non-Markovian Stochastic Process Research Articles

Non-Markovian diffusion equations and processes: Analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K ( t ) . The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l ( t ) . We then consider the subordinated process Y ( t ) = X ( l ( t ) ) where X ( t ) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y ( t ) is governed by a non-Markovian Fokker–Planck equation which involves the memory kernel K ( t ) . We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.

A delay stochastic process with applications in molecular biology

Molecular processes of cell differentiation often involve reactions with delays. We develop a mathematical model that provides a basis for a rigorous theoretical analysis of these processes as well as for direct simulation. A discrete, stochastic approach is adopted because several molecules appear in small numbers only. Our model is a non-Markovian stochastic process. The main theoretical results include a constructive proof of the existence of the process and a derivation of the rates for initiation and completion of reactions with delays. These results guarantee that the stochastic process is a consistent and realistic description of the molecular system. They also serve as a theoretical justification of recent work on delay stochastic simulation. We apply our model to an important process in developmental biology, the formation of somites in the vertebrate embryo. Simulation of the molecular oscillator controlling this process reveals major differences between stochastic and deterministic models.

Counting Statistics of Non-Markovian Quantum Stochastic Processes

We derive a general expression for the cumulant generating function (CGF) of non-Markovian quantum stochastic transport processes. The long-time limit of the CGF is determined by a single dominating pole of the resolvent of the memory kernel from which we extract the zero-frequency cumulants of the current using a recursive scheme. The finite-frequency noise is expressed not only in terms of the resolvent, but also initial system-environment correlations. As an illustrative example we consider electron transport through a dissipative double quantum dot for which we study the effects of dissipation on the zero-frequency cumulants of high orders and the finite-frequency noise.

Noise-induced coupling and phase transition in initially homogeneous bistable system

We show that a colored spatial noise induces a heterogeneous behavior and coupling of initially uncoupled single bistable units. A formal approximation reduces a non-Markovian stochastic process described by the initial set of equations into Markovian process in terms of Langevin equation, for which a simple piecewise linear emulation was used to represent the nonlinear deterministic force. It turned out that the coupling leads to a phase transition due to the noise-induced diffusive term. As an example, a typical bistable noisy system with symmetric double-well potential was studied.

A multiserver retrial queue: regenerative stability analysis

We consider a multiserver retrial GI/G/m queue with renewal input of primary customers, interarrival time ? with rate $\lambda=1/\mathsf{E}\tau$ , service time S, and exponential retrial times of customers blocked in the orbit. In the model, an arriving primary customer enters the system and gets a service immediately if there is an empty server, otherwise (if all m servers are busy) he joins the orbit and attempts to enter the system after an exponentially distributed time. Exploiting the regenerative structure of the (non-Markovian) stochastic process representing the total number of customers in the system (in service and in orbit), we determine stability conditions of the system and some of its variations. More precisely, we consider a discrete-time process embedded at the input instants and prove that if $\rho=:\lambda\mathsf{E}S and $\mathsf{P}(\tau>S)>0$ , then the regeneration period is aperiodic with a finite mean. Consequently, this queue has a stationary distribution under the same conditions as a standard multiserver queue GI/G/m with infinite buffer. To establish this result, we apply a renewal technique and a characterization of the limiting behavior of the forward renewal time in the (renewal) process of regenerations. The key step in the proof is to show that the service discipline is asymptotically work-conserving as the orbit size increases. Included are extensions of this stability analysis to continuous-time processes, a retrial system with impatient customers, a system with a general retrial rate, and a system with finite buffer for waiting primary customers. We also consider the regenerative structure of a multi-dimensional Markov process describing the system.

Memory effects in single-molecule spectroscopy

From the time series of LH2 optical single-molecule fluorescence excitation spectra of Rhodospirillum molischianum the memory function of the Mori–Zwanzig equation for the optical intensity is derived numerically. We show that the time dependence of the excited states is determined by at least three different non-Markovian stochastic processes with decay constants for the Mori–Zwanzig kernel on the order of 1 – 5 min - 1 . We suggest that this decay stems from the conformational motion of the protein scaffold of LH2.

The non-Markovian approach to the valuation and hedging of European contingent claims on power with scaling spikes

We present and further develop a new approach to modeling power prices with spikes proposed earlier by the author. In contrast to other approaches, we model power prices with spikes as a non-Markovian stochastic process that allows for modeling spikes directly as self-reversing jumps. We show how this approach can be used to value European contingent claims on power with spikes as well as to value and dynamically hedge European contingent claims on forwards on power for power with spikes in a practically important special case of the scaling probability distribution for the magnitude of spikes.

Some aspects of fractional diffusion equations of single and distributed order

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order β ∈ (0, 1). The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of sub-diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one. Then the fundamental solution is still a probability density of a non-Markovian process that, however, is no longer self-similar but exhibits a corresponding distribution of time-scales.

Regular and stochastic behavior of Parkinsonian pathological tremor signals

Regular and stochastic behavior in the time series of Parkinsonian pathological tremor velocity is studied on the basis of the statistical theory of discrete non-Markov stochastic processes and flicker-noise spectroscopy. We have developed a new method of analyzing and diagnosing Parkinson's disease (PD) by taking into consideration discreteness, fluctuations, long- and short-range correlations, regular and stochastic behavior, Markov and non-Markov effects and dynamic alternation of relaxation modes in the initial time signals. The spectrum of the statistical non-Markovity parameter reflects Markovity and non-Markovity in the initial time series of tremor. The relaxation and kinetic parameters used in the method allow us to estimate the relaxation scales of diverse scenarios of the time signals produced by the patient in various dynamic states. The local time behavior of the initial time correlation function and the first point of the non-Markovity parameter give detailed information about the variation of pathological tremor in the local regions of the time series. The obtained results can be used to find the most effective method of reducing or suppressing pathological tremor in each individual case of a PD patient. Generally, the method allows one to assess the efficacy of the medical treatment for a group of PD patients.

Derivatives Use: Valuation and hedging of contingent claims on power with spikes: A non-Markovian approach

A new approach to modelling spikes in power prices proposed in earlier papers by the author is presented and further developed. By contrast with the standard approaches, power prices with spikes are modelled as a non-Markovian stochastic process that allows for modelling spikes directly as self-reversing jumps. It is also shown how this approach can be used to value and hedge European contingent claims on power with spikes.

Volatility, persistence, and survival in financial markets

We study the temporal fluctuations in time-dependent stock prices (both individual and composite) as a stochastic phenomenon using general techniques and methods of nonequilibrium statistical mechanics. In particular, we analyze stock price fluctuations as a non-Markovian stochastic process using the first-passage statistical concepts of persistence and survival. We report the results of empirical measurements of the normalized qth-order correlation functions fq(t), survival probability S(t), and persistence probability P(t) for several stock market dynamical sets. We analyze both minute-to-minute and higher-frequency stock market recordings (i.e., with the sampling time deltat of the order of days). We find that the fluctuating stock price is multifractal and the choice of deltat has no effect on the qualitative multifractal behavior displayed by the 1/q dependence of the generalized Hurst exponent Hq associated with the power-law evolution of the correlation function fq(t) approximately tHq. The probability S(t) of the stock price remaining above the average up to time t is very sensitive to the total measurement time tm and the sampling time. The probability P(t) of the stock not returning to the initial value within an interval t has a universal power-law behavior P(t) approximately t(-theta), with a persistence exponent theta close to 0.5 that agrees with the prediction theta=1-H2. The empirical financial stocks also present an interesting feature found in turbulent fluids, the extended self-similarity.

Local evolution equations for non-Markovian processes

A Fokker–Planck equation approach for the treatment of non-Markovian stochastic processes is proposed. The approach is based on the introduction of fictitious trajectories sharing with the real ones their local structure and initial conditions. Different statistical quantities are generated by different construction rules for the trajectories, which coincide only in the Markovian case. The merits and limitations of the approach are discussed and applications to transport in ratchets and to anomalous diffusion are illustrated.

Modeling power forward prices for power with spikes: a non-Markovian approach

We obtain an analytical expression for the power forward prices in the case when the dynamics of the power spot prices with spikes are described by the non-Markovian stochastic process proposed earlier by the author. We also show how the power forward prices far from the maturity time of the forward contracts on power do not exhibit spikes while the power spot prices do.

Non-Markovian stochastic processes: Colored noise

We survey classical non-Markovian processes driven by thermal equilibrium or nonequilibrium (nonthermal) colored noise. Examples of colored noise are presented. For processes driven by thermal equilibrium noise, the fluctuation-dissipation relation holds. In consequence, the system has to be described by a generalized (integro-differential) Langevin equation with a restriction on the damping integral kernel: Its form depends on the correlation function of noise. For processes driven by nonequilibrium noise, there is no such a restriction: They are considered to be described by stochastic differential (Ito- or Langevin-type) equations with an independent noise term. For the latter, we review methods of analysis of one-dimensional systems driven by Ornstein-Uhlenbeck noise.

Generating functional analysis of minority games with real market histories

It is shown how the generating functional method of De Dominicis can be used to solve the dynamics of the original version of the minority game (MG), in which agents observe real as opposed to fake market histories. Here one again finds exact closed equations for correlation and response functions, but now these are defined in terms of two connected effective non-Markovian stochastic processes: a single effective agent equation similar to that of the `fake' history models, and a second effective equation for the overall market bid itself (the latter is absent in `fake' history models). The result is an exact theory, from which one can calculate from first principles both the persistent observables in the MG and the distribution of history frequencies.

The Characteristic Analysis of Weather and Climate Time Series

By using the result of the time-fractional diffusion equation that presented a self-similar non-Markovian stochastic process, this paper proposes that some views on the relationship between weather and climate on the basis of the observational data from Beijing and Jinan: weather could be interpreted as the white noise, and the fractional derivative of order q of climate (0≤ q ≤ 1) is Weather. Further to analyze the climatic discrete models, which is in conjunction with the persistence of time series, it is found that good memory of a climate time series, and long-tail of probability density function.

Valuation and hedging of European contingent claims on power with spikes: a non-Markovian approach

A new approach to modeling spikes in power prices proposed earlier by the author is presented and further developed. In contrast to the standard approaches, power prices with spikes as a non-Markovian stochastic process are modeled that allows for modeling spikes directly as self-reversing jumps. It is shown how this approach can be used to value and hedge European contingent claims on power with spikes. It is also shown that the values of European contingent claims on power with spikes satisfy the Cauchy problem for a certain linear evolution equation. In this way, the values of European contingent claims on power with spikes can be represented in terms of the Green's function for this Cauchy problem and the Green's function itself can be interpreted in terms of the values of the Arrow-Debreu securities on power with spikes.

Upper Bound of Time Derivative of Information Entropy in Non-Markovian Stochastic Process

Based on the Fokker-Planck description of non-Markovian stochastic process in higher dimension and the Schwartz inequality principle, we have calculated upper bound of rate of entropy change for the thermodynamically closed systems. The interplay of frictional memory kernel and noise-correlation time reveals extremal nature of the upper bound.

Non-Markov stochastic dynamics of real epidemic process of respiratory infections

The study of social networks and especially of stochastic dynamics of diseases spread in human population has recently attracted considerable attention in statistical physics. In this work we present a new statistical method of analyzing the spread of epidemic processes of grippe and acute respiratory track infections (ARTI) by means of the theory of discrete non-Markov stochastic processes. We use the results of our last theory (Phys. Rev. E 65 (2002) 046107) to study statistical memory effects, long-range correlation and discreteness in real data series, describing the epidemic dynamics of human ARTI infections and grippe. We have carried out the comparative analysis of the data of the two infections (grippe and ARTI) in one of the industrial districts of Kazan, one of the largest cities of Russia. The experimental data are analyzed by the power spectra of the initial time correlation function and the memory functions of junior orders, the phase portraits of the four first dynamic variables, the three first points of the statistical non-Markov parameter and the locally averaged kinetic and relaxation parameters. The received results give an opportunity to provide a strict quantitative description of regular and stochastic components in epidemic dynamics of social networks taking into account their time discreteness and effects of statistical memory. They also allow to reveal the degree of randomness and predictability of the real epidemic process in the specific social network.

Time to absorption in discounted reinforcement models

Reinforcement schemes are a class of non-Markovian stochastic processes. Their non-Markovian nature allows them to model some kind of memory of the past. One subclass of such models are those in which the past is exponentially discounted or forgotten. Often, models in this subclass have the property of becoming trapped with probability 1 in some degenerate state. While previous work has concentrated on such limit results, we concentrate here on a contrary effect, namely that the time to become trapped may increase exponentially in 1/ x as the discount rate, 1− x, approaches 1. As a result, the time to become trapped may easily exceed the lifetime of the simulation or of the physical data being modeled. In such a case, the quasi-stationary behavior is more germane. We apply our results to a model of social network formation based on ternary (three-person) interactions with uniform positive reinforcement.