M. Zähle

Harmonic calculus on fractals—A measure geometric approach II

Differentiation of functions w.r.t. finite atomless measures with compact support on the real line is introduced. The related harmonic calculus is similar to that of the classical Lebesgue case. As an application we obtain the Weyl exponent for the spectral asymptotics of the Laplacians w.r.t. linear Cantor-type measures with arbitrary weights.

On the Link Between Fractional and Stochastic Calculus

Methods of classical fractional calculus are applied to generalized Stieltjes and stochastic integration theory. Under these aspects we also consider stochastic differential equations driven by processes with generalized quadratic variations. The paper gives a survey on this approach.

Normal cycles of Lipschitz manifolds by approximation with parallel sets

Abstract It is shown that sufficiently close inner parallel sets and closures of the complements of outer parallel sets to a d-dimensional Lipschitz manifold in R d with boundary have locally positive reach and the normal cycle of the Lipschitz manifold can be defined as limit of normal cycles of the parallel sets in the flat seminorms for currents, provided that the normal cycles of the parallel set have locally bounded mass. The Gauss–Bonnet formula and principal kinematic formula are proved for these normal cycles. It is shown that locally finite unions of non-osculating sets with positive reach of full dimension, as well as the closures of their complements, admit such a definition of normal cycle.

Mixed curvature measures for sets of positive reach and a translative integral formula

Mixed curvature measures for sets of positive reach are introduced and a translative version of the principal kinematic formula from integral geometry is proved. This is an extension of a known result from convex geometry. Integral representations of the mixed curvature measures in various particular cases of dimension two and three are derived.

Fractional derivatives of Weierstrass-type functions

For a special type of fractional differentiation, formulas for the Weierstrass and the Weierstrass-Mandelbrot functions are shown. For integer valued parameters λ, a well conditioned numerical procedure for computing the derivatives in the mean of these functions is derived and used to compute some values.

Lipschitz-Killing curvatures of self-similar random fractals

For a large class of self-similar random sets F in R^d geometric parameters C_k(F), k=0,...,d, are introduced. They arise as a.s. (average or essential) limits of the volume C_d(F(\epsilon)), the surface area C_{d-1}(F(\epsilon)) and the integrals of general mean curvatures over the unit normal bundles C_k(F(\epsilon)) of the parallel sets F(\epsilon) of distance \epsilon, rescaled by \epsilon^{D-k}, as \epsilon\rightarrow 0. Here D equals the a.s. Hausdorff dimension of F. The corresponding results for the expectations are also proved.

Fractional differentiation in the self-affine case. III. The density of the Cantor set

We compute the fractional density of the middle-third Cantor measure explicitly. Its numerical value is 0.62344.

Approximation and characterization of generalised Lipschitz-Killing curvatures

A differential-geometric and measure-geometric analogue to Hadwigers characterization of linear combinations of Minkowski functionals of convex bodies as continuous additive euclidean invariants is given. The equivalent of the quermassintegrals are generalised Lipschitz-Killing curvatures and measures. By means of polyhedral approximation with respect to flat seminorms of associated normal cycles the general problem may be reduced to the classical case.

Potential spaces and traces of Lévy processes on h-sets

Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brownian motion in ℝn with generator f(−Δ) are investigated. Estimates for the related Rieszand Bessel-type kernels of order s are derived which include the classical case f(r) = rα/2 with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernstein functions f potential representations of the trace spaces, the trace Dirichlet forms, and the trace processes on fractal h-sets are derived. Here we suppose the trace condition ∫01r−(n+1)f(r−2)−1h(r) dr < ∞ on f and the gauge function h.

LONG RANGE DEPENDENCE, NO ARBITRAGE AND THE BLACK–SCHOLES FORMULA

A bond and stock model is considered where the driving process is the sum of a Wiener process W and a continuous process Z with zero generalized quadratic variation. By means of forward integrals a hedge against Markov-type claims is constructed. If Z is independent of W under some natural assumptions on Z and the admissible portfolio processes the model is shown to be arbitrage free. The fair price of the above claims agrees with that in the classical case Z ≡ 0. In particular, the Black–Scholes formula remains valid for non-semimartingale models with long range dependence.