The improved LaSalle-type theorems for stochastic functional differential equations

By Yi Shen, Qi Luo, Xuerong Mao. 2006

82 citations recorded by the source corpus.

Abstract The main aim of this paper is to improve some results obtained by Mao [X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud. 7 (2000) 307–328]. Our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above. For example, we need only the local Lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions. To guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation (SFDE) under the weaker conditions imposed in this paper, we establish a generalised existence-and-uniqueness theorem which covers a wider class of nonlinear SFDEs as demonstrated by the examples discussed in this paper. Moreover, from our improved results follow some new criteria on the stochastic asymptotic stability for SFDEs.

Subfields: Applied Mathematics, Finance, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/17c2e6a0f917d739fafef8e2ca86e32344a9fed0

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In Applied Mathematics

1. Fuke Wu
2. Yumei Huang
3. Zhiguo Yang
4. Xiaofeng Zong
5. Daoyi Xu

In Finance

1. Fuke Wu
2. Yumei Huang
3. Zhiguo Yang
4. Daoyi Xu

In Numerical Analysis

1. Fuke Wu
2. Yumei Huang
3. Zhiguo Yang
4. Daoyi Xu

Citations in the corpus

• Existence–uniqueness and continuation theorems for stochastic functional differential equations (2008, methodology): “Mao [19, Theorem 2.5, p. 153] obtained the global existence–uniqueness of solutions of the initial value problem (5) and (6) if B(t, xt ) and σ(t, xt ) satisfy the quasi-local Lipschitz condition1: for each k = 1,2, . . . , there is a constant ck > 0 such that ∣∣B(t,ϕ) − B(t,ψ)∣∣ ∨ ∣∣σ(t, ϕ) − σ(t,ψ”
• Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations (2016, background): “There are many results on stability theorems for stochastic differential systems (see Mao’s book [4,5]).”