Passage times of random walks and Lévy processes across power law boundaries

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup _n→∞(S_n/n^κ) is almost surely zero, finite or infinite when 1/2<κ<1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes.This work is partially supported by ARC Grant DP0210572.     

Relative stability of trimmed sums

Summary This paper gives extensions of Mori's strong law for ^(r) S _n =S _n –X _n ^(1)} ...–X _n ^(r) , where S _n =X₁+X₂+...+X _n ,X _i are iidrv's and (X _i ⁿ ()) is (X _i ) arranged in decreasing order of absolute magnitude. The methods differ from Mori's. Continuity of the distribution of the X _i is assumed throughout. Necessary and sufficient conditions for relative stability (^(r) S _n /B _n ±1 a.s. for some B _n ), including a generalised condition of Spitzer's and a dominated ergodic theorem, are proved. A one-sided version of the relative stability results is also given. A theorem of Kesten's is generalised to show that if (^(r) S _n –A _n )/B _n is bounded almost surely for constants A _n ,B _n + then for some _n . A corollary to this is that if ¦ ^(r) S _n ¦/B _n is bounded away from 0 and + a.s. then ^(r) S _n is relatively stable. This generalises a result of Chow and Robbins, apart from the continuity assumption.     

Characterization of a novel <Emphasis Type="Italic">Aspergillus niger</Emphasis> beta-glucosidase tolerant to saccharification of lignocellulosic biomass products and fermentation inhibitors

Properties of beta-glucosidase produced by Aspergillus niger URM 6642 recently isolated from the Atlantic rainforest biome and its potential tolerance to saccharification of lignocellulosic biomass products and fermentation inhibitors was evaluated. The fungus was cultivated under solid state culture conditions at 37°C with different agro-industrial wastes. High levels of beta-glucosidase (3778.9 U g^?1)from A. niger were obtained with rice meal as substrate under solid state culture conditions after ten days. Optimum pH for this particular beta-glucosidase activity was 4.0 although it was stable in the range of 4.0 to 7.0. The half-life (T_½) of beta-glucosidase at 55°C is 3 h. However, at the optimum temperature of the enzyme, 65°C, T_½ is 20 min. The enzyme showed tolerance to various compounds such as glucose, xylose, 5-hydroxymethyl furfural, furfural, coumarin, ethanol and acetic acid. Therefore, beta-glucosidase from the novel A. niger species may be of potential use in the saccharification of lignocellulosic biomass, as well as an additional enzyme supplement in cellulase cocktails used to increase the yield of fermentable sugars.     

Finite approximation schemes for Lévy processes,and their application to optimal stopping problems

This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0,T]$[0,T]$, and having a finite number of states, for a pure jump Lévy process _Lt$L_t$. The sequences of discrete processes converge to the original process, as the time interval becomes finer and the number of states grows larger, in various modes of weak and strong convergence, according to the way they are constructed. An important feature is that the filtrations generated at each stage by the approximations are sub-filtrations of the filtration generated by the continuous time Lévy process. This property is useful for applications of these results, especially to optimal stopping problems, as we illustrate with an application to American option pricing. The rates of convergence of the discrete approximations to the underlying continuous time process are assessed in terms of a “complexity” measure for the option pricing algorithm.