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Journal articles on the topic 'Random time change of Brownian motion and symmetry'
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1
Székely, B., and T. Szabados. "Strong approximation of continuous local martingales by simple random walks." Studia Scientiarum Mathematicarum Hungarica 41, no. 1 (March 2004): 101–26. http://dx.doi.org/10.1556/012.2004.41.1.6.
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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].
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Lerche, Hans Rudolf, and Ilse Maahs. "Sequential Detection of Drift Change for Brownian Motion with Unknown Sign." gmj 15, no. 4 (December 2008): 713–30. http://dx.doi.org/10.1515/gmj.2008.713.
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Abstract We study tests of power one for the following change-point problem. Suppose one observes a process 𝑊 which is either a Brownian motion without drift or a Brownian motion that has zero drift up to a random time τ after which with equal probability the drift becomes either θ or –θ, where the value of θ > 0 is known. The distribution of τ is also assumed to be known. We search for a stopping time 𝑇* that minimizes an appropriate Bayes risk and give a solution that is asymptotically optimal, when the cost of observation tends to zero.
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Gwynne, Ewain, Jason Miller, and Scott Sheffield. "The Tutte Embedding of the Poisson–Voronoi Tessellation of the Brownian Disk Converges to $$\sqrt{8/3}$$-Liouville Quantum Gravity." Communications in Mathematical Physics 374, no. 2 (November 4, 2019): 735–84. http://dx.doi.org/10.1007/s00220-019-03610-5.
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Abstract Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a $$\sqrt{8/3}$$8/3-Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on $$\mathbb {R}^2$$R2 is the $$\epsilon \rightarrow 0$$ϵ→0 limit of simple random walk on $$\epsilon \mathbb {Z}^2$$ϵZ2? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the $$\lambda \rightarrow \infty $$λ→∞ limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is $$\lambda $$λ times the associated area measure. Among other things, this implies that as $$\lambda \rightarrow \infty $$λ→∞ the Tutte embedding (a.k.a. harmonic embedding) of the discretized Brownian disk converges to the canonical conformal embedding of the continuum Brownian disk, which in turn corresponds to $$\sqrt{8/3}$$8/3-LQG. Along the way, we obtain other independently interesting facts about conformal embeddings of Brownian surfaces, including information about the Euclidean shapes of embedded metric balls and Voronoi cells. For example, we derive moment estimates that imply, in a certain precise sense, that these shapes are unlikely to be very long and thin.
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BAYLY, PHILIP V., and LAWRANCE N. VIRGIN. "EXPERIMENTAL EVIDENCE OF DIFFUSIVE DYNAMICS AND “RANDOM WALKING” IN A SIMPLE DETERMINISTIC MECHANICAL SYSTEM: THE SHAKEN PENDULUM." International Journal of Bifurcation and Chaos 02, no. 04 (December 1992): 983–88. http://dx.doi.org/10.1142/s0218127492000586.
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An experimental model of a simple pendulum, harmonically shaken, displays chaotic dynamics. Moreover, in strongly excited chaotic regimes the time series of total angular displacement, which is rarely examined, wanders unboundedly, displaying a power spectrum which falls off as 1/fα over several decades. This behavior corresponds to deterministic diffusion, which has been found in simulations of nonlinear maps with periodic translational symmetry. The displacement time series obtained by sampling the pendulum displacement once per cycle is self-affine and quantitatively similar to Brownian motion.
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Henderson, Vicky, and Rafał Wojakowski. "On the equivalence of floating- and fixed-strike Asian options." Journal of Applied Probability 39, no. 2 (June 2002): 391–94. http://dx.doi.org/10.1239/jap/1025131434.
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There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.
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Henderson, Vicky, and Rafał Wojakowski. "On the equivalence of floating- and fixed-strike Asian options." Journal of Applied Probability 39, no. 02 (June 2002): 391–94. http://dx.doi.org/10.1017/s0021900200022592.
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There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.
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CRIENS, DAVID. "A NOTE ON REAL-WORLD AND RISK-NEUTRAL DYNAMICS FOR HEATH–JARROW–MORTON FRAMEWORKS." International Journal of Theoretical and Applied Finance 23, no. 03 (May 2020): 2050020. http://dx.doi.org/10.1142/s021902492050020x.
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We show that for time-inhomogeneous Markovian Heath–Jarrow–Morton models driven by an infinite-dimensional Brownian motion and a Poisson random measure an equivalent change of measure exists whenever the real-world and the risk-neutral dynamics can be defined uniquely and are related via a drift and a jump condition.
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FEDOTOV, SERGEI, and ABBY TAN. "LONG MEMORY STOCHASTIC VOLATILITY IN OPTION PRICING." International Journal of Theoretical and Applied Finance 08, no. 03 (May 2005): 381–92. http://dx.doi.org/10.1142/s0219024905003013.
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The aim of this paper is to present a stochastic model that accounts for the effects of a long-memory in volatility on option pricing. The starting point is the stochastic Black–Scholes equation involving volatility with long-range dependence. We define the stochastic option price as a sum of classical Black–Scholes price and random deviation describing the risk from the random volatility. By using the fact that the option price and random volatility change on different time scales, we derive the asymptotic equation for this deviation involving fractional Brownian motion. The solution to this equation allows us to find the pricing bands for options.
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Kendall, Wilfrid S. "Symbolic computation and the diffusion of shapes of triads." Advances in Applied Probability 20, no. 4 (December 1988): 775–97. http://dx.doi.org/10.2307/1427360.
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This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape.The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the inequalities requisite if they are to arise as squared-side-lengths from a point-set in n-space. The D. G. Kendall theorem on the diffusion of shape identifies the statistics of the diffusion arising (under a time-change) as the shape of a triangle whose vertices diffuse by Brownian motion in 2-space or 3-space.Symbolic Itô calculus is used to give a new proof of the Clifford-Green theorem, and to generalize the D. G. Kendall theorem to the case of triangles in higher-dimensional space whose vertices diffuse either according to Brownian motion or according to an Ornstein–Uhlenbeck process.
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Kendall, Wilfrid S. "Symbolic computation and the diffusion of shapes of triads." Advances in Applied Probability 20, no. 04 (December 1988): 775–97. http://dx.doi.org/10.1017/s0001867800018371.
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This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape. The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the inequalities requisite if they are to arise as squared-side-lengths from a point-set in n-space. The D. G. Kendall theorem on the diffusion of shape identifies the statistics of the diffusion arising (under a time-change) as the shape of a triangle whose vertices diffuse by Brownian motion in 2-space or 3-space. Symbolic Itô calculus is used to give a new proof of the Clifford-Green theorem, and to generalize the D. G. Kendall theorem to the case of triangles in higher-dimensional space whose vertices diffuse either according to Brownian motion or according to an Ornstein–Uhlenbeck process.
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Hainaut, Donatien. "Calendar Spread Exchange Options Pricing with Gaussian Random Fields." Risks 6, no. 3 (August 8, 2018): 77. http://dx.doi.org/10.3390/risks6030077.
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Most of the models leading to an analytical expression for option prices are based on the assumption that underlying asset returns evolve according to a Brownian motion with drift. For some asset classes like commodities, a Brownian model does not fit empirical covariance and autocorrelation structures. This failure to replicate the covariance introduces a bias in the valuation of calendar spread exchange options. As the payoff of these options depends on two asset values at different times, particular care must be taken for the modeling of covariance and autocorrelation. This article proposes a simple alternative model for asset prices with sub-exponential, exponential and hyper-exponential autocovariance structures. In the proposed approach, price processes are seen as conditional Gaussian fields indexed by the time. In general, this process is not a semi-martingale, and therefore, we cannot rely on stochastic differential calculus to evaluate options. However, option prices are still calculable by the technique of the change of numeraire. A numerical illustration confirms the important influence of the covariance structure in the valuation of calendar spread exchange options for Brent against WTI crude oil and for gold against silver.
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V. Gapeev, Pavel, and Hessah Al Motairi. "Perpetual American Defaultable Options in Models with Random Dividends and Partial Information." Risks 6, no. 4 (November 6, 2018): 127. http://dx.doi.org/10.3390/risks6040127.
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We present closed-form solutions to the perpetual American dividend-paying put and call option pricing problems in two extensions of the Black–Merton–Scholes model with random dividends under full and partial information. We assume that the dividend rate of the underlying asset price changes its value at a certain random time which has an exponential distribution and is independent of the standard Brownian motion driving the price of the underlying risky asset. In the full information version of the model, it is assumed that this time is observable to the option holder, while in the partial information version of the model, it is assumed that this time is unobservable to the option holder. The optimal exercise times are shown to be the first times at which the underlying risky asset price process hits certain constant levels. The proof is based on the solutions of the associated free-boundary problems and the applications of the change-of-variable formula.
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OIDA, KAZUMASA. "THE BIRTH AND DEATH PROCESSES OF HYPERCYCLE SPIRALS." Advances in Complex Systems 06, no. 04 (December 2003): 515–35. http://dx.doi.org/10.1142/s0219525903001018.
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The behavior of hypercycle spirals in a two-dimensional cellular automaton model is analyzed. Each spiral can be approximated by an Archimedean spiral with center, width, and phase change according to Brownian motion. A barrier exists between two spirals if the phase synchronization hypothesis is taken into account, and the occurrence rate of pair decay (simultaneous disappearance of two spirals) can be explained through a random walk simulation with the barrier. Simulation experiments show that adjacent species violation is necessary to create new spirals. A hypercycle system can live for a long time if spirals in the system are somewhat unstable, since new spirals cannot emerge when existing spirals are too stable.
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GRACE ELIZABETH RANI, T. G., and G. JAYALALITHA. "COMPLEX PATTERNS IN FINANCIAL TIME SERIES THROUGH HIGUCHI’S FRACTAL DIMENSION." Fractals 24, no. 04 (December 2016): 1650048. http://dx.doi.org/10.1142/s0218348x16500481.
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This paper analyzes the complexity of stock exchanges through fractal theory. Closing price indices of four stock exchanges with different industry sectors are selected. Degree of complexity is assessed through Higuchi’s fractal dimension. Various window sizes are considered in evaluating the fractal dimension. It is inferred that the data considered as a whole represents random walk for all the four indices. Analysis of financial data through windowing procedure exhibits multi-fractality. Attempts to apply moving averages to reduce noise in the data revealed lower estimates of fractal dimension, which was verified using fractional Brownian motion. A change in the normalization factor in Higuchi’s algorithm did improve the results. It is quintessential to focus on rural development to realize a standard and steady growth of economy. Tools must be devised to settle the issues in this regard. Micro level institutions are necessary for the economic growth of a country like India, which would induce a sporadic development in the present global economical scenario.
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Gauthier, Andrea, Stuart Jantzen, Gaël McGill, and Jodie Jenkinson. "Molecular Concepts Adaptive Assessment (MCAA) Characterizes Undergraduate Misconceptions about Molecular Emergence." CBE—Life Sciences Education 18, no. 1 (March 2019): ar4. http://dx.doi.org/10.1187/cbe.17-12-0267.
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This paper discusses the results of two experiments assessing undergraduate students’ beliefs about the random nature of molecular environments. Experiment 1 involved the implementation of a pilot adaptive assessment ( n = 773) and focus group discussions with undergraduate students enrolled in first- through third-year biology courses; experiment 2 involved the distribution of the redesigned adaptive assessment to the same population of students in three consecutive years ( n = 1170). The overarching goal of the study was to provide a detailed characterization of learners’ perceptions and beliefs regarding molecular agency, environments, and diffusion and whether or not those beliefs change over time. Our results indicated that advanced learners hold as many misconceptions as novice learners and that confidence in their misconceptions increases as they advance through their undergraduate education. In particular, students’ understanding of random/Brownian motion is complex and highly contextual, suggesting that the way in which we teach biology does not adequately remediate students’ preconceived notions of molecular agency and may actually reinforce them.
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TATOM, FRANK B. "THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS." Fractals 03, no. 01 (March 1995): 217–29. http://dx.doi.org/10.1142/s0218348x95000175.
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The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).
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Su, Hongsheng, Zonghao Ding, and Xingsheng Wang. "Prediction of Wind Turbine Gearbox Oil Temperature Based on Stochastic Differential Equation Modeling." Mathematics 12, no. 17 (September 9, 2024): 2783. http://dx.doi.org/10.3390/math12172783.
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Aiming at the problem of high failure rate and inconvenient maintenance of wind turbine gearboxes, this paper establishes a stochastic differential equation model that can be used to fit the change of gearbox oil temperature and adopts an iterative computational method and Markov-based modified optimization to fit the prediction sequence in order to realize the accurate prediction of gearbox oil temperature. The model divides the oil temperature change of the gearbox into two parts, internal aging and external random perturbation, adopts the approximation theorem to establish the internal aging model, and uses Brownian motion to simulate the external random perturbation. The model parameters were calculated by the Newton–Raphson iterative method based on the gearbox oil temperature monitoring data. Iterative calculations and Markov-based corrections were performed on the model prediction data. The gearbox oil temperature variations were simulated in MATLAB, and the fitting and testing errors were calculated before and after the iterations. By comparing the fitting and testing errors with the ordinary differential equations and the stochastic differential equations before iteration, the iterated model can better reflect the gear oil temperature trend and predict the oil temperature at a specific time. The accuracy of the iterated model in terms of fitting and prediction is important for the development of preventive maintenance.
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Hoang, Chau, Tuan Anh Phan, and Jianjun Paul Tian. "Stochastic Models for Ontogenetic Growth." Axioms 13, no. 12 (December 9, 2024): 861. https://doi.org/10.3390/axioms13120861.
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Based on allometric theory and scaling laws, numerous mathematical models have been proposed to study ontogenetic growth patterns of animals. Although deterministic models have provided valuable insight into growth dynamics, animal growth often deviates from strict deterministic patterns due to stochastic factors such as genetic variation and environmental fluctuations. In this study, we extend a general model for ontogenetic growth proposed by West et al. to stochastic models for ontogenetic growth by incorporating stochasticity using white noise. According to data variance fitting for stochasticity, we propose two stochastic models for ontogenetic growth, one is for determinate growth and one is for indeterminate growth. To develop a universal stochastic process for ontogenetic growth across diverse species, we approximate stochastic trajectories of two stochastic models, apply random time change, and obtain a geometric Brownian motion with a multiplier of an exponential time factor. We conduct detailed mathematical analysis and numerical analysis for our stochastic models. Our stochastic models not only predict average growth well but also variations in growth within species. This stochastic framework may be extended to studies of other growth phenomena.
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Su, Hongsheng, Dantong Wang, and Xuping Duan. "Condition Maintenance Decision of Wind Turbine Gearbox Based on Stochastic Differential Equation." Energies 13, no. 17 (August 31, 2020): 4480. http://dx.doi.org/10.3390/en13174480.
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Maintenance decision analysis is necessary to ensure the safe and stable operation of wind turbine equipment. To address gearboxes with a high failure rate in wind turbines, this paper establishes a new stochastic differential equation model of gearbox state transition to maximize the utilization of gearboxes. This model divides the state of the gearbox into two parts: internal degradation and external random interference. Weibull distribution and polynomial approximation were used to construct the internal degradation model of the gearbox. The external random interference is simulated by Brownian motion. On the basis of the analysis of monitoring data, the parameters of the gearbox state model were solved using the Newton–Raphson iterative method and entropy method. The state change of the gearbox was simulated in MATLAB, and the residual value between the predicted state and the real state was calculated. Compared with the state transformation model constructed by the traditional ordinary differential equation and the gamma distribution, the Weibull polynomial approximation stochastic model can better reflect the state of the device. With reliability set as the decision goal, the maintenance time of the gearbox is predicted, and the validity of the model is verified through case analysis.
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Abasto, D. F., M. Mohseni, S. Lloyd, and P. Zanardi. "Exciton diffusion length in complex quantum systems: the effects of disorder and environmental fluctuations on symmetry-enhanced supertransfer." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, no. 1972 (August 13, 2012): 3750–70. http://dx.doi.org/10.1098/rsta.2011.0213.
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Symmetric couplings among aggregates of n chromophores increase the transfer rate of excitons by a factor n 2 , a quantum-mechanical phenomenon called ‘supertransfer’. In this work, we demonstrate how supertransfer effects induced by geometrical symmetries can enhance the exciton diffusion length by a factor n along cylindrically symmetric structures, consisting of arrays of rings of chromophores, and along spiral arrays. We analyse both closed-system dynamics and open quantum dynamics, modelled by combining a random bosonic bath with static disorder. In the closed-system case, we use the symmetries of the system within a short-time approximation to obtain a closed analytical expression for the diffusion length that explicitly reveals the supertransfer contribution. When subject to disorder, we show that supertransfer can enhance excitonic diffusion lengths for small disorders and characterize the crossover from coherent to incoherent motion. Owing to the quasi-one-dimensional nature of the model, disorder ultimately localizes the excitons, diminishing but not destroying the effects of supertransfer. When dephasing effects are included, we study the scaling of diffusion with both time and number of chromophores and observe that the transition from a coherent, ballistic regime to an incoherent, random-walk regime occurs at the same point as the change from supertransfer to classical scaling.
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Cairoli, Andrea, Rainer Klages, and Adrian Baule. "Weak Galilean invariance as a selection principle for coarse-grained diffusive models." Proceedings of the National Academy of Sciences 115, no. 22 (May 14, 2018): 5714–19. http://dx.doi.org/10.1073/pnas.1717292115.
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How does the mathematical description of a system change in different reference frames? Galilei first addressed this fundamental question by formulating the famous principle of Galilean invariance. It prescribes that the equations of motion of closed systems remain the same in different inertial frames related by Galilean transformations, thus imposing strong constraints on the dynamical rules. However, real world systems are often described by coarse-grained models integrating complex internal and external interactions indistinguishably as friction and stochastic forces. Since Galilean invariance is then violated, there is seemingly no alternative principle to assess a priori the physical consistency of a given stochastic model in different inertial frames. Here, starting from the Kac–Zwanzig Hamiltonian model generating Brownian motion, we show how Galilean invariance is broken during the coarse-graining procedure when deriving stochastic equations. Our analysis leads to a set of rules characterizing systems in different inertial frames that have to be satisfied by general stochastic models, which we call “weak Galilean invariance.” Several well-known stochastic processes are invariant in these terms, except the continuous-time random walk for which we derive the correct invariant description. Our results are particularly relevant for the modeling of biological systems, as they provide a theoretical principle to select physically consistent stochastic models before a validation against experimental data.
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Bukowicki, Marek, Marta Gruca, and Maria L. Ekiel-Jeżewska. "Dynamics of elastic dumbbells sedimenting in a viscous fluid: oscillations and hydrodynamic repulsion." Journal of Fluid Mechanics 767 (February 12, 2015): 95–108. http://dx.doi.org/10.1017/jfm.2015.31.
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AbstractHydrodynamic interactions between two identical elastic dumbbells settling under gravity in a viscous fluid at low Reynolds number are investigated using the point-particle model. The evolution of a benchmark initial configuration is studied, in which the dumbbells are vertical and their centres are aligned horizontally. Rigid dumbbells and pairs of separate beads starting from the same positions tumble periodically while settling. We find that elasticity (which breaks the time-reversal symmetry of the motion) significantly affects the system dynamics. This is remarkable when taking into account that elastic forces are always much smaller than gravity. We observe oscillating motion of the elastic dumbbells, which tumble and change their length non-periodically. Independently of the value of the spring constant, a horizontal hydrodynamic repulsion appears between the dumbbells: their centres of mass move apart from each other horizontally. This motion is fast for moderate values of the spring constant $k$, and slows down when $k$ tends to zero or to infinity; in these limiting cases we recover the periodic dynamics reported in the literature. For moderate values of the spring constant, and different initial configurations, we observe the existence of a universal time-dependent solution to which the system converges after an initial relaxation phase. The tumbling time and the width of the trajectories in the centre-of-mass frame increase with time. In addition to its fundamental significance, the benchmark solution presented here is important to understanding general features of systems with a larger number of elastic particles, in regular and random configurations.
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FERNÁNDEZ-MARTÍNEZ, M., M. A. SÁNCHEZ-GRANERO, M. J. MUÑOZ TORRECILLAS, and BILL MCKELVEY. "A COMPARISON OF THREE HURST EXPONENT APPROACHES TO PREDICT NASCENT BUBBLES IN S&P500 STOCKS." Fractals 25, no. 01 (February 2017): 1750006. http://dx.doi.org/10.1142/s0218348x17500062.
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Since the pioneer contributions due to Vandewalle and Ausloos, the Hurst exponent has been applied by econophysicists as a useful indicator to deal with investment strategies when such a value is above or below [Formula: see text], the Hurst exponent of a Brownian motion. In this paper, we hypothesize that the self-similarity exponent of financial time series provides a reliable indicator for herding behavior (HB) in the following sense: if there is HB, then the higher the price, the more the people will buy. This will generate persistence in the stocks which we shall measure by their self-similarity exponents. Along this work, we shall explore whether there is some connections between the self-similarity exponent of a stock (as a HB indicator) and the stock’s future performance under the assumption that the HB will last for some time. With this aim, three approaches to calculate the self-similarity exponent of a time series are compared in order to determine which performs best to identify the transition from random efficient market behavior to HB and hence, to detect the beginning of a bubble. Generalized Hurst Exponent, Detrended Fluctuation Analysis, and GM2 algorithms have been tested. Traditionally, researchers have focused on identifying the beginning of a crash. We study the beginning of the transition from efficient market behavior to a market bubble, instead. Our empirical results support that the higher (respectively the lower) the self-similarity index, the higher (respectively the lower) the mean of the price change, and hence, the better (respectively the worse) the performance of the corresponding stock. This would imply, as a consequence, that the transition process from random efficient market to HB has started. For experimentation purposes, S&P500 stock Index constituted our main data source.
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Kislov, A. V., and A. F. Glazovsky. "Simulation of the dynamics of the Hansbreen tidal glacier (Svalbard) based on the stochastic model." Ice and Snow 59, no. 4 (December 1, 2019): 452–59. http://dx.doi.org/10.15356/2076-6734-2019-4-441.
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The dynamics of the Hansbreen tidal glacier (Svalbard) is manifested at different time scales. In addition to the long-term trend, there are noticeable inter-annual fluctuations. And the last ones are precisely the subject of this work. Based on general conclusions of the theory of temporal dynamics of the massive inertial objects, the observed inter-annual changes in the length of the glacier can be explained as a result of the accumulation of anomalies of the heat fluxes and water flows. In spite the fact that the initial model of glacier dynamics is deterministically based on the physical law of conservation of ice mass (the so-called the «minimal model» was used), the model of length change is interpreted as stochastic. From this standpoint, it is the Langevin equation, which includes the effect of random temperature anomalies that can be interpreted as a white noise. From a mathematical point of view, this process is analogous to Brownian motion, i.e. the length of the Hansbreen glacier randomly fluctuates in the vicinity of its stable equilibrium position. Based on the Langevin equation, we passed to the Fokker–Planck equation, the solution of which allowed us to obtain the distribution function of the probabilities of interannual fluctuations of glacier length, which is close to the normal law. It was shown that the possible range of the variability covers the observed interval of the length fluctuations. The pdf is close to normal distribution.
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Zhang, Haoyan, and Pingping Jiang. "On some properties of sticky Brownian motion." Stochastics and Dynamics, December 14, 2020, 2150037. http://dx.doi.org/10.1142/s0219493721500374.
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In this paper, we investigate a generalization of Brownian motion, called sticky skew Brownian motion, which has two interesting characteristics: stickiness and skewness. This kind of processes spends a lot more time at its sticky points so that the time they spend at the sticky points has positive Lebesgue measure. By using time change, we obtain an SDE for the sticky skew Brownian motion. Then, we present the explicit relationship between symmetric local time and occupation time. Some basic probability properties, such as transition density, are studied and we derive the explicit expression of Laplace transform of transition density for the sticky skew Brownian motion. We also consider the first hitting time problems over a constant boundary and a random jump boundary, respectively, and give some corollaries based on the results above.
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Yan, Hao, and Hanshuang Chen. "Breakdown of arcsine law for resetting brownian motion." Physica Scripta, November 6, 2023. http://dx.doi.org/10.1088/1402-4896/ad0a2e.
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Abstract For a one-dimensional Brownian motion starting from the origin, the cumulative distribution of the occupation time $V$ staying above the origin obeys the celebrated arcsine law. In this work, we show how the law is modified for a resetting Brownian motion, where the Brownian is reset to the position $x_r$ at random times but with a constant rate $r$. When $x_r$ is exactly equal to zero, we derive the exact expression of the probability distribution $P_r(V|0,t)$ of $V$ during time $t$, and the moments of $V$ as functions of $r$ and $t$. $P_r(V|0,t)$ is always symmetric with respect to $V=t/2$ for arbitrary value of $r$, but the probability density of $V$ at $V=t/2$ increases with the increase of $r$. Interestingly, $P_r(V|0,t)$ at $V=t/2$ changes from a minimum to a local maximum at a critical value $R^*\approx 0.742338$, where $R=rt$ denotes the average number of resetting during time $t$. Moreover, we consider the case when $x_r$ is 
a random variable and is distributed by a function $g(x_r)$, where $g(x_r)$ is assumed to be symmetric with respect to zero and possesses its maximum at zero. We derive the general expressions of the moments of $V$ when the variance of $x_r$ is low. The mean value of $V$ is always equal to $t/2$, but the fluctuation in $x_r$ leads to an increase in the second and third moments of $V$. Our results provide a quantitative understanding of how stochastic resetting destroys the persistence of Brownian motion.
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Buijsman, Wouter. "Efficient circular Dyson Brownian motion algorithm." Physical Review Research 6, no. 2 (June 10, 2024). http://dx.doi.org/10.1103/physrevresearch.6.023264.
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Circular Dyson Brownian motion describes the Brownian dynamics of particles on a circle (periodic boundary conditions), interacting through a logarithmic, long-range two-body potential. Within the log-gas picture of random matrix theory, it describes the level dynamics of unitary (“circular”) matrices. A common scenario is that one wants to know about an initial configuration evolved over a certain interval of time, without being interested in the intermediate dynamics. Numerical evaluation of this is computationally expensive as the time-evolution algorithm is accurate only on short time intervals because of an underlying perturbative approximation. This work proposes an efficient and easy-to-implement improved circular Dyson Brownian motion algorithm for the unitary class (Dyson index β=2, physically corresponding to broken time-reversal symmetry). The algorithm allows one to study time evolution over arbitrarily large intervals of time at a fixed computational cost, with no approximations being involved. Published by the American Physical Society 2024
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Di Crescenzo, Antonio, Claudio Macci, Barbara Martinucci, and Serena Spina. "Analysis of random walks on a hexagonal lattice." IMA Journal of Applied Mathematics, November 18, 2019. http://dx.doi.org/10.1093/imamat/hxz026.
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Abstract We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a two-dimensional Brownian motion is also discussed. Furthermore, we obtain some results on its asymptotic behaviour making use of large deviation theory. Finally, we investigate the first-passage-time problem of the random walk through a vertical straight line. Under suitable symmetry assumptions, we are able to determine the first-passage-time probabilities in a closed form, which deserve interest in applied fields.
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Bonaccorsi, Stefano, and Mirko D’Ovidio. "Sticky Brownian motions on star graphs." Fractional Calculus and Applied Analysis, September 18, 2024. http://dx.doi.org/10.1007/s13540-024-00336-7.
Full textAbstract:
AbstractThis paper is concerned with the construction of Brownian motions and related stochastic processes in a star graph, which is a non-Euclidean structure where some features of the classical modeling fail. We propose a probabilistic construction of the Sticky Brownian motion by slowing down the Brownian motion when in the vertex of the star graph. Later, we apply a random change of time to the previous construction, which leads to a trapping phenomenon in the vertex of the star graph, with characterization of the trap in terms of a singular measure $$\varPhi $$ Φ . The process associated to this time change is described here and, moreover, we show that it defines a probabilistic representation of the solution to a heat equation type problem on the star graph with non-local dynamic conditions in the vertex that can be written in terms of a Caputo-Džrbašjan fractional derivative defined by the singular measure $$\varPhi $$ Φ . Extensions to general graph structures can be given by applying to our results a localisation technique.
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Gu, Chu-Jun. "Mass Interaction Principle as a Foundational Framework for Quantum Mechanics." Global Journal of Human-Social Science, May 18, 2024, 27–157. http://dx.doi.org/10.34257/gjsfravol24is3pg27.
Full textAbstract:
This paper proposes mass interaction principle (MIP) as: the particles will be subjected to the random frictionless Brownian motion by the collision of space time particle(STP) ubiquitous in spacetime. The change in the amount of action of the particles during each collision is an integer multiple of the Planck constant h. The motion of particles under the action of STP is a Markov process. Under this principle, we infer that the statistical inertial mass of a particle is a statistical property that characterizes the difficulty of particle diffusion in spacetime. Within the framework of MIP, all the essences of quantum mechanics are derived, which proves that MIP is the origin of quantum mechanics. Due to the random collisions between STP and the matter particles, matter particles are able to behave exactly as required by the supervisor and shepherd for all microscopic behaviors of matter particles. More importantly, we solve a world class puzzle about the anomalous magnetic moment of muon in the latest experiment,and give a self-consistent explanation to the lifetime discrepancy of muon between standard model prediction and experiments at the same time. Last but not least, starting from MIP, we prove the principle of entropy increasing and clarify the physical root of entropy at absolute zero. Within the framework of MIP, we comprehensively discussed the Copenhagen interpretation. It leads to an important conclusion that Copenhagen interpretation is unnecessary for the quantum mechanical system. We found that Maxwell’s classical electromagnetic theory is applicable to the microscopic world not as previously thought. The key is that Brownian motion and classical electromagnetic theory must be combined together to completely solve the problem of electrons outside the nucleus not radiating electromagnetic waves.
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Glover, Kristoffer, and Goran Peskir. "Quickest Detection Problems for Ornstein–Uhlenbeck Processes." Mathematics of Operations Research, July 7, 2023. http://dx.doi.org/10.1287/moor.2021.0186.
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Consider an Ornstein–Uhlenbeck process that initially reverts to zero at a known mean-reversion rate β0, and then after some random/unobservable time, this mean-reversion rate is changed to β1. Assuming that the process is observed in real time, the problem is to detect when exactly this change occurs as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the process prior to the change of its mean-reversion rate. The solution is expressed in terms of a stopping time that minimises the probability of a false early detection and the expected delay of a missed late detection. Allowing for both positive and negative values of β0 and β1 (including zero), the problem and its solution embed many intuitive and practically interesting cases. For example, the detection of a mean-reverting process changing to a simple Brownian motion ([Formula: see text] and [Formula: see text]) and vice versa ([Formula: see text] and [Formula: see text]) finds a natural application to pairs trading in finance. The formulation also allows for the detection of a transient process becoming recurrent ([Formula: see text] and [Formula: see text]) as well as a recurrent process becoming transient ([Formula: see text] and [Formula: see text]). The resulting optimal stopping problem is inherently two-dimensional (because of a state-dependent signal-to-noise ratio), and various properties of its solution are established. In particular, we find the somewhat surprising fact that the optimal stopping boundary is an increasing function of the modulus of the observed process for all values of β0 and β1.
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Pérez-Guerrero, Daniela, José Luis Arauz-Lara, Erick Sarmiento-Gómez, and Guillermo Iván Guerrero-García. "On the Time Transition Between Short- and Long-Time Regimes of Colloidal Particles in External Periodic Potentials." Frontiers in Physics 9 (April 22, 2021). http://dx.doi.org/10.3389/fphy.2021.635269.
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The dynamics of colloidal particles at infinite dilution, under the influence of periodic external potentials, is studied here via experiments and numerical simulations for two representative potentials. From the experimental side, we analyzed the motion of a colloidal tracer in a one-dimensional array of fringes produced by the interference of two coherent laser beams, providing in this way an harmonic potential. The numerical analysis has been performed via Brownian dynamics (BD) simulations. The BD simulations correctly reproduced the experimental position- and time-dependent density of probability of the colloidal tracer in the short-times regime. The long-time diffusion coefficient has been obtained from the corresponding numerical mean square displacement (MSD). Similarly, a simulation of a random walker in a one dimensional array of adjacent cages with a probability of escaping from one cage to the next cage is one of the most simple models of a periodic potential, displaying two diffusive regimes separated by a dynamical caging period. The main result of this study is the observation that, in both potentials, it is seen that the critical time t*, defined as the specific time at which a change of curvature in the MSD is observed, remains approximately constant as a function of the height barrier U0 of the harmonic potential or the associated escape probability of the random walker. In order to understand this behavior, histograms of the first passage time of the tracer have been calculated for several height barriers U0 or escape probabilities. These histograms display a maximum at the most likely first passage time t′, which is approximately independent of the height barrier U0, or the associated escape probability, and it is located very close to the critical time t*. This behavior suggests that the critical time t*, defining the crossover between short- and long-time regimes, can be identified as the most likely first passage time t′ as a first approximation.
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Bressloff, Paul C. "Generalized Itô's lemma and the stochastic thermodynamics of diffusion with resetting." Journal of Physics A: Mathematical and Theoretical, October 8, 2024. http://dx.doi.org/10.1088/1751-8121/ad8495.
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Abstract Methods from the theory of stochastic processes are increasingly being used to extend classical thermodynamics to mesoscopic non-equilibrium systems. One characteristic feature of these systems is that averaging the stochastic entropy with respect to an ensemble of stochastic trajectories leads to a second law of thermodynamics that quantifies the degree of departure from thermodynamic equilibrium. A well known mechanism for maintaining a diffusing particle out of thermodynamic equilibrium is stochastic resetting. In its simplest form, the position of the particle instantaneously resets to a fixed position $x_0$ at a sequence of times generated from a Poisson process of constant rate $r$. Within the context of stochastic thermodynamics, instantaneous resetting to a single point is a unidirectional process that has no time-reversed equivalent. Hence, the average rate of entropy production calculated using the Gibbs-Shannon entropy cannot be related to the degree of time-reversal symmetry breaking. The problem of unidirectionality can be avoided by considering resetting to a random position or diffusion in an intermittent confining potential. In this paper we show how stochastic entropy production along sample paths of diffusion processes with resetting can be analyzed in terms of extensions of It\^o's formula for stochastic differential equations (SDEs) that include both continuous and discrete processes. First, we use the stochastic calculus of jump-diffusion processes to calculate the rate of stochastic entropy production for instantaneous resetting, and show how previous results are recovered upon averaging over sample trajectories. Second, we formulate single-particle diffusion in a switching potential as a hybrid SDE (hSDE) and develop a hybrid extension of It\^o's stochastic calculus to derive a general expression for the rate of stochastic entropy production. We illustrate the theory by considering overdamped Brownian motion in an intermittent harmonic potential. Finally, we calculate the average rate of entropy production for a population of non-interacting Brownian particles moving in a common switching potential. In particular, we show that the latter induces statistical correlations between the particles, which means that the total entropy is not given by the sum of the 1-particle entropies.
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Parthasarathy, Harish. "Some New Apsects of Quantum Gravity." Qeios, March 27, 2024. http://dx.doi.org/10.32388/pb17oh.
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We have proposed the quantization of the gravitational field in a synchronous reference frame taking as independent position fields, the six spatial components of the metric tensor. The Einstein-Hilbert Lagrangian is quadratic in the space time derivatives of these metric tensor components and hence in particular, the momentum fields become linear functions of the space-time derivatives of the position fields. It is this fact that gives a simple form to the Hamiltonian density of the gravitational field in a synchronous frame, this simple form of the Hamiltonian being a quadratic function of the momentum fields with a shift that is linear in the spatial derivatives of the metric, very much like the Hamiltonian of a non-relativistic particle moving in a vector potential. Differential equations for the gravitational field propagator are derived and we explain how approximations to this propagator can be derived and used to deduce the graviton propagator corrections caused by nonlinear interactions of the graviton field with itself. We explain how this corrected graviton propagator can be used to deduce how much mass the graviton acquires due to these self-interactions of cubic and higher order. We then consider the important problem involving the coupling of a nonlinear field theory described by its Lagrangian density to a quantum noisy bath and explain how the resulting Hamiltonian of the field plus bath can be used to derive the Hudson-Parthasarathy noisy Schrodinger equation (HPS) which is a quantum stochastic differential equation for the joint unitary evolution of the field interacting with the noisy bath. We explain this in the context of gravity coupled to a noisy bath like a noisy electromagnetic field. The HPS equation contains linear as well as quadratic terms in the white bath noise with the linear terms representing quantum annihilation and creation/quantum Brownian motion process differentials and the quadratic terms representing quantum conservation/Poisson processes differentials. Finally we explain how using Feynman path integrals for fields for evaluating the quantum effective action produced by higher order cumulants of the current field, we can calculate corrections to the quantum effective action produced by higher order cumulants of the current field and hence demonstrate how gauge symmetries of the classical action get broken when we pass over to the quantum effective action with additional symmetry breaking terms produced by the presence of higher order cumulants of the current. This kind of approximate symmetry breaking is known to give masses to massless particles or more generally, corrections to the masses of already massive particles and we illustrate this idea in the context of interactions of the gravitational field with a random electromagnetic field being regarded as the current. This interaction is the standard Maxwell action used in general relativity. The drawback of our approach to quantum gravity is that is its not diffeomorphic invariant since we have chosen our frame to be always synchronous. Further work on how one can incorporate interactions of the gravitational field with a random non-Abelian gauge field is in progress which becomes important because it generates non only quadratic but also cubic and fourth degree terms in the gauge field when it interacts with gravity.
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GAO Yi-Wen, WANG Ying, TIAN Wen-De, and CHEN Kang. "Dynamic behavior of an active polymer chain in the spatially-modulated driven field." Acta Physica Sinica, 2022, 0. http://dx.doi.org/10.7498/aps.71.20221367.
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Active polymers exhibit very rich dynamic behaviors due to their deformable long-chain architecture. In this paper, we perform Langevin dynamics simulations to study the behavior of a single self-propelled polymer chain in a plane (two dimensions) whose activity can be tuned by external field. We consider a spatially on-off periodic field along x direction, i.e. the plane is patterned into stripes of alternating active and passive regions. The width <i>d</i> of the stripes (half period length) plays a key role in determining the kinetic behavior of a flexible polymer chain. When <i>d</i> 》 2<i>R</i><sub><i>g</i>0</sub>(<i>R</i><sub><i>g</i>0</sub> is the radius of gyration of the passive flexible chain in the random coil state), the polymer chain can stay for a long time in either the active or the passive regions and moves mainly by slow Brownian diffusion; when 2<i>R<sub>L</sub></i> < <i>d</i> < 2 <i>R</i><sub><i>g</i>0</sub> (<i>R<sub>L</sub></i> is the radius of the spiral formed by the self-propelled polymer chain), the polymer chain could stay entirely in one region but cross-regional motion happens frequently; when <i>d</i> < 2<i>R<sub>L</sub></i>, the polymer chain does not stay entirely in one region and keeps moving cross-regionally accompanied by the stretching of the parts in active regions. Along with the change of the kinetic behavior of the polymer chain as <i>d</i> varies, the long-time diffusive coefficient changes by as many as two orders of magnitude and other statistical quantities such as spatial density distribution, mean total propelling force, characteristic size and orientation all show non-monotonic variation. In addition, we find four typical processes of the cross-regional motion of a flexible chain. For a semiflexible polymer chain, the cross-regional motion is accompanied by buckling behavior and the width <i>d</i> affects greatly on the degree of buckling and the continuity of the motion. Our work suggests a new idea for tuning and controlling the dynamic behavior of active polymers and is expected to provide a reference for the design and the potential applications of chain-like active materials.