Qigui Yang

Almost sure and moment stability properties of fractional order Black-Scholes model

We deal with the stability problem of the fractional order Black-Scholes model driven by fractional Brownian motion (fBm). First, necessary and sufficient conditions are established for almost sure asymptotic stability and pth moment asymptotic stability by means of the largest Lyapunov exponent and the pth moment Lyapunov exponent, respectively. Moreover, we are able to present large deviations results for this fractional process. In particular, for the first time it is found that the Hurst parameter affects both stability conclusions and large deviations. Interestingly, large deviations always happen for the considered system when 1/2 < H < 1. This fact is due to the long-range dependence (LRD) property of the fBm. Numerical simulation results are presented to illustrate the above findings.

The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion

This paper deals with the Ornstein-Uhlenbeck (O-U) process driven by the fractional Brownian motion (fBm). Based on the fractional Itô formula, we present the corresponding fBm-driven Fokker-Planck equation for the nonlinear stochastic differential equations driven by an fBm. We then apply it to establish the evolution of the probability density function (PDF) of the fBm-driven O-U process. We further obtain the closed form of such PDF by combining the Fourier transform and the method of characteristics. Interestingly, the obtained PDF has an infinite variance which is significantly different from the classical O-U process. We reveal that the fBm-driven O-U process can describe the heavy-tailedness or anomalous diffusion. Moreover, the speed of the sub-diffusion is inversely proportional to the viscosity coefficient, while is proportional to the Hurst parameter. Finally, we carry out numerical simulations to verify the above findings.

Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion

Little seems to be known about evaluating the stochastic stability of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) via stochastic Lyapunov technique. The objective of this paper is to work with stochastic stability criterions for such systems. By defining a new derivative operator and constructing some suitable stochastic Lyapunov function, we establish some sufficient conditions for two types of stability, that is, stability in probability and moment exponential stability of a class of nonlinear SDEs driven by fBm. We will also give an example to illustrate our theory. Specifically, the obtained results open a possible way to stochastic stabilization and destabilization problem associated with nonlinear SDEs driven by fBm.

Fractional noise destroys or induces a stochastic bifurcation

Little seems to be known about the stochastic bifurcation phenomena of non-Markovian systems. Our intention in this paper is to understand such complex dynamics by a simple system, namely, the Black-Scholes model driven by a mixed fractional Brownian motion. The most interesting finding is that the multiplicative fractional noise not only destroys but also induces a stochastic bifurcation under some suitable conditions. So it opens a possible way to explore the theory of stochastic bifurcation in the non-Markovian framework.

Dynamics of the stochastic Lorenz chaotic system with long memory effects

Little seems to be known about the ergodic dynamics of stochastic systems with fractional noise. This paper is devoted to discern such long time dynamics through the stochastic Lorenz chaotic system (SLCS) with long memory effects. By a truncation technique, the SLCS is proved to generate a continuous stochastic dynamical system Λ. Based on the Krylov-Bogoliubov criterion, the required Lyapunov function is further established to ensure the existence of the invariant measure of Λ. Meanwhile, the uniqueness of the invariant measure of Λ is proved by examining the strong Feller property, together with an irreducibility argument. Therefore, the SLCS has exactly one adapted stationary solution.

Bifurcation dynamics of the tempered fractional Langevin equation

Tempered fractional processes offer a useful extension for turbulence to include low frequencies. In this paper, we investigate the stochastic phenomenological bifurcation, or stochastic P-bifurcation, of the Langevin equation perturbed by tempered fractional Brownian motion. However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework. We first derive the spectral density function of the considered system based on the generalized Parsevals formula and the Wiener-Khinchin theorem. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosive-like, unimodal, and bimodal kurtosis. Therefore, our procedures in this paper are not merely comparable in scope to the existing theory of Markovian systems but also provide a possible approach to discern P-bifurcation dynamics in the non-Markovian settings.

Almost sure and moment stability properties of LTI stochastic dynamic systems driven by fractional Brownian motion

We deal with the stability problem of the scalar linear time invariant (LTI) stochastic system driven by fractional Brownian motion (fBm). Firstly, the necessary and sufficient conditions are provided for the almost sure asymptotic stability and pth moment asymptotic stability by means of the largest Lyapunov exponent and the Lyapunov exponent of the pth mean, respectively. Furthermore, we obtain the large deviations results for this fractional process. It has been shown that the Hurst parameter affects the stability conclusions and the large deviations. Interestingly, the large deviations always happen for the considered system when 1/2 <; H<;1. This is due to the long-range dependence (LRD) of the fBm.

Robust controllability of interval fractional order linear time invariant stochastic systems

We deal with the robust controllability problem of the fractional order linear time invariant (FO-LTI) stochastic systems with interval coefficients. We present a necessary and sufficient condition for the controllability problem for the case when there is no interval uncertainty. Based on the concept of linear independency of interval vectors, we formulate the approach to check the robust controllability of interval FO-LTI stochastic systems by employing some simple but very effective sufficient condition for checking the linear independency of interval vectors. Finally, an illustrative example is presented. We show that this interval FO-LTI stochastic system is weakly controllable, while the corresponding deterministic system without noise perturbation is uncontrollable.