Mikael Signahl

On Error Rates in Normal Approximations and Simulation Schemes for Lévy Processes

Let X=(X(t) : t≥0) be a Lévy process. In simulation, one often wants to know at what size it is possible to truncate the small jumps while retaining enough accuracy. A useful tool here is the Edgeworth expansion. We provide a third order expansion together with a uniform error bound, assuming third Lévy moment is 0. We next discuss approximating X in the finite variation case. Truncating the small jumps, we show that, adding their expected value, and further, including their variability by approximating by a Brownian motion, gives successively better results in general. Finally, some numerical illustrations involving a normal inverse Gaussian Lévy process are given.

Colour classification by neural networks in graphic arts

This paper presents a hierarchical modular neural network for colour classification in graphic arts, capable of distinguishing among very similar colour classes. The network performs analysis in a rough to fine fashion, and is able to achieve a high average classification speed and a low classification error. In the rough stage of the analysis, clusters of highly overlapping colour classes are detected. Discrimination between such colour classes is performed in the next stage by using additional colour information from the surroundings of the pixel being classified. Committees of networks make decisions in the next stage. Outputs of members of the committees are adaptively fused through the BADD defuzzification strategy or the discrete Choquet fuzzy integral. The structure of the network is automatically established during the training process. Experimental investigations show the capability of the network to distinguish among very similar colour classes that can occur in multicoloured printed pictures. The classification accuracy obtained is sufficient for the network to be used for inspecting the quality of multicoloured prints.

The Cauchy problem for the wave equation with Levy noise initial data

In this paper we study the Cauchy problem for the wave equation with spacetime Levy noise initial data in the Kondratiev space of stochastic distributions. We prove that this problem has a strong and unique C2-solution, which takes an explicit form. Our approach is based on the use of the Hermite transform.

Hilbert Space Embeddings for Gelfand–Shilov and Pilipović Spaces

We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, \(\mathcal{H}\), and its dual, \(\mathcal{H}^\prime\) . We prove that for such quasi-Banach space \(\mathcal{B}\), there are convenient Hilbert spaces, \(\mathcal{H}_{k}, k=1,2\), with normalized Hermite functions as orthonormal bases and such that \(\mathcal{B}\) lies between \(\mathcal{H}_1\; \mathrm{and}\;\mathcal{H}_2\), and the latter spaces lie between \(\mathcal{H}\; \mathrm{and}\;\mathcal{H}^\prime\).

Remarks on mapping properties for the Bargmann transform on modulation spaces

We investigate the mapping properties for the Bargmann transform and prove that this transform is isometric and bijective from modulation spaces to convenient Banach spaces of analytic functions.

Sensitivity analysis via simulation in the presence of discontinuities

Abstract.In this paper we address the problem of estimating the mean derivative when the entity containing the parameter has jumps. The methods considered are finite differences, infinitesimal perturbation analysis and the likelihood ratio score function. We calculate the difference between the differentiated mean and the mean derivative. In case of finite differences, we compute the stepsize in the simulation that asymptotically minimizes the mean square error. We also show that the two latter methods, infinitesimal perturbation analysis and likelihood ratio score function, are mathematically equivalent.