Narrow Results

In this paper, we investigate asymptotics for the minimal spanning acycles (MSAs) of the (Alpha)-Delaunay complex on a stationary Poisson process on ${ℝ}^d,d\ge 2$. MSAs are topological (or higher-dimensional) generalizations of minimal spanning trees. We establish a central limit theorem (CLT) for total weight of the MSA on a Poisson Alpha-Delaunay complex. Our approach also allows us to establish CLTs for the sum of birth times and lifetimes in the persistent diagram of the Delaunay complex. The key to our proof is in showing the so-called weak stabilization of MSAs which proceeds by establishing suitable chain maps and uses matroidal properties of MSAs. In contrast to the proof of weak-stabilization for Euclidean minimal spanning trees via percolation-theoretic estimates, our weak-stabilization proof is algebraic in nature and provides an alternative proof even in the case of minimal spanning trees.

As a supplement of our previous work [10], we consider the localized region of the random Schrödinger operators on l²(Z^d) and study the point process composed of their eigenvalues and corresponding localization centers. For the Anderson model we show that, this point process in the natural scaling limit converges in distribution to the Poisson process on the product space of energy and space. In other models with suitable Wegner-type bounds, we can at least show that limiting point processes are infinitely divisible.

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an L^p-L^q bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

We consider the Schrödinger operator with random decaying potential on ${\ell }^2(Z^d)$ and showed that, (i) Integrated Density of States (IDS) coincide with that of free Laplacian in general cases, (ii) the set of extremal eigenvalues, after rescaling, converges to an inhomogeneous Poisson process, under certain condition on the single-site distribution, and (iii) there are “border-line” cases, such that we have Poisson statistics in the sense of (ii) above if the potential does not decay, while we do not if the potential does decay.

Checkpoint policies have been studied by several researchers. In this paper, we propose a new model for evaluating the database recovery actions and for determining an optimum checkpoint interval. Taking account of unsuccessful rollback recovery after system failure, we derive the stationary availability as a function of the checkpoint interval under the assumption that the failures occur at a Poisson process. We further discuss the optimum checkpoint interval which maximizes the availability above. The influence of unsuccessful rollback recovery on the system is illustrated by numerical examples.

The compensated Poisson noise is expressed as a composite sum (splitting) of creation and annihilation operators, whose probabilistic interpretation relies on time changes. We construct an Itô table for this decomposition and obtain continuous and discrete time realizations of Lévy processes on the finite difference algebra and on , e.g. the space–time dual of the Poisson process (compensated gamma process), and the continuous binomial process.

We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition P we assign the weight w(P) = p^e(P)q^e'(P), where e(P) and e'(P) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

The stock jumps of the underlying assets underpinning the Margrabe options have been studied by Cheang and Chiarella [Cheang, GH and Chiarella C (2011). Exchange options under jump-diffusion dynamics. Applied Mathematical Finance, 18(3), 245–276], Cheang and Garces [Cheang, GHL and Garces LPDM (2020). Representation of exchange option prices under stochastic volatility jump-diffusion dynamics. Quantitative Finance, 20(2), 291–310], Cufaro Petroni and Sabino [Cufaro Petroni, N and Sabino P (2020). Pricing exchange options with correlated jump diffusion processes. Quantitate Finance, 20(11), 1811–1823], and Ma et al. [Ma, Y, Pan D and Wang T (2020). Exchange options under clustered jump dynamics. Quantitative Finance, 20(6), 949–967]. Although the authors argue that they explored stock jumps under Hawkes processes, those processes are the Poisson process in their applications. Thus, they studied Hawkes processes in-between two assets while this study explores Hawkes process within any asset. Furthermore, the Poisson process can be flipped into Hawkes process and vice versa. In terms of hedging, this study uses specific Greeks (rho and phi) while some of the mentioned studies used other Greeks (Delta, Theta, Vega, and Gamma). Moreover, hedging is carried out under static and dynamic environments. The results illustrate that the jumpy Margrabe option can be extended to complex barrier option and waiting to invest option. In addition, hedging strategies are robust both under static and dynamic environments.

This paper introduces a class of stochastic processes making jumps around the circle. These circular processes are the wrapped versions of the Poisson, the negative binomial, the binomial processes and of extensions thereof obtained by compounding with a secondary frequency distribution. Their prevailing application is for modeling planar motions. These processes can be weakly stationary, can have uniform stationary distribution, can have independent or stationary circular increments. Their autocovariance functions, one-dimensional trigonometric moments and wrapped distributions are obtained. For the wrapped Poisson process, it is shown that the formula for the one-dimensional distribution can be obtained either by discrete Fourier transform or by wrapping the Poisson distribution around the circle and by simplifying it with the generalized hyperbolic function. Some numerical illustrations and comparisons are provided.

A one-dimensional (1D) point process, if considered as a random measure, can be represented by a sum, at most countable, of Delta Dirac measures concentrated at some random points. The integration with respect to the point process leads to the definition of the continuous wavelet transform of the process itself. As a possible choice of the mother wavelet, we propose the Mexican hat and the Morlet wavelet in order to implement the energy density of the process as a function of two wavelet parameters. Such mathematical tool works as a microscope to process an in-depth analysis of some classes of processes, in particular homogeneous, cluster, and locally scaled processes.

This letter considers quantum random bit generators which use continuous wave (CW) or pulse lasers with Poisson characteristics. The statistical characteristics of randomly generated bits are investigated, and the efficiency of the random generators is analyzed via the bit rate.

In this paper, we analyze the impact of savings withdrawals on a bank’s capital holdings under Basel III capital regulation. We examine the interplay between savings withdrawals and the investment strategies of a bank, by extending the classical mean–variance paradigm to investigate the bankers optimal investment strategy. We solve this via an optimization problem under a mean–variance paradigm, subject to a quadratic optimization function which incorporates a running penalization cost alongside the terminal condition. By solving the Hamilton–Jacobi–Bellman (HJB) equation, we derive the closed-form expressions for the value function as well as the banker’s optimal investment strategies. Our study provides a novel insight into the way banks allocate their capital holdings by showing that in the presence of savings withdrawals, banks will increase their risk-free asset holdings to hedge against the forthcoming deposit withdrawals whilst facing short-selling constraints. Moreover, we show that if the savings depositors of the bank are more stock-active, an economic expansion will imply a greater reduction in bank savings. As a result, the banker will reduce his/her loan portfolio and will depend on high stock returns with short-selling constraints to conform to Basel III capital regulation.

Most small businesses face uncertain demand for their products and services. The revenue they earn is most likely to be of stochastic nature. They face difficulties in making a fixed loan repayment throughout the life of the loan as they earn stochastic revenues. If banks are to lend to this market using the fixed loan repayment schedule/regime, it is highly likely that these businesses will default on their repayments several times in period $n$. If default rate is high in a loan portfolio, it means that the bank has to set up a higher bad loan provision, thereby tying up its capital. This reduces the lending business, profitability and growth of the financial institution. This is one of the reasons why financial institutions perceive these small businesses as a high risk market. Only financial institutions with a very high risk appetite will tap into this market. We are proposing revenue-based lending as a solution to the problem. Stochastic repayments will reduce periodic defaults. This reduction in periodic defaults will reduce the bad loan provision thereby making more funds available for lending. We want to show that the bad loan provision for the bank will be higher for a fixed loan repayment compared to the stochastic loan repayment.

The main purposes of this introduction chapter are (i) to give an overview of the following 109 papers, which discuss investment analysis, portfolio management, and financial derivatives; (ii) to classify these 109 chapters into nine topics; and (iii) to classify the keywords in terms of chapter numbers.

This chapter extends the Margrabe formula such that it is suitable for accounting for any type jump of stocks. Despite the fact that prices of an exchange option are characterized by jumps, it seems no study has explored those price jumps of an exchange option. The jump in this chapter is illustrated by a Poisson process. Moreover, the Poisson process can be extended into Cox process in case there is more than one jump. The results illustrate that incompleteness in an exchange option leads to a premium which in turn increases an option value while hedging strategies reveal mixed-bag type of results.

We study properties of thinning and Markow chain thinning of renewal processes. Among others, for some special renewal processes we investigate whether these processes can be obtained through Markov chain thinning.

The chapter deals with mechanistic biologically motivated modeling of metastasis. A general methodology for such a modeling is outlined, and a comprehensive mathematical model of individual natural history of metastatic cancer allowing for interaction between the primary tumor and metastases is formulated. This model is applied to computing the distribution of the sizes of detectable (or all) metastases in a given host site at any time post-diagnosis. A parametric version of the model for exponentially growing primary and secondary tumors and exponentially distributed metastasis promotion times is fully developed, and identifiability properties of this model are established.

Two stochastic models, the multistage model and a geometric model, are presented to describe formation and growth of cancer precursor lesions. Both models are applied to data from animal experiments by maximum likelihood methods. The models are used to test biological hypotheses about the process of formation and growth of preneoplastic lesions and to describe the dose-response relationship for model parameters.

This chapter measures how the risk associated with foreign direct investment in the prosperous, liberal economies of Hong Kong and Taiwan is affected by the prospect of reunification with the poor, politically and economically backward mainland. This China factor is modeled as a jump to a higher political risk level. I find that the China factor effect is substantial on Taiwan but still almost six times higher on Hong Kong. Nevertheless, Hong Kong’s political risk is considerably lower than China’s, thereby confirming the intuition that Hong Kong could be a lower-risk backdoor avenue to the potentially lucrative Chinese market.

In this chapter, we introduce a variety of discrete probability distributions and continuous probability distributions that are commonly used in economics and finance. Examples of discrete probability distributions include Bernoulli, Binomial, Negative Binomial, Geometric and Poisson distributions. Examples of continuous probability distributions include Beta, Cauchy, Chi-square, Exponential, Gamma, generalized Gamma, normal, lognormal, Weibull, and uniform distributions. The properties of these distributions as well as their applications in economics and finance are discussed. We also show some important techniques of obtaining moments and MGF's for various probability distributions.