Limit theorems for prices of options written on semi-Markov processes

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

Domination for subordinated positive semigroups

We first give a necessary and sufficient condition for -f(B) to be dominated by -f(A), where f is a completely Bernstein function, B and A are C0-semigroup generators. Then we prove that there is no domination relationship between a semigroup and the subordinated semigroup if additional conditions are satisfied.

Time-Non-Local Pearson Diffusions

In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence.

Subordination for sequentially equicontinuous equibounded $$C_0$$-semigroups

We consider operators A on a sequentially complete Hausdorff locally convex space X such that $$-A$$ - A generates a (sequentially) equicontinuous equibounded $$C_0$$ C 0 -semigroup. For every Bernstein function f we show that $$-f(A)$$ - f ( A ) generates a semigroup which is of the same ‘kind’ as the one generated by $$-A$$ - A . As a special case we obtain that fractional powers $$-A^{\alpha }$$ - A α , where $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) , are generators.

Self-similar cauchy problems and generalized Mittag-Leffler functions

By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. This work could be seen as an-in depth analysis of a specific class, the one with the self-similarity property, of the general inverse of increasing Markov processes introduced in [15].

Lifshitz tail for continuous Anderson models driven by Lévy operators

We investigate the behavior near zero of the integrated density of states for random Schrödinger operators [Formula: see text] in [Formula: see text], [Formula: see text], where [Formula: see text] is a complete Bernstein function such that for some [Formula: see text], one has [Formula: see text], [Formula: see text], and [Formula: see text] is a random nonnegative alloy-type potential with compactly supported single site potential [Formula: see text]. We prove that there are constants [Formula: see text] such that [Formula: see text] where [Formula: see text] is the common cumulative distribution function of the lattice random variables [Formula: see text]. For typical examples of [Formula: see text] the constants [Formula: see text] and [Formula: see text] can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, the large class of operator monotone functions of the Laplacian. This class includes both local and nonlocal kinetic terms such as the Laplace operator, its fractional powers, the quasi-relativistic Hamiltonians and many others.

Sufficient conditions for symmetric matrices to have exactly one positive eigenvalue

Let A = [aij] be a real symmetric matrix. If f : (0, ∞) → [0, ∞) is a Bernstein function, a sufficient condition for the matrix [f (aij)] to have only one positive eigenvalue is presented. By using this result, new results for a symmetric matrix with exactly one positive eigenvalue, e.g., properties of its Hadamard powers, are derived.

Efficient simulation of Brown‒Resnick processes based on variance reduction of Gaussian processes

Brown‒Resnick processes are max-stable processes that are associated to Gaussian processes. Their simulation is often based on the corresponding spectral representation which is not unique. We study to what extent simulation accuracy and efficiency can be improved by minimizing the maximal variance of the underlying Gaussian process. Such a minimization is a difficult mathematical problem that also depends on the geometry of the simulation domain. We extend Matheron's (1974) seminal contribution in two directions: (i) making his description of a minimal maximal variance explicit for convex variograms on symmetric domains, and (ii) proving that the same strategy also reduces the maximal variance for a huge class of nonconvex variograms representable through a Bernstein function. A simulation study confirms that our noncostly modification can lead to substantial improvements among Gaussian representations. We also compare it with three other established algorithms.