram.com/CosineIntegral.html. This function is not difficult to numerically approximate.

We can then make this integral easier to solve numerically by applying the integration by parts transformation. "a" here will be Sin(1/x) and "b" will be the exponential. First we compute the Integral of a. First I define the integral of the oscillating function:



Here Ci is the previously mentioned CosineIntegral function.  The full transformed integral is:



oI still oscillates, but not as widely as the previous function. This integral is easy to evaluate numerically. If we take out the analytic component and just focus on the integral, we can see we basically just transformed the function in the graph above into an analytically evaluable expression and the integral of this function:

I'm always kind of amazed what kinds of functions this technique can be applied to.

Some fun consequences of the previous post on Cauchy distributions

Look at my kinda rant on stack exchange here.

Essentially, the theory of diversification for stocks is completely different if you assume that the differentials of stocks are Cauchy distributed instead of Normally distributed. In fact, as I kinda point out, diversification doesn't even seem to make any sense under the conditions that stocks are levy processes.

There were a number of good answers and its gonna take me a while to go through the recommended reading.

Stocks are not Wiener Processes.

The controversy over the distribution that best fits the change of stock prices is apparently fairly recent. In turns out that stocks are actually Levy processes. I was stumbled across this fact while trying to fit the data onto the normal distribution and failing. Having read somewhere that the differential in stock prices is normal (see Black Sholes model), I assumed it would be at least a reasonable fit. After failing, I programmatically tried a ton of random distributions till a suitable fit was found.

Here is a histogram of the daily closing differences for General Electric. On top of it is superimposed a fit of the histogram with first the Normal distribution and then the Cauchy distribution. The fit was found using maximum likelihood estimation.

One of these is a better fit. Which do you think?

It blows my mind how terrible of a fit the Normal distribution is - why did it take Mandelbrot and Nassim Taleb to bring this fact up? In fact, running a simple test for normality on the data shows an incredibly small chance of it being normally distributed.

Of course this is not proof that the differential is Cauchy distributed. For that however you have to simply look at the properties of each distribution. For example the sum of two Cauchy random variables is another Cauchy random variable with parameters equal to the sum of the two previous parameters. Define beforehand the properties of stocks and you can derive the behavior of the distribution which should match it.

Using the normal distribution is fine if you are making some kind of approximation. However whenever an approximation is made, you have to ask how good it will be and under what conditions it fails. It looks like this hasn´t been seriously tried until recently.