Nonlinear diffusions and the KPP equation
July 13, 2015 by zulfahmed
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


12:54 PM (13 hours ago)




Dear Professor Donoho,
I learned that the form in which I had been seeking for volatility dynamics is equation in one dimension studied by KolmogorovPetrovskyPiskunov in the 1930s and in multidimensions by AronsonWeinberger and many others
(d/dt) u = (d/dx)^2 u + f(u)
It is clear that I must study the research more carefully in order to understand when there are phase transitions to turbulence precisely for f(u); we know this is the case when f(u) = kappa*u*du/dx as in NavierStokes but for f(u) = kappa*u^gamma, gamma>0 I have more study to do with the attached papers for guidance. I thank you for patience, sir, as I am selfstudied and did not know the nonlinear parabolic equation literature.
[1] Aronson, D. G.; Weinberger, H. F. Multidimensional nonlinear diffusions arising in population genetics. Adv. in Math. 30 (1978), no. 1, 33–76.
[17] Kolmogorov, A. N.; Petrovsky, I. G.; Piskunov, N. S. Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin Université d’Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), Série Internationale, Section A 1 (1937), 1–26.
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


1:13 PM (13 hours ago)




Henry McKean addressed the KPP equation in 1975 which is now the top on my queue for study. My goal is to gain some insight about solutions and more importantly the phase transition to unstable and turbulent behavior which remains murky in my mind for these. The application envisioned is for the models of univariate volatility series in finance with the end goal of control of turbulent dynamics in global financial volatility.
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


3:13 PM (11 hours ago)




A probabilistic interpretation of the nonlinear equation
(d/dt + (Laplacian))u = beta *u – alpha*u^2
is available that is new to me but wellknown based on the sign of beta (see snapshot and attached paper). Theoretical results regarding range of growth of these processes is sqrt(2*beta)*t, i.e. if the stochastic sample path X_t of the solution process stays in radius R, and M_t = inf{ R: X(s,B^c(0,R))=0 } then M_t/t > sqrt(2*beta). This is not exactly applicable necessary but it gives us some intuition regarding how a nonlinear F(u) can be handled in the situation where the nonlinearity is quadratic. Even in this case theoretical results are relatively recent.
nonexistence results for semilinear parabolic equations
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


4:07 PM (10 hours ago)




Dear Professor Donoho,
I have finally located some theory for nonexistence results for global solutions for semilinear parabolic equations of the type
(d/dt + (Laplacian))u = u^{1+alpha} = F(u)
u(t=0,x) = a(x)
on R^d. I am interested in these cases to phase transition to turbulence with alpha as a parameter. If alpha*d > 2, there are global solutions but when alpha*d < 2 there are no global solutions. This requires an adjustment of intuition for I had been working with the erroneous ideas that the size of F determined the solvability of these semilinear parabolic equations.
Attachments area
Preview attachment nonexistenceglobalsolutionssemilinearparabolicsugitani1974.pdf
nonexistenceglobalsolutionssemilinearparabolicsugitani1974.pdf
Sugitani’s 1974 nonexistence theorem: reduction of nonexistence to ODE
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


7:48 PM (6 hours ago)




Dear Professor Donoho,
The technique employed by Sugitani to show that semilinear parabolic equations do not have global existence relies on properties of the scaling properties of stable distributions and integrating out the space variable and then examining an ordinary differential equation by comparison. This line of reasoning may be valuable for models of global financial volatility intuitively since scaling laws are known to hold in empirical data although this is clearly still work in progress.
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>






Correction: nonexistence applies for F(u) terms that are restricted fractional powers of u. This issue seems important for understanding whether diffusion models capture the data exactly and whether the correction would lead to different quantitative behaviour than the assumptions underlying current quantiative finance.
Sugitani’s integral formulation for semilinear parabolic equation
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


8:37 PM (5 hours ago)



Dear Professor Donoho,
Sugitani’s 1974 paper gives us the way in which analysts restructure the semilinear parabolic equations with ARBITRARY term F including fractional powers of the Laplacian for R^d which solves a very useful problem of analytic study for these and perhaps gives us a simple method of solving these numerically.
The integral equation is in terms of the symmetric stable density. This is an extremely useful reformulation because it gives us a representation that can be analyzed for essentially arbitrary F.
exercise to regain footing in analysis: extend Sugitani1974 to fractional time
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


1:29 AM (1 hour ago)




Dear Professor Donoho,
As an exercise to find solid mathematical footing, I am taking on the exercise of studying and extending Sugitani’s results attached paper from 1974 to (a) fractional time, (b) RieszFeller derivative and arbitrary reaction term F. It is clearer to me that in order to understand how to model continuoustime financial series one needs to have some intuition for a range of reactiondiffusion equations for R^d. I have a short paragraph to begin the work which now stays close to Sugitani as I regain my analytic intuitions from my time at MIT 15 years ago working with Dan Stroock.
Preview attachment reactiondiffusionsugitani1974.pdf
reactiondiffusionsugitani1974.pdf
Preview attachment nonexistenceglobalsolutionssemilinearparabolicsugitani1974.pdf
nonexistenceglobalsolutionssemilinearparabolicsugitani1974.pdf
stochastic solutions for fractional KPP and other equations
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>






Dear Professor Donoho,
Stochastic solutions for fractional nonlinear equations are useful for our project of testing from phase transitions to turbulence for continuoustime volatility. So the techniques needed for doing the exercise of taking nonexistence result of Sugitani 1974 is contained in the the 2008 paper
http://arxiv.org/pdf/0803.4457v1.pdf
We replace d/dt of Sugitani 1974 with a Caputo time derivative the space derivative to RieszFeller and take Laplace transform for time, Fourier transform for space, solve inverse transforms. The details will take some time but this is quite tractable and perhaps a good exercise for me to work out the details.
Then we would have stochastic processes from a richer class than diffusions with which to model continuoustime finance series and have a clearer idea of whether diffusion models are appropriate on empirical data.
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