Tomoyuki Ichiba

Hybrid Atlas models

We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stable capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of such models we discuss properties of various investment strategies, including the so-called growth-optimal and universal portfolios.

ON COLLISIONS OF BROWNIAN PARTICLES

We examine the behavior of n Brownian particles diffusing on the real line, with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the currentconfigura- tion of particles. Sufficient conditions are established forthe absence of triple collisions, as well as for the presence of (infinitely-many) triple collisions among the particles. As an application to the Atlas model of equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains.

Stability in a Model of Interbank Lending

We propose a simple model of the banking system and analyze stochastic stability of interbank lending. The monetary reserves of banks are modeled as a system of interacting Feller diffusions. The model is simple enough for mathematical analysis, yet captures how lending preferences of banks affect possible multiple bank failures. In our model we quantify the lending preference from one bank to another as a function of all the reserves and find an extreme example that only

On Probabilistic Excitement of Sports Games

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Assessing Substitution Variation Across Sites in Grass Chloroplast DNA

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Yet another condition for absence of collisions for competing Brownian particles

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Skew-Unfolding the Skorokhod Reflection of a Continuous Semimartingale

banks can survive after the multiple bank failures. This banking system induces a class of random graph processes in continuous time exhibiting some stability property. Our analysis reveals quantities which can be used as indicators by regulators to assess the systemic risk.

Convergence rates for rank-based models with applications to portfolio theory

In this paper we introduce a quantitative measure of the excitement of sports games. This measure can be thought of as the variability of the expectancy of winning as a game progresses. We illustrate the concept of excitement at soccer games for which the theoretical win expectancy can be well approximated from a Poisson model of scoring. We show that in the Poisson model, higher scoring rates lead to increased expected excitement. Given a particular strength of a team, the most exciting games are expected with opponents who are slightly stronger. We apply this theory to the FIFA World Cup 2006 games, where the winning expectancy was independently estimated by betting markets. Thus, it was possible to compute the expected and the realized excitement of each given game from the trading data.

Strong solutions of stochastic equations with rank-based coefficients

We assess the similarity of base substitution processes, described by empirically derived 4 × 4 matrices, using chi-square homogeneity tests. Such significance analyses allow us to assess variation in sequence evolution across sites and we apply them to matrices derived from noncoding sites in different contexts in grass chloroplast DNA. We show that there is statistically significant variation in rates and patterns of mutation among noncoding sites in different contexts and then demonstrate a similar and significant influence of context on substitutions at fourfold degenerate sites of coding regions from grass chloroplast DNA. These results show that context has the same general effect on substitution bias in coding and noncoding DNA: the A+T content of flanking bases is correlated with rate of substitution, transition bias, and GC → AT pressure, while the number of flanking pyrimidines on a single strand is correlated with a mutational bias, or skew, toward pyrimidines. Despite the similarity in general trends, however, when we compare coding and noncoding matrices we find that there is a statistically significant difference between them even when we control for context. Most noticeably, fourfold degenerate sites in coding sequences are undergoing substitution at a higher rate and there are also significant differences in the relationship between pyrimidines skew and the number of flanking pyrimidines. Possible reasons for the differences between coding and noncoding sites are discussed. Furthermore, our analysis illustrates a simple statistical way for comparing substitution processes across sites allowing us to better study variation in evolutionary processes across a genome.

A second-order stock market model

Consider a finite system of rank-based competing Brownian particles, where the drift and diffusion of each particle depend only on its current rank relative to other particles. We present a simple sufficient condition for absence of multiple collisions of a given order, continuing the earlier work by Bruggeman and Sarantsev (2015). Unlike in that paper, this new condition works even for infinite systems.