
12:54 PM (2 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.
2 Attachments
Preview attachment AronsonWeinberger1978.pdf
AronsonWeinberger1978.pdf
Preview attachment kpphamelnadirashvili1998.pdf
kpphamelnadirashvili1998.pdf

1:13 PM (1 hour 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.
Attachments area
Preview attachment McKean_1975_CPAM.pdf
McKean_1975_CPAM.pdf

3:13 PM (0 minutes 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.
2 Attachments
Preview attachment Screenshot 20150712 14.53.54.png
Screenshot 20150712 14.53.54.png
Preview attachment englandersupercriticalbm.pdf
englandersupercriticalbm.pdf
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