Tamás Szabados

Strong approximation of fractional Brownian motion by moving averages of simple random walks

The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to Mandelbrot and van Ness (SIAM Rev. 10 (1968) 422) as a self-similar Gaussian process W(H)(t) with stationary increments. Here self-similarity means that , where H[set membership, variant](0,1) is the Hurst parameter of fractional Brownian motion. F.B. Knight gave a construction of ordinary Brownian motion as a limit of simple random walks in 1961. Later his method was simplified by Revesz (Random Walk in Random and Non-Random Environments, World Scientific, Singapore, 1990) and then by Szabados (Studia Sci. Math. Hung. 31 (1996) 249-297). This approach is quite natural and elementary, and as such, can be extended to more general situations. Based on this, here we use moving averages of a suitable nested sequence of simple random walks that almost surely uniformly converge to fractional Brownian motion on compacts when . The rate of convergence proved in this case is O(N-min(H-1/4,1/4) log N), where N is the number of steps used for the approximation. If the more accurate (but also more intricate) Komlos et al. (1975,1976) approximation is used instead to embed random walks into ordinary Brownian motion, then the same type of moving averages almost surely uniformly converge to fractional Brownian motion on compacts for any H[set membership, variant](0,1). Moreover, the convergence rate is conjectured to be the best possible O(N-H log N), though only O(N-min(H,1/2) log N) is proved here.

Stochastic Integration Based on Simple, Symmetric Random Walks

Abstract A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and technically less demanding than other existing ones. In a large part of the theory one has a.s. uniform convergence on compacts. In particular, the method gives a.s. convergence for the stochastic integral of a finite variation function of the integrator, which is not càdlàg in general.

T Cells Survey the Stability of the Self: A Testable Hypothesis on the Homeostatic Role of TCR-MHC Interactions

In the lifetime of an individual, every single gene will have undergone mutation on about 1010 separate occasions. Nevertheless, cancer occurs mainly with advancing age. Here, we hypothesize that the evolutionary pressure driving the creation of the T cell receptor (TCR) repertoire was primarily the homeostatic surveillance of the genome. The subtly variable T cells may in fact constitute an evolutionary link between the invariable innate and hypervariable B cell systems. The new model is based on the homeostatic role of T cells, suggesting that molecular complementarity between the positively selected TCR and the self peptide-presenting major histocompatibility complex molecules establishes and regulates homeostasis, strictly limiting variations of its components. Notwithstanding, the ‘homeostatic role of T cells’ model offers a more realistic explanation as to how a naïve clonal immune system can cope with the much faster replicating pathogens, despite a limited repertoire that is capable of facing only a small fraction of the vast antigenic universe at a time.

An exponential functional of random walks

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

Some Aspects of Complementarity in the Immune System

The burden of this paper is the suggestion that the defence capacity of the immune system is rather limited. It cannot stand in readiness to deal with a practically endless diversity and abundance of microbes. In contrast to conventional thinking the current model proposes: (1) The core idea that cells of the immune system are basically and constantly interconnected with host cells (e.g., through TCR-MHC interactions) and that foreign antigens (peptides) may tend to obstruct such interactions. Peptides presented during a viral infection typically decrease complementarity between the structures that are the products of the major histocompatibility complex (MHC) genes (or other genes related to it) and T cells. The altered MHC profile exposes infected cells to a polyclonal immune attack from other T cells such that tissue destruction occurs in an allograft rejection-like fashion. This may explain why a substantial portion of T cell numbers is activated when only a small number of specific T cells is ‘obstructed’ from functioning by the presence of nonself peptides. (2) Phagocytes ‘see’ targets even in a non-immune host because complement distribution associated with polyreactive natural antibodies magnifies sensitization differences between pathogens and host cells. (3) There is only a probability that hypermutation will successfully change the genome in some B cell clones to produce high affinity antibodies that prevent the re-infection of the host by the same pathogen, but cannot conquer primary infections. (4) The history of the development of the immune responses suggests that during prolonged interaction between host and microbes in our natural habitat, carried on over many generations, the adaptive antibody population may facilitate the evolution of the natural antibody repertoire. The model predicts that microbes, which are not a part of the local environment, may invade the organism without significant resistance. The model is discussed in various interactions for survival in the context of infection and tumorigenicity.

Strong approximation of continuous local martingales by simple random walks

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

Self-intersection Local Time of Planar Brownian Motion Based on a Strong Approximation by Random Walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka–Rosen–Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.

Moments of an exponential functional of random walks and permutations with given descent sets

AbstractThe exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ1 + ξ1ξ2 + ξ1ξ2ξ3 + ⋯ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μk = E(ξk) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle.

Least cost safety inventory for large transfer lines

The term transfer line represents a synchronized serial production line where the output from each production stage is equal to the output of the line. The efficiency of the line is the fraction of good units produced over an infinite time horizon. Loss of production due to breakdowns or defective items at any stage stops the entire line temporarily and this often results in low efficiency. The efficiency can be increased by adding safety inventories to decouple the stages. This paper presents a model and a simulation method for determining the safety inventories needed at each stage to ensure that a prespecified efficiency is met and that the carrying cost of the total safety inventory is minimized. The solution procedure is applicable to large transfer lines and can accommodate any theoretical or empirical probability distribution for breakdowns and production of defective items.

Autoimmune T-cells induced by low dose immune checkpoint blockade could be a powerful therapeutic tool in cancer through activation of eliminative inflammation and immunity

1 Department of Probability, Alfred Renyi Institute of Mathematics, The Hungarian Academy of Sciences, 1053 Budapest, Reáltanoda str. 13-15. Hungary; 2 Department of Anesthesiology, Emergency and Intensive Care Medicine, University of Göttingen, Robert-Koch-Str. 40, Göttingen, 37075, Germany; 3 The Cambridge Stem Cell-Gene Therapy, Cultivated RBC Research Initiative A ; 4 Department of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp 3, Budapest, 1521, Hungary; 5 Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Kende u. 13-17., 1111 Budapest, Hungary; 6 European Laboratory for Particle Physics (CERN) Geneva 23 CH-1211 CH-1211, Switzerland.