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Switch process induced by the sine-Wiener noises in the gene transcriptional regulatory system were investigated by numerical simulations. Results show that (i) Only multiplicative noise which is the fluctuation of degradation reaction rate can induce a switch process (on → off) as the intensity of multiplicative noise increases for the case of uncorrelated noises; (ii) For the case of correlated noises, the switch process can be induced not only by intensity of multiplicative noise but also by intensity of additive noise which is the fluctuation of synthesis reaction rate and the cross-correlation intensity between the multiplicative and the additive noise can also induce the switch process; (iii) A successive switch process (on → off → on) occurs accompanied by a noise intensity increase when the correlation intensity takes larger value.

The anomalous diffusion and ergodicity breaking of Brownian particles in heterogeneous environments (space-dependent diffusion coefficient D ( x ) = D 0 | x | α ) subject to Levy noise are investigated, by both the analytically theory and the numerically simulating schemes. Within the tailored parameter, the competition of the power-law exponent α and the Levy index μ ensures the long tail of probability density function with the form P ( x ) ∼ 1 | x | μ ∕ p + 1 with p = 2 2 − α , falling rapidly when μ ∕ p > 2 . Meanwhile, we observe the subdiffusive motion that is characterized by the scale of ensemble averaged mean-square displacement (MSD), 〈 x 2 ( t ) 〉 ∝ t κ with κ = 2 p μ 1 . Furthermore, we also observe the weak ergodicity breaking which means the inequivalence between individual time averaged MSD and the corresponding ensemble averaged MSD. We quantify the weak ergodicity breaking behavior in terms of the dimensionless parameters E B and EB . For 1 ≤ μ 2 (except for μ = 2 ), the ergodicity breaking enhances as the α increases, but weakens as μ or D 0 . Overall, the EB follows the from, EB ∝ ( Δ ∕ T ) ( 1 2 − 2 p μ ) , in the large time. Additionally, the anomalous diffusion above might be particularly relevant for ultracold atoms diffusion in heterogeneous media.

The synchronicity between the stock and the stock-index in a market system is investigated. The results show that: (i) the synchronicity between the stock and the stock-index increases with the rising degree of market information capitalized into stock prices in certain range; (ii) the synchronicity decreases for large firm-specific information; (iii) the stock return synchronicity is small compared to the big noise trading, however the variance noise facilitates the synchronization within the tailored realms. These findings may be helpful in understanding the effect of market information on synchronicity, especially for the response of firm-specific information and noise trading to synchronicity.

This article investigates the trading time risk (TTR) of stock investment in the case of stock price drop of Dow Jones Industrial Average (ˆDJI) and Hushen300 data (CSI300), respectively. The escape time of stock price from the maximum to minimum in a data window length (DWL) is employed to measure the absolute TTR, the ratio of the escape time to data window length is defined as the relative TTR. Empirical probability density functions of the absolute and relative TTRs for the ˆDJI and CSI300 data evidence that (i) whenever the DWL increases, the absolute TTR increases, the relative TTR decreases otherwise; (ii) there is the monotonicity (or non-monotonicity) for the stability of the absolute (or relative) TTR; (iii) there is a peak distribution for shorter trading days and a two-peak distribution for longer trading days for the PDF of ratio; (iv) the trading days play an opposite role on the absolute (or relative) TTR and its stability between ˆDJI and CSI300 data.

We investigate the herd behavior of stock prices in a finance system with the Heston model. Based on parameter estimation of the Heston model obtained by minimizing the mean square deviation between the theoretical and empirical return distributions, we simulate mean residence time of positive return (MRTPR). Plots of MRTPR against the amplitude or mean reversion of volatility demonstrate a phenomenon of herd behavior for a positive cross correlation between noise sources of the Heston model. Also, for a negative cross correlation, a phenomenon of herd behavior is observed in plots of MRTPR against the long-run variance by increasing amplitude or mean reversion of volatility.

We study the coherent resonance phenomenon of stock market returns and volatility and discuss the effectiveness of financial market based on the coherent resonance theory of statistical physics. In this paper, we use the Heston model to give the series describing the return and volatility. Considering the limited data of CSI 300 index, the Bayesian estimation parameters of the proposed model are given by combining Bayesian parameter estimation and realized volatility. The characteristic correlation time (CCT) and normalized fluctuations of pulse durations R p are employed to further explore the coherent resonance behavior of stock market price returns and volatility. Based on the stochastic simulation of the estimated parameters, we can observe the phenomena of coherent resonance-like and anti-coherent resonance in CCT and R p as a function of long-run variance and amplitude of volatility. We can also observe the critical phenomenon, there is a critical mean reversion of volatility. On both sides of the critical value, the opposite characteristics of coherent and anti-coherent resonance can be observed. The enhanced long-run variance of volatility and weakened amplitude of volatility both cause coherent-like resonance in CCT and R p vs. the mean reversion of volatility. In addition, it can be seen that the market efficiency is lower when coherent resonance occurs, and higher when anti-coherence resonance occurs.

In this paper we analyze the correlation structure between U.S. markets and Asian markets. Our motivation being analysis of the effect of the 2008 financial crisis on Asian markets. For this purpose, we apply the linear correlations coefficient, the mutual information and the transfer entropy between U.S. stock market indices and eleven major Asia stock market indices. Then, with the overlapping windows for whole sample, and non-overlapping windows for three sub-samples, we quantify the dynamically evolving results of correlation structure between U.S. markets and Asian markets, and study the effect of the 2008 financial crisis on Asian markets by measuring the adherence between post and pre-crisis. Subsequently, for each blue-chip, we propose to study the 2008 financial crisis by measuring the adherence between post and pre-crisis. From this analysis, we can get a dependent structure map of Asian markets and U.S. markets, as well as may better forecasting of risk.

The roles of capital flow in an ensemble composed of sub-markets are investigated. A modified Heston model and recycled noises are employed to describe the dynamics of stock price and capital flow in the ensemble, respectively. The mean escape times of two sub-markets with a cubic nonlinearity are calculated by using numerical simulation. The results evidence that (i) there is a worst delay time or rate of capital inflow concerning the minimal stability of stock price and an optimal delay time or rate of capital outflow concerning the maximal stability of stock price when λ ≤ 0 ( λ denotes strength of correlation between two Wiener processes of the stock price and the volatility); (ii) when λ > 0 , the stability of stock price is maximally enhanced by an optimal delay time or rate of capital inflow and reduced by a worst rate of capital outflow, but monotonously strengthened by delay time of capital outflow.

The effects of periodic force on the stability of the metastable state in logistic model are investigated. The expression of the mean first-passage time (MFPT) from the metastable state to the stable state is derived. Based on the expression, the effects of periodic force on the MFPT were analyzed. The results indicate that: (i) For the case of the multiplicative noise induced transition, the multiplicative noise and the periodic force weaken the stability of the metastable state; (ii) For the case of the additive noise induced transition, the stability of the metastable state is weakened by the additive noise when the periodic force takes value of the first half cycle, while the stability of the metastable state is enhanced by the additive noise when the periodic force takes value of the second half cycle; (iii) For the case of the correlation between the multiplicative and the additive noise induced transition, the correlation between the multiplicative and the additive noises and the periodic force weaken the stability of the metastable state.

Immunotherapy is one of the most recent approaches in cancer therapy. A mathematical model of tumor–immune interaction, subject to a periodic immunotherapy treatment (imitated by a periodic signal), correlative and bounded stochastic fluctuations and time delays, is investigated by numerical simulations for its signal power amplification (SPA). Within the tailored parameter regime, the synchronous response of tumor growth to the immunotherapy, stochastic resonance (SR), versus both the noises and delays is obtained. The details are as follows (i) the peak values of SPA versus the noise intensity ( A ) in the proliferation term of tumor cells decrease as the frequency of periodic signal increases, i.e. an increase of the frequency restrains the SR; (ii) an increase of the amplitude of periodic signal restrains the SR versus A , but boosts up the SR versus the noise intensity B in the immune term; (iii) there is an optimum cross-correlated degree between the two bounded noises, at which the system exhibits the strongest SR versus the delay time τ α (the reaction time of tumor cell population to their surrounding environment constraints); (iv) upon increasing the delay time τ α , double SR versus the delay time τ β (the time taken by both the tumor antigen identification and tumor-stimulated proliferation of effectors) emerges. These results may be helpful for an immunotherapy treatment for the sufferer.

The returns and risks of investment portfolio in a financial system was investigated by constructing a theoretical model based on the Heston model . After the theoretical model and analysis of portfolio were calculated and analyzed, we find the following: (i) The statistical properties (i.e., the probability distribution, the variance and loss rate of equity portfolio return) between simulation results of the theoretical model and the real financial data obtained from Dow Jones Industrial Average are in good agreement; (ii) The maximum dispersion of the investment portfolio is associated with the maximum stability of the equity portfolio return and minimal investment risks; (iii) An increase of the investment period and a worst investment period are associated with a decrease of stability of the equity portfolio return and a maximum investment risk, respectively.

The auto-correlation function and the cross-correlation function of an autonomous stochastic system with time delays are investigated. We obtain the distribution curves of the auto-correlation function C x ( s ) and C y ( s ) , and the cross-correlation function C x y ( s ) and C y x ( s ) of the stochastic dynamic variables by the stochastic simulation method. The delay time changes prominently the behaviors of the dynamical variables of an autonomous stochastic system, which makes the auto-correlation and the cross-correlation of the autonomous stochastic system alternate oscillate periodically from positive to negative, or from negative to positive, decrease gradually, and finally tends to zero with the decay time. The delay time and the noise strength have important impacts for the auto-correlation and the cross-correlation of the autonomous stochastic delay system. The delay time enhances the auto-correlation and the cross-correlation, on the contrary, the noise strength lowers the auto-correlation and the cross-correlation. Under the time delay, by comparison we further show differences of the auto-correlation and the cross-correlation between the dynamical variables x and y .

Corporation financial stability was investigated by the method of Econophysics and Bayesian approach in this paper. A stochastic predator–prey model was built to describe the corporation financial condition. The mean limiting extinction time was proposed to measure the corporation financial stability. The model parameters were estimated by using Bayesian approach with real corporation financial data. Experimental results show that: (i) an increase of system risk corresponds to a decrease of the corporation stability; (ii) the optimal initial financial condition enhances the corporation stability to its maximum extent; (iii) under the optimal initial financial condition, the optimal system parameters and system risk strength can be observed.

The transport properties of an active Brownian particle with a time-delayed feedback and an external bias are investigated theoretically. By virtue of the perturbation theory for small delay, analytical expressions for the mean velocity and effective diffusion coefficient are derived. There exists a critical absolute value of the bias, below and above which the delay, respectively, enhances and weakens the diffusion, for a fixed noise intensity. The effects of delay observed above are more pronounced for weaker noise. These results are further verified via direct numerical simulations.

The effects of the delay time on the stability of a market model are investigated, by using a modified Heston model with a cubic nonlinearity and cross-correlated noise sources. These results indicate that: (i) There is an optimal delay time τ o which maximally enhances the stability of the stock price under strong demand elasticity of stock price, and maximally reduces the stability of the stock price under weak demand elasticity of stock price; (ii) The cross correlation coefficient of noises and the delay time play an opposite role on the stability for the case of the delay time τ o and the same role for the case of the delay time > τ o . Moreover, the probability density function of the escape time of stock price returns, the probability density function of the returns and the correlation function of the returns are compared with other literatures.

Stochastic resonance (SR) induced by the bounded noise in an asymmetric bistable system is investigated. Based on stochastic simulation, the signal power amplification S P A is derived for the case of the additive modulated signal. The simulation results indicate that: (1) the SR phenomenon is weakened by the asymmetry of the system; (2) increasing the frequency ω restricts the SR of the system; (3) there is an optimum value of amplitude A matching the strongest SR phenomenon.

A model of tumor–immune system interplay with time delays and cross-correlated sine-Wiener noises is investigated by numerical simulations for the stationary probability distribution (SPD) and stationary mean value 〈 x 〉 s t of tumor cell population. Results show that (i) the structure of the SPD transfers from bimodal to unimodal as the reaction time ( τ α ) of the cell population to environmental constraints increases; (ii) conversely, as the time ( τ β ) of immune system to develop specific immune competence and the correlation time ( τ ) of noises increase, the structure of the SPD transfers from unimodal to bimodal; (iii) as the cross-correlated intensity ( λ ′ ) between noises increases, the transitions induced by τ are suppressed.

The stochastic resonance in an underdamped quartic double-well potential with time delayed feedback is studied numerically. The signal power amplification is employed to characterize the stochastic resonance of the system. Simulation results indicate that: (i) for moderate frequency of the periodic driving, the stochastic resonance is decreased monotonically by increasing the delay time, but at high frequency, the reverse-resonance is induced to transform into a stochastic resonance by time delay; (ii) the damping coefficient has a critical value for which the stochastic resonance is optimum; (iii) a stochastic multi-resonance emerges when the signal power amplification is a function of the driving frequency.

The effects of strong noise on the decline and propagation processes of a population in the Malthus–Verhulst model with time delay are investigated by a stochastic simulation. Time delays in two different processes are concurrent in ecosystems. The simulation results indicate that: The stability of the population is enhanced by the decreasing multiplicative noise intensity and the increasing delay time. The replacement of old individuals with young ones is accelerated by an increasing multiplicative noise intensity, an increasing additive noise intensity and a decreasing delay time. An increasing multiplicative noise intensity will drive the population of species to fluctuate more largely.

A time-delayed tumor cell growth model with correlated noises is investigated. In the condition of small delay time, the stationary probability distribution is derived and the stationary mean value ( 〈 x 〉 s t ) and normalized variance λ 2 of the tumor cell population and state transition rate ( κ ) between two steady states are numerically calculated. The results indicate that: (i) The delay time ( τ ) enhances the coherence resonance in 〈 x 〉 s t as a function of the multiplicative noise intensity ( D ) and increases 〈 x 〉 s t as a function of the additive noise intensity ( α ), i.e., τ enhances fluctuation of the system, however, the strength ( λ ) of correlations between multiplicative and additive noise plays a contrary role to τ on these; (ii) τ enhances the coherence resonance in κ as a function of D and increases κ as a function of α , i.e., τ speeds up the rate of state transition, however, λ also plays a contrary role to τ on these.

The effects of correlations between additive and multiplicative noises on the state variable correlation function of a bistable system driven by cross-correlated noises were studied. Applying an approximate Fokker–Planck equation to the system, the analytic expression for the state variable correlation function of the system is derived by projection operator method. From numerical computations we find that the normalized correlation function C ( s ) decreases exponentially as the decay time s increases and decreases with increasing cross-correlated strength λ , however C ( s ) increases with increasing cross-correlated time τ . λ and τ play opposing roles in the C , i.e., λ enhances the rate of fluctuation decay of dynamical variable and the τ slows down the rate of fluctuation decay of dynamical variable.