Ok Hobbes,

Time to get busy. I’m getting into fractional differential equations. Ready? Ok normal differential equations are like this:Solve

y'(t) = lambda*y(t)

y(0) =1

This one you divide by y(t) and integrate in time noting y’/y = (log(y))’ by the chain rule. This is the template for mathematician’s easiest ever differential equation that actually has uses. The solution is y=exp(lambda*t). In the case of a fractional parameter 0< alpha <1, the fractional differential equation equivalent is

y^{(alpha)}(t) = lambda*y(t)

y(0) = 1

This one is solved by

E_{alpha}( lambda*t )

where E_{alpha} is not the exponential but the Mittag-Leffler function also known as ‘fractional exponential’ because of this analogy to basic ODE solved by exponential function. So this is the fundamental importance of Mittag-Leffler functions. But these are specializations of a two-parameter family E_{alpha,beta}(z) defined by

E_{alpha,beta}(z) = \int_Loop kernel(s,alpha,beta) ds

where the kernel(s,alpha,beta) is known. We have some other tools: we have a formula called ‘integration by parts’ that occurs simply from a product rule for derivatives. So we can deal with products of functions and so on.

Now these fractional differential equations sound like all sorts of other terms from the ‘stochastic’ school but it is an exact non-noisy situation that simply happens to have solutions that are not smooth as silk. The solutions of ordinary differential equations have a well-developed mathematical theory of existence of solutions, uniqueness, and so on and so do fractional differential equations which are actually integral equations with various power law weights stuck in. Let’s assume that existence and regularity and uniqueness — the uniqueness of the solution is crucial for knowing without thinking too much that there is pure determinism in these and no stochasticity. Now geeky engineer types are very good at solving these; much better than I want to become although probably cannot avoid it. What’s really interesting to me is that there are a bunch of people who came up with solving these by an iterative improvement of solutions by a scheme like

Let’s say u_k is the k-th iterate. Let

u_{k+1} = u_k + expression like 1/gamma(1+alpha)( (d/dt)^alpha u_k + other terms in the fractional differential equation)

If you actually think this is fun, read the attached paper but the issue is really that this is a numerical scheme that solves these iteratively and therefore I should code up the solution to solve these equations numerically.

We’re interested in solving for the equation parameters looking at denoised data for volatility for which we have the noisy data in R code already. We can bring this easily to octave or wherever. This is a good idea too. Hmm, let’s leave this to the side and look at our model of volatility that is deterministic now. Of course the denoising method will not be perfect since the theory always holds for sufficiently large samples in statistics. Now let’s consider the model we want to fit to the data to see if we can evade doing too much work here:((d/dt)^{alpha} – Laplacian)u = F(u) = beta + gamma*u^delta

Let’s stare at this for a while while we smoke weed.

## Leave a Reply