Nguyen Tien Dung

Semimartingale approximation of fractional Brownian motion and its applications

The aim of this paper is to provide a semimartingale approximation of a fractional stochastic integration. This result leads us to approximate the fractional Black-Scholes model by a model driven by semimartingales, and a European option pricing formula is found.

An approximate approach to fractional stochastic integration and its applications

The aim of this paper is to introduce an approximation approach to fractional stochastic integration. Based on obtained result we find explicit solution of some fractional stochastic differential equations and study the ruin probability in the ALM model. 2000 AMS Classification: 26A33, 91B30.

On exponential stability of linear Levin-Nohel integro-differential equations

The aim of this paper is to investigate the exponential stability for linear Levin-Nohel integro-differential equations with time-varying delays. To the best of our knowledge, the exponential stability for such equations has not yet been discussed. In addition, since we do not require that the kernel and delay are continuous, our results improve those obtained in Becker and Burton [Proc. R. Soc. Edinburgh, Sect. A: Math. 136, 245-275 (2006)]; Dung [J. Math. Phys. 54, 082705 (2013)]; and Jin and Luo [Comput. Math. Appl. 57(7), 1080-1088 (2009)].

New stability conditions for mixed linear Levin-Nohel integro-differential equations

For the mixed Levin-Nohel integro-differential equation, we obtain new necessary and sufficient conditions of asymptotic stability. These results improve those obtained by Becker and Burton [“Stability, fixed points and inverse of delays,” Proc. - R. Soc. Edinburgh, Sect. A 136, 245–275 (2006)]10.1017/S0308210500004546 and Jin and Luo [“Stability of an integro-differential equation,” Comput. Math. Appl. 57(7), 1080–1088 (2009)]10.1016/j.camwa.2009.01.006 when b(t) = 0 and supplement the 32-stability theorem when a(t, s) = 0. In addition, the case of the equations with several delays is discussed as well.

On Delayed Logistic Equation Driven by Fractional Brownian Motion

In this paper we use the fractional stochastic integral given by Carmona et al. [1] to study a delayed logistic equation driven by fractional Brownian motion which is a generalization of the classical delayed logistic equation . We introduce an approximate method to find the explicit expression for the solution. Our proposed method can also be applied to the other models and to illustrate this, two models in physiology are discussed.

Asymptotic behavior of linear advanced differential equations

Abstract In this article, we consider a general class of linear advanced differential equations, and obtain explicitly sufficient conditions of convergence and exponential convergence to zero. A necessary condition is provided as well.

Stochastic dynamics of a delayed bistable system with multiplicative noise

In this paper we investigate the properties of a delayed bistable system under the effect of multiplicative noise. We first prove the existence and uniqueness of the positive solution and show that its moments are uniformly bounded. Then, we study stochastic dynamics of the solution in long time, the lower and upper bounds for the paths and an estimate for the average value are provided.

AN ASYMPTOTIC STABILITY THEOREM FOR QUASILINEAR DELAY DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS

In this paper, we provide new necessary and sufficient conditions of the asymptotic stability for a class of quasilinear differential equations with several delays and oscillating coefficients. Our results are established by means of fixed point theory and improve those obtained in [J. R. Graef, C. Qian and B. Zhang, Asymptotic behavior of solutions of differential equations with variable delays, Proc. London Math. Soc.81 (2000) 72–92; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal.63 (2005) e233–e242].