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The stochastic time-delayed system of credit risk contagion driven by correlated Gaussian white noises is investigated. Novikov’s theorem, the time-delay approximation, the path-integral approach, and first-order perturbation theory are used to derive time-delayed Fokker-Planck model and the stationary probability distribution function of the dynamical system of credit risk contagion in the financial market. Using the method of numerical simulation, the Hopf bifurcation and chaotic behaviors of credit risk contagion are analyzed when time-delay and nonlinear resistance coefficient are varied and the effects of time-delay, nonlinear resistance and the intensity and the correlated degree of correlated Gaussian white noises on the stationary probability distribution of credit risk contagion are investigated. It is found that, as the infectious scale of credit risk and the wavy frequency of credit risk contagion are increased, the stability of the system of credit risk contagion is reduced, the dynamical system of credit risk contagion gives rise to chaotic phenomena, and the chaotic area increases gradually with the increase in time-delay. The nonlinear resistance only influences the infectious scale and range of credit risk, which is reduced when the nonlinear resistance coefficient increases. In addition, the curve of the stationary probability distribution is monotone decreasing with the increase in parameters value of time-delay, nonlinear resistance, and the intensity and the correlated degree of correlated Gaussian white noises.

In the last few years, complex nonlinear dynamics analysis approach has provided an alternative scientific methodology to understand the highly complex nonlinear dynamics of the modern economic and financial systems [

Recently, stochastic systems with time-delay have attracted much attention in many fields, such as biological systems [

In addition, noise is prevalent in the financial system [

In this paper, the complex stochastic dynamics phenomena of credit risk contagion with time-delay and correlated noises are investigated. We extend the method presented in [

In this paper we will report a microscopic model of credit risk contagion in the financial market with time-delay and correlated noises, which aims at modelling the complex economic and social phenomena and investigating the complex nonlinear dynamics behavior in the process of credit risk contagion by considering the effect of time-delay and correlated noises.

The time-delay in the process of information gathering, recognizing, and transmission will be inevitable in the financial system. Particularly, in the process of remote transmission, the time-delay will be more evident [

In the extant literature on finance, a number of studies adopt Gaussian white noise to analyze its effect on the economic and financial system, such as [

We let

We assume that

According to Novikov’s theorem [

According to literature [

Thus, (

Similarly, we can obtain

We assume that

Namely,

So, the time-delayed Fokker-Planck equation driven by correlated noises can be rewritten as

In (

Equation (

Time process diagrams of credit risk contagion in the financial market under the influence of time-delay

Bifurcation and chaotic properties of the dynamical system of credit risk contagion in the financial market when the time-delay

In the real financial market, the effects of the nonlinear resistance coefficient on the infectious scale and range of credit risk have a decisive influence. Figure

Time process diagrams of credit risk contagion in the financial market under the influence of the nonlinear resistance coefficient

Bifurcation and chaotic properties of the dynamical system of credit risk contagion in the financial market when the time-delay

In (

We assume that the stationary solution is

According to (

The stationary probability distribution function of credit risk contagion

The stationary probability distribution function of credit risk contagion

Figure

The three-dimensional diagram of the stationary probability distribution function of credit risk contagion

In this paper the complex stochastic dynamics phenomena of credit risk contagion with time-delay driven by correlated Gaussian white noises are investigated. The time-delayed Fokker-Planck model of credit risk contagion driven by correlated Gaussian white noises is obtained by using Novikov’s theorem. Moreover, the stationary probability distribution function of the dynamical system of credit risk contagion is derived by combining the methods of the time-delay approximation, the path-integral approach, and first-order perturbation theory. Using the method of numerical simulation, the Hopf bifurcation and chaotic behaviors of credit risk contagion are analyzed when time-delay and nonlinear resistance coefficient are varied. The numerical simulations show that, with the increase in time-delay, the infectious scale and range of credit risk are increased gradually and the wavy frequency of credit risk contagion is increased acutely. Moreover, the stability of the system of credit risk contagion is reduced, the dynamical system of credit risk contagion gives rise to chaotic phenomena, and the chaotic area increases gradually with the increase in time-delay. In addition, the effects of time-delay, nonlinear resistance, and the intensity and the correlated degree of correlated Gaussian white noises on the stationary probability distribution of credit risk contagion are investigated by the numerical simulations. It is found that the curve of the stationary probability distribution is monotone decreasing with the increase in parameters value of time-delay, nonlinear resistance, and the intensity and the correlated degree of correlated Gaussian white noises.

However, further studies could expand the present work to cover other topics. For example, in the real world, types of noises usually affect the process of credit risk contagion and its dynamics behavior, such as Poissonian white noise, combined Gaussian and Poissonian noise, and colored noise. Therefore, our future study will focus on these issues.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to express their gratitude to the referees for their invaluable comments. This work was supported by the National Natural Science Foundation of China Grant (nos. 70932003, 71201023, 71371051, and 71301078), the Humanities and Social Science Youth Foundation of the Ministry of Education of China (no. 12YJC630101), the Funding of Jiangsu Innovation Program for Graduate Education (no. CXZZ12-0131).