Optimal approximation of anticipating SDEs

In this article, we analyse the optimal approximation of anticipating stochastic differential equations, where the integral is interpreted in Skorohod sense. We derive optimal rate of convergence for the mean squared error at the terminal point and an asymptotically optimal scheme for a class of linear anticipating SDEs. Although alternative proof techniques are needed, our results can be seen as generalizations of the corresponding results for It\=o SDEs. As a key tool we carry over optimal approximation from vectors of correlated Wiener integrals to a general class of random vectors, which cover the solutions of the Skorohod SDEs

Simplex optimization of headspace factors for headspace gas chromatography determination of residual solvents in pharmaceutical products

The purpose of the present work was to find optimum conditions of headspace gas chromatography (HS-GC) determination of residual solvents which usually appear in pharmaceutical products. Two groups of solvents were taken into account in the present examination. Group I consisted of isopropanol, n-propanol, isobutanol, n-butanol and 1,4-dioxane and group II included cyclohexane, n-hexane and n-heptane. The members of the groups were selected in previous investigations in which experimental design and chemometric methods were applied. Four factors were taken into consideration in optimization which describe HS conditions: sample volume, equilibration time, equilibrium temperature and NaCl concentration in a sample. The relative GC peak area served as an optimization criterion which was considered separately for each analyte. Sequential variable size simplex optimization strategy was used and the progress of optimization was traced and visualized in various ways simultaneously. The optimum HS conditions appeared different for the groups of solvents tested, which proves that influence of experimental conditions (factors) depends on analyte properties. The optimization resulted in significant signal increase (from seven to fifteen times)

Grouping of residual solvents present in pharmaceuticals using experimental planning and chemometric methods

The main effects of six experimental factors on the efficiency of HS (headspace) extraction in headspace gas chromatography – flame ionization detector (HS-GC-FID) determination of twenty organic solvents routinely used in production of pharmaceuticals were obtained on the basis of the results of experiments carried out according to the Plackett-Burman factorial design. The effects were used as a basis for grouping the solvents into five groups, the solvents belonging to a group responded similarly to changes of HS conditions. To this end, visualization approaches were used as well as chemometric methods: cluster analysis (CA) and principal component analysis (PCA). Moreover, the most important HS experimental factors were selected for further optimization of the HS-GC determination procedure

A Wick functional limit theorem and applications to fractional Brownian motion

The Wick product is a well-known tool in stochastic analysis to construct stochastic integrals with respect to Gaussian processes beyond semimartingales. Similarly, on disturbed random walks one can define a discrete counterpart. In this thesis we prove that weak convergence of central limit theorems carries over to applications of Wick products. Thus, the analogy of the discrete and continuous Wick calculus finds its expression in particular in convergence results. These convergences range to a functional limit theorem for Gaussian processes. Due to an extension of Sottinen\u27s Donsker-type approximation of the fractional Brownian motion (Finance and Stochastics. (5), 343-355 (2001)) to all Hurst parameters, we can also approximate processes of fractional geometric Brownian motion type. Based on this, we examine the convergence of solutions of Wick difference equations to solutions of corresponding Wick-Ito stochastic differential equations. We determine the asymptotical computational costs of the difference equations and illustrate it on examples for the fractional Black-Scholes model. Moreover, we provide the equivalence conditions for convergence of discrete S-transforms to continuous S-transforms. In particular, this convergence is represented in terms of the Wiener chaos decompositions.Das Wick Produkt ist ein bekanntes Werkzeug der stochastichen Analysis, um stochastische Integrale bezüglich Gaußschen Prozessen jenseits von Semimartingalen zu konstruieren. Gleichermaß en kann auf gestörten Irrfahrten ein diskretes Pendant definiert werden. Wir zeigen, dass die schwache Konvergenz von zentralen Grenzwertsätzen unter der Anwendung der Wick Produkte erhalten bleibt. Damit findet die Analogie der diskreten und stetigen Wick Kalküle gerade in Konvergenzresultaten ihren Ausdruck. Diese Konvergenzen reichen bis zu einem Funktionalen Grenzwertsatz für Gaußsche Prozesse. Die Erweiterung von Sottinens Donsker-Approximation der fraktionalen Brownschen Bewegung (Finance and Stochastics. (5), 343-355 (2001)) für alle Hurst-Parameter befähigt uns, Prozesse von der Art der fraktionalen geometrischen Brownschen Bewegung zu approximieren. Basierend darauf untersuchen wir die Konvergenz von Lösungen von Wick-Differenzen Gleichungen gegen Lösungen von zugehörigen Wick-Ito stochastischen Differentialgleichungen. Wir bestimmen den asymptotischen Rechenaufwand der Simulation dieser Differenzengleichungen und erläutern dies an Beispielen des fraktionalen Black-Scholes Modells. Zudem charakterisieren wir Äquivalenzen zur Konvergenz der diskreten gegen die stetigen S-Transformierten. Insbesondere ist diese Konvergenz darstellbar in Form von Wiener Chaos-Zerlegungen

Zagrożenia związane z działaniem człowieka w systemie ochrony informacji niejawnych

The author describes the threats related to human activity in the system of protecting classified information. The publication also presents the pillars of the classified information protection system and its simplified model in organizational units.Autor zwrócił uwagę na zobowiązania międzynarodowe Polski, szczególnie te, związane z dostosowaniem regulacji prawnych do standardów obowiązujących w państwach członkowskich NATO dotyczących bezpieczeństwa informacji, które zaowocowały wprowadzeniem do porządku prawnego w Polsce Ustawy z dnia 5 sierpnia 2010 r. o ochronie informacji niejawnych. Zostało podkreślone, iż zgodnie z przepisami tej ustawy, każda jednostka organizacyjna przetwarzająca informacje niejawne zobowiązana jest do zorganizowania systemu ochrony tych informacji przed nieuprawnionym ujawnieniem. Zostały wymienione podstawowe zagrożenia dla bezpieczeństwa klasyfikowanych informacji, szczególnie te, które wynikają z działania człowieka. Zdaniem autora, trudno jest wyeliminować wszystkie zagrożenia związane z działaniem człowieka, pomimo dotkliwych sankcji karnych, jakie grożą za ujawnienie lub wykorzystywanie wbrew przepisom cytowanej ustawy informacji niejawnych. Należy jednak podjąć próby, aby pojawiające się zagrożenia i ich możliwe skutki ograniczyć do minimum, Szczegółową analizę omawianej problematyki autor zawarł w swojej dysertacji, która wkrótce zostanie zaprezentowana