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This is for the onedimensional case, but there are later results for higher dimensions and RieszFeller fractionalizing of the Laplacian as well.
We can now return to our setting in R^d where we are interested in modifications of the centurylong tradition in quantitative finance of modeling volatility using minor variations of sample paths of diffusions by adding a nonlinear forcing term F. We can use these formulae directly in the solution of the NONLINEAR diffusion equations whose sample paths I believe will ultimately address two important shortcomings of this state of affairs: (a) possibly provide quantitative basis for phase transition to turbulence in global financial volatility, (b) fit empirical data of global financial volatility in a large complex network. Goals (a) and (b) are ambitious goals but at this stage of the Volatility Storms project, I am focusing on getting some solid mathematical foundations for nonlinear fractional diffusions in R^d.
Let p(t,x) be the Green function (1) or (2) or their generalization to R^d with RieszFeller fractional derivative for the space part. Then the equation (please forgive the switching of meaning of alpha, beta, etc. which are due to mapping to our notation but the ideas are clear)
yesterday we looked at Adomian polynomials to address solving this for alpha=1 which can be done with p(t,x) the symmetric Levy stable density. We can now replace p(t,x) with the Green functions above or their generalizations in the integral representation of (3)
Preview attachment mainardifundamentalsolutionsdiffusionwave.pdf
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