Khasminskii-type theorem for a class of stochastic functional differential equations

1 Introduction

In general, in order for a stochastic differential equation to have a unique global solution for any given initial data, the coefficients of the equation are generally required to satisfy the linear growth condition (see, e.g., [1]) or a non-Lipschitz condition and linear growth condition (see, e.g., [2,3,4]), so the linear growth condition plays an important role in avoiding explosion in the finite time. However, many important equations in practice do not satisfy the linear growth condition, such as stochastic Lotka-Volterra systems (see, e.g., [5]). So it is necessary to establish more general existence-and-uniqueness theorems. There are many results of the solution of a stochastic functional differential equation (SFDE) without jumps. Xuerong and Rassias (see, e.g., [6]) examined the global solutions of SFDE under a more general condition, which has been introduced by Khasminskii. By this idea, Yi et al. (see, e.g., [7]) considered existence-and-uniqueness theorems of global solutions to SFDE. Qi et al. (see, e.g., [8]) established the existence-and-uniqueness theorems of global solutions to SFDE under local Lipschitz condition and Khasminskii-type conditions. Minghui et al. (see, e.g., [9]) established existence-and-uniqueness theorems for SFDE where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Lyapunov-type functions. Then, Fuke (see, e.g., [10]) considered the existence-and-uniqueness theorems of global solutions to neutral SFDE with the local Lipschitz condition but without the linear growth condition. Later, Fuke established the Khasminskii-type theorems for SFDE with finite delay. Quanxin (see, e.g., [11]) found the pth moment exponential stability of impulsive SFDEs with Markovian switching. Recently, Quanxin and his cooperators (see, e.g., [12]) studied the Razumikhin stability theorem for a class of impulsive stochastic delay differential systems.

For SFDE with jumps, Wei et al. (see, e.g., [13]) found the existence and uniqueness of solutions to a general neutral SFDE with infinite delay and Lévy jumps in the phase space C g under the local Carathêodory-type conditions and gave the exponential estimation and almost surely asymptotic estimation of solutions. By using the Razumikhin method and Lyapunov functions, Quanxin (see, e.g., [14]) obtained several Razumikhin-type theorems to prove the pth moment exponential stability of the suggested system and further discussed the pth moment exponential stability of stochastic delay differential equations with Lévy noise and Markov switching.

For a class of nonlinear stochastic differential delay equations with Poisson jump and time-dependent delay, Haidan and Quanxin (see, e.g., [15]) proved that the considered stochastic system has a unique global solution and investigated the pth moment exponential stability and the almost surely exponential stability of solutions under the local Lipschitz condition and a new nonlinear growth condition, which are weaker than those in the previous literature, by virtue of the Lyapunov function and the semi-martingale convergence theorem.

In this paper, we consider the existence-and-uniqueness theorems of global solutions to neutral SFDE with Markovian switching and Lévy jumps and establish the Khasmiskii-type theorem in the spirit of Minghui et al. (see, e.g., [9]). The main difficulty comes from the neutral term and the Lévy jumps term. After placing some assumptions on the neutral term and jump term, we obtained the existence and uniqueness of the solution by elementary inequality, the Gronwall inequality, the Burkh o ¨ lder-Davis-Gundy inequality, and the Itô formula.

This paper is organized as follows. We will establish the Khasminiskii-type existence-and-uniqueness theorems for neutral SFDEs with Markovian switching and Lévy jumps in Section 2. We will proceed to consider a special class of neutral SFDEs with Markovian switching and Lévy jumps, namely, neutral stochastic differential delay equations with variable delays in Section 3. An example is given in Section 4 to illustrate our results throughout the paper.

2 The Khasminskii-type theorem for SFDEs with Markovian switching and jumps

Throughout this paper, unless otherwise specified, we use the following notations. Let ∣ x ∣ be the Euclidean norm of a vector x ∈ R n . Let R + be the family of nonnegative real numbers. If A is a matrix, its trace norm is denoted by ∣ A ∣ = trace ( A T A ) . Let τ > 0 . Let C ( [ − τ , 0 ] ; R n ) be the family of continuous functions from [ − τ , 0 ] to R n with supremum norm ‖ φ ‖ = sup − τ ≤ θ ≤ 0 ∣ φ ( θ ) ∣ , which is a Banach space. Let ( Ω , ℱ , { ℱ t } t ≥ 0 , P ) be a complete probability space with a filtration { ℱ t } t ≥ 0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ 0 contains all P-null sets). Let p ≥ 1 and L ℱ t p ( [ − τ , 0 ] ; R n ) be the family of ℱ t -measurable C ( [ − τ , 0 ] , R n ) -valued random variables ϕ such that E ‖ ϕ ‖ p < ∞ . Let W ( t ) = ( W 1 ( t ) , … , W m ( t ) ) T be an m-dimensional Brownian motion defined on the probability space. Let p ¯ = { p ¯ ( t ) , t ≥ 0 } be a stationary and R n -valued Poisson point process. Then, for A ∈ ℬ ( R n − { 0 } ) , here ℬ ( R n − { 0 } ) denotes the Borel σ -field on R n − { 0 } , and we define the Poisson counting measure N associated with p ¯ by

For simplicity, we denote N ( t , A ) = N ( ( 0 , t ] × A ) . It is well known that there exists a σ -finite measure π , such that

This measure π is called the Lévy measure. Moreover, by Doob-Meyer’s decomposition theorem, there exists a unique { ℱ t } t ≥ 0 -adapted martingale N ˜ ( t , A ) and a unique { ℱ t } t ≥ 0 -adapted natural increasing process N ˆ ( t , A ) such that

Here, N ˜ ( t , A ) is called the compensated Lévy jumps and N ˆ ( t , A ) = π ( A ) t is called the compensator.

Let r ( t ) , t ≥ 0 , be a right-continuous Markovian chain on the probability space taking values in a finite state space S = { 1 , 2 , … , N } with generator Γ = ( r i j ) N × N given by

Here, Δ > 0 and γ i j ≥ 0 , i ≠ j , is the transition rate of the Markovian chain from i to j , while γ i i = − ∑ i ≠ j γ i j . We assume that the Markovian chain, Brownian motion, and Lévy jumps are independent. For Z ∈ ℬ ( R n − { 0 } ) , π ( Z ) < ∞ , consider a nonlinear neutral SFDE with Markovian switching and Lévy jump,

with the initial data x 0 = ξ ∈ C ( [ − τ , 0 ] ; R n ) . Here, G : C ( [ − τ , 0 ] ; R n ) × S → R n , f : C ( [ − τ , 0 ] ; R n ) × R n × R + × S → R n , g : C ( [ − τ , 0 ] ; R n ) × R n × R + × S → R n × m , h : C ( [ − τ , 0 ] ; R n ) × R n × R + × S × Z → R n , and for θ ∈ [ − τ , 0 ] , x t ( θ ) = x ( t + θ ) . Now we denote by C 1 , 2 ( [ − τ , ∞ ) × R n ; R + ) the family of all continuous nonnegative function V ( t , x ) defined on [ − τ , ∞ ) × R n , such that they are continuously twice differentiable in x and once in t . Given V ∈ C 1 , 2 ( [ − τ , ∞ ) × R n ; R + ) , define the function L V : R + × C ( [ − τ , 0 ] ; R n ) → R by

where