Narrow Results

In this paper, we investigate how particles settle in a one-dimensional deterministic model. We leverage a model for deterministic Brownian motion and introduce boundary conditions to simulate particle settling. By adjusting parameters in the Jerk equation, we explore how different diffusion, and the size of the settling space affect the particles’ behavior. We analyze how these parameters influence the settling time and its relationship to the dispersion medium. Additionally, we examine statistical properties as follows: the average settling time, the probability distributions for mean displacement, and how the probability distribution of settling times changes with the size of the space.

We consider a generalization of the classical results obtained by Freidlin and Wentzell to consider the case of time-dependent dissipative drifts. We show the convergence of diffusions with multiplicative noise in the zero limit of a diffusivity parameter to the related dynamical systems. We also derive the solution to the associated transport equation as an application.

We study the perpetual American call option pricing problem in a model of a financial market in which the firm issuing a traded asset can regulate the dividend rate by switching it between two constant values. The firm dividend policy is unknown for small investors, who can only observe the prices available from the market. The asset price dynamics are described by a geometric Brownian motion with a random drift rate modeled by a continuous time Markov chain with two states. The optimal exercise time of the option for small investors is found as the first time at which the asset price hits a boundary depending on the current state of the filtering dividend rate estimate. The proof is based on an embedding of the initial problem into a two-dimensional optimal stopping problem and the analysis of the associated parabolic-type free-boundary problem. We also provide closed form estimates for the rational option price and the optimal exercise boundary.

We construct a Dirichlet form associated with the infinite dimensional Lévy–Laplace operator. We show that there exists a natural connection between this form and a Markov process.

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

In this study, we apply empirical methodologies, which are essentially curve fitting techniques, and use cross-sectional bond price data to estimate and analyze the Taiwanese government bond (TGB) term structure of interest rates. We choose two economic models: the Vasicek model and the CIR model, and one mathematical model: the B-spline approximation function, as the discount bond function to extract the term structure from market coupon bond prices. To assess the fitting performances and investigate the economic information content of the term structure fitting models, we compare the estimation errors and examine whether trading mis-priced bonds according the fitting model, can provide excess returns. The hypothesis is that the mathematical model can fit the term structure better than the economic models. But the economic models, which contain economic information, are able to explain the term structure dynamics. Thus the economic models can perform better in identifying mis-priced bonds and in predicting excess trading returns, than the mathematical model. Using the methodologies in this study, we can investigate the term structure fitting problems and look at the economic information content of the term structure fitting models.

This study examines an environmental management strategy to effectively cap greenhouse gas (GHG) emissions in an electricity market. To do so, we model a stochastic impulse two-person, nonzero-sum game between a power plant, a representative of electricity production within a country whose primary aim is to maximize profits, and government who is motivated to minimize the social and environmental cost of pollution. We assume that the power plant’s competitive price is equal to the marginal cost whilst the government’s running cost is linear. When the uncontrolled output of GHG emissions evolves as a Geometric Brownian motion, we provide a more dynamic and robust depiction of the interplay between government and the energy sector. We provide solutions to the impulse control problem derived via the quasi-variational inequalities (QVIs). We then present a sufficiency criterion for the existence of a Nash equilibrium for the optimal policy. Ultimately, our use of short-run price competition characterized by strategic supplies for renewable and fossil resources and inclusion of endogenous constraints on production capacity provides a more robust model and an effective framework for the development of policy that allows governments to meet emissions targets whilst guaranteeing energy supply.

In this paper, the stochastic moment stability and almost-sure stability of a planner gyropendulum system with synchronous motor under the white noises are investigated. By applying the theory of diffusion process, an eigenvalue problem for the moment Lyapunov exponent is formulated. Then, through a perturbation method and a Fourier cosine series expansion, the second-order expansion of the moment Lyapunov exponent is solved, which is just the leading eigenvalue of an infinite matrix. Finally, the convergence and validity of the procedure are numerically verified, and the effects of system and noise parameters on the moment Lyapunov exponent are discussed. It was found that the increase in both the noise intensity and coefficient of the synchronous motor torque will weaken the stability of the gyropendulum system, and when they reach certain values, the system becomes unstable. In addition, according to the relationship between the moment Lyapunov exponent and maximal Lyapunov exponent, the stable thresholds are also given.

The purpose of this paper is to construct both strong and weak solutions (in certain functional classes) of the Cauchy problem for a class of systems of nonlinear parabolic equations via a unified stochastic approach. To this end we give a stochastic interpretation of such a system, treating it as a version of the backward Kolmogorov equation for a two-component Markov process with coefficients depending on the distribution of its first component. To extend this approach and apply it to the construction of a generalized solution of a system of nonlinear parabolic equations, we use results from Kunita's theory of stochastic flows.

Non-hierarchical cluster analysis for panel data is known to be hampered by structural preservation, computational complexity and efficiency, and dependency problems. Resolving these issues becomes increasingly important as efficient collection and maintenance of panel data make application more conducive.

To address some computational issues and structural preservation, Bonzo [3] presented a stochastic version of Kosmelj and Batagelj's approach [16] to clustering panel data. The method used a probability link function (instead of the usual distance functions) in defining cluster inertias with the aim of preserving the clusters' probabilistic structure. Formulating clustering as an optimization problem, the objective function allows the application of heuristic and stochastic optimization techniques.

In this paper, we present a modified heuristic for adaptive simulated annealing (ASA) by perturbing the state vector's sampling distribution, specifically, by perturbing the drift of a diffusion process. Such an approach has been used to hasten convergence towards global optimum at equilibrium for diversely complex, combinatorial, and large-scale systems. The perturbed ASA (PASA) heuristic is then embedded in a genetic algorithm (GA) procedure to hasten and improve the stochastic local search process. The PASA-GA hybrid can be further modified and improved such as by explicit parallel implementation.

There is an ever-increasing interest in investigating dynamics in time-varying graphs (TVGs). Nevertheless, so far, the notion of centrality in TVG scenarios usually refers to metrics that assess the relative importance of nodes along the temporal evolution of the dynamic complex network. For some TVG scenarios, however, more important than identifying the central nodes under a given node centrality definition is identifying the key time instants for taking certain actions. In this paper, we thus introduce and investigate the notion of time centrality in TVGs. Analogously to node centrality, time centrality evaluates the relative importance of time instants in dynamic complex networks. In this context, we present two time centrality metrics related to diffusion processes. We evaluate the two defined metrics using both a real-world dataset representing an in-person contact dynamic network and a synthetically generated randomized TVG. We validate the concept of time centrality showing that diffusion starting at the best ranked time instants (i.e., the most central ones), according to our metrics, can perform a faster and more efficient diffusion process.

The relativistic diffusion process and friction have been studied, especially in the framework of $f(R)$-gravity theory. The study of relativistic diffusion and friction processes based on $f(R)$-gravity is an alternative solution to solve the incompatibility problem emerging in the attempt to couple between the Fokker–Planck equation [FPE] to the Einstein field equation [EFE] encountered by Calogero. The energy–momentum tensor of the cosmological scalar field as proposed by Calogero is replaced by the presence of additional terms in the field equation of $f(R)$-gravity. The additional energy–momentum tensor in the field equation of $f(R)$-gravity in this context is regarded to compensate for the presence of the diffusion and another process like friction. The additional energy–momentum tensor is also regarded as due to the so-called curvature fluid or background fluid. Here we assume the presence of interaction between matter and the background fluid in the form of physical processes like diffusion, friction, etc. We also assume that there is ‘interplay’ between diffusion process and friction. In other words, the diffusion process and friction are not independent. As examples, we consider some viable models of $f(R)$ that satisfy both cosmological and local gravity constraints, i.e. $f(R)=R+\mathrm{\Lambda }{R}^2,f(R)=R-\frac{{{ℳ}}^4}{R}$, and $f(R)=R-{ℰ}{r}_{c}ln(1+\frac{R}{r_c})$. Furthermore, we apply it to explain the diffusion and friction processes in the expanding universe by considering the Friedmann–Lemaitre–Robertson–Walker (FLRW) model.

In this paper, we consider a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. We analyze long-time behavior of densities of the distributions of the solution. We prove that the densities can converge in $L^1$ to an invariant density or can converge weakly to a singular measure under appropriate conditions.

In this paper, we develop a new structural model that allows for a distinction between default and liquidation to be made. Default occurs when firm’s asset value process crosses a bankruptcy barrier. Here, we do not assume that default immediately triggers liquidation. Instead, the firm is allowed to continue operating even if it is in default. Liquidation is triggered as soon as the firm’s asset value has cumulatively spent a prespecified amount of time below the default barrier or has dropped below the liquidation barrier. The proposed model includes the Black–Cox model as a limiting case. A semi-analytical formula of the liquidation probability is derived for the case where firm’s asset value follows a geometric Brownian motion. Nonlinear volatility diffusion models are discussed as well.

In literature, a GARCH-jump mixture model, namely, the GARCH-jump model with autoregressive conditional jump intensity (GARJI) model, in which two conditional independent processes, i.e., a diffusion and a compounded Poisson process are used to account for stock price movements caused by normal and extreme event news arrivals, individually, is developed by Chan and Maheu (2002, 2004) to describe the volatility clustering and leverage effect phenomenon. The resulting model is less efficient and provides only ex post filter for the probability of the occurrences of large price movements. A more informative and parsimonious model, however, the VG NGARCH model, is proposed and calibrated in this study. Being an extension of the variance-gamma model developed by Madan et al. (1998), the proposed VG NGARCH model incorporates an autoregressive structure on the conditional shape parameters, which describes the news arrival rates of different impact sizes on the price movements, and an ex ante prediction for the occurrences of large price movements is provided. The performance of the proposed VG NGARCH model is compared to the GARJI model based on daily stock prices of five component financial companies in S&P 500, namely, Bank of America, Wells Fargo, J.P. Morgan Chase, CitiGroup, and AIG, respectively, from January 3, 2006 to December 31, 2009. The goodness of fit of the VG NGARCH model and its ability to predict the probabilities of large price movements are demonstrated by comparing with the benchmark GARJI model.

The objects studied in survival and event history analysis are stochastic phenomena developing over time. It is therefore natural to use the highly developed theory of stochastic processes. We argue that this theory should be used more in event history analysis. Some specific examples are treated: Markov chains, martingale-based counting processes, birth type processes, diffusion processes and Lévy processes. Some less well known applications are given, with the internal memory of the process as a connecting issue.

In a repeated game with imperfect public information, the set of equilibria depends on the way that the distribution of public signals varies with the players' actions. Recent research has focused on the case of “frequent monitoring,” where the time interval between periods becomes small. Here we study a simple example of a commitment game with a long-run and short-run player in order to examine different specifications of how the signal distribution depends upon period length. We give a simple criterion for the existence of efficient equilibrium, and show that the efficiency of the equilibria that can be supported depends in an important way on the effect of the player's actions on the variance of the signals, and whether extreme values of the signals are “bad news” of “cheating” behavior, or “good news” of “cooperative” behavior.

In this paper, we review diffusion phenomena in stochastically perturbed Hamiltonian systems, with the aim of defining the framework to use Nekhoroshev-like estimates as prototypes for the form of the diffusion coefficient. A discussion of the features of this framework is carried out. More importantly, the results of numerical simulations based on the proposed models are compared against the experimental data from recent measurements performed at the CERN Large Hadron Collider (LHC) of the extent of phase space where bounded motions occur. The main conclusions are presented and discussed in detail together with future steps.

This paper considers strategic entry decisions in a duopoly market when the underlying state variable follows a diffusion with volatility that depends on the current state variable. The extension to this case is more than marginal, since empirical studies have suggested that the volatility is indeed non-constant in real options practices. It is shown that, even in the extended model, three types of equilibria exist in the case of strategic substitution, as for the geometric Brownian case, when the revenue functions are linear. Also, the presence of strategic interactions may push a firm with cost advantage to invest earlier, and the firm value as well as the optimal threshold for the investment decision increases as the market uncertainty increases.

We study the perpetual American call option pricing problem in a model of a financial market in which the firm issuing a traded asset can regulate the dividend rate by switching it between two constant values. The firm dividend policy is unknown for small investors, who can only observe the prices available from the market. The asset price dynamics are described by a geometric Brownian motion with a random drift rate modeled by a continuous time Markov chain with two states. The optimal exercise time of the option for small investors is found as the first time at which the asset price hits a boundary depending on the current state of the filtering dividend rate estimate. The proof is based on an embedding of the initial problem into a two-dimensional optimal stopping problem and the analysis of the associated parabolic-type free-boundary problem. We also provide closed form estimates for the rational option price and the optimal exercise boundary.