Sornette Math

Sornette Math
An appendix to Sornette crashes
Exponential growth is characterized by the relationship:
[1] B(x) = B(0) e^αx where B(0) and α are constants.
The differential equation satisfied by this function is:
[2] dB/dx = α B
Now we introduce nonlinearity by assuming that [2] is an approximation to the "real world".
In fact, we'll assume that
[3] dB/dt = f(B)
Expanding f(B) in a Taylor series (about B = 0) and recognizing that, for small B-values, [2] should pertain, we get:
[4] dB/dx = α B - η B³ + something involving higher odd powers of B
Note that we include only terms with odd exponents.

>You can do that?
Why not?
We ignore all powers greater than the 3rd, so we have our simplified, but nonlinear differential equation:
[4A] dB/dx = α B - η B³
To solve this equation we check our old calculus textbook and rewrite it as:
[5] [1/B + η B/(α - η B²) ] dB = α dx
Integrating, the solution is:
[6] B²/(α - η B²) = C₀ e^2αx where C₀ is a constant.
For sanitary reasons, we'll call the right-side R. Then, solving for B(x) we get:
[7] B² = α R / (1 + η R)
We can rewrite [7] as:
[8] B² = K₀ Q / (1 + Q) where Q = η R = C₁ e^2αx
Note that [8] can also be written:
[8A] B² = K₁ dQ/dx / (1 + Q) since dQ/dx = 2α Q.
This can further be rewritten like so:
[8B] B² = K₁ d/dx [log (1 + Q)]
>Huh? What happened to C₀ and who are all the Ks and ...?
They're all constants ... so we'll just give 'em names that sound like constants. Got it?
>No!
Then sleep while I continue.
>zzzZZZ
Okay, now we have a function B(x) that grows sorta exponentially (as in [1]) but with a nonlinear twist), so now ...
>Does this have anything to do with stock prices?
Patience!
We'll assume our stock price is like our function B ... but with an oscillatory component.
In fact, we'll write:
[9] F(x) = B(x) cos(a + ωx)
where "a" is some constant.
See? Oscillations via a good ol' trig function.
Aah, but we now introduce another ...
>Don't tell me. Another nonlinearity?
Yes, indeed. We'll rewrite [9]:
[10] F(x) = B(x) cos(φ(x))
where φ(x) looks like a + ωx for small x (so as to agree with [9]). In other words:
[11] dφ/dx = w (approximately)
We now introduce into [11] our next nonlinearity, like so:
[12] dφ/dx = ω + κ B²
We regard this (again) as a series expansion where we retain only the dominant nonlearity: namely, the term in B².
It's now easy to solve ...
To integrate [12] we first note [8B]. We get:
[14] φ = a + ω x + b log (1 + Q)
where "b" is some constant.

>You forgot equation [13]!
That's unlucky, ain't it?
When B is small, then Q is small and since log[1] = 0, then [14] says that φ ≈ a + ω x and that's what we like to see, eh?
>I have no idea.
I didn't think so.
Altogether now, using [8] and [10] and [14]::
[A] F(x) = K SQRT{Q/(1+Q)} cos{a + ω x + b log (1 + Q)}
where
Q = C₁ e^2αx and K, C₁ and α are constants.

>That's it?
Hardly.
We've been assuming that F(x) is some stock price.
Following Sornette, we'll actually fit one of these sexy curves to log[F(x)] rather than to F itself.
Further, note that Q = C₁ e^2αx.
In keeping with our transformation to logarithmic variables, we'll let x = log(t) and ...
>It's about time that ... uh, time appeared.
Pay attention.
Note that Q = C₁ e^2αx becomes Q = C₁ t^2α if x = log(t).
If we absorb the constant C₁ into some scaled variable T, we can write: Q = T^2α.
>You can do that?
Why not? I can measure time in seconds, months or light years.
>Light years is a length!
Oops, sorry 'bout that.
Anyway, making this substitution in [A] gives:
[B] F(x) = K T^α / SQRT(1+T^2α) cos{a + ω log(T) + b log (1 + T^2α)}
where there are a bunch of constants.

>That's it?
Yes, that's it. We have the form of sexy function with which to fit the logarithm of stock prices.
For even sexier stuff, read Sornette's paper.
Now we can relax ...