Stability of Stochastic Differential Equations

Mao, Xuerong

doi:10.1533/9780857099402.107

Cited by 109 publications

(151 citation statements)

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“…In order to provide a better understanding of this expression, in the following subsections we study the convergence conditions of the MSE expression derived in Theorem 1 and the residual MSE after convergence. For a deeper insight into different forms for convergence of stochastic systems, the reader can refer to [31] and [32].…”

Section: Mse Analysis With Equal Link Probabilitiesmentioning

confidence: 99%

Mean Square Convergence of Consensus Algorithms in Random WSNs

Pereira

Pagès-Zamora

2010

IEEE Trans. Signal Process.

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Abstract-Distributed consensus algorithms for estimation of parameters or detection of events in wireless sensor networks have attracted considerable attention in recent years. A necessary condition to achieve a consensus on the average of the initial values is that the topology of the underlying graph is balanced or symmetric at every time instant. However, communication impairments can make the topology vary randomly in time, and instantaneous link symmetry between pairs of nodes is not guaranteed unless an acknowledgment protocol or an equivalent approach is implemented. In this paper, we evaluate the convergence of the consensus algorithm in the mean square sense in wireless sensor networks with random asymmetric topologies. For the case of links with equal probability of connection, a closed form expression for the mean square error of the state along with the dynamical range and the optimum value of the link weights that guarantee convergence are derived. For the case of links with different probabilities of connection, an upper bound for the mean square error of the state is derived. This upper bound can be computed for any time instant and can be employed to compute a link weight that reduces the convergence time of the algorithm.Index Terms-Asymmetric links, mean average consensus, mean square convergence, random topology, wireless sensor networks.

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Section: Mse Analysis With Equal Link Probabilitiesmentioning

confidence: 99%

Mean Square Convergence of Consensus Algorithms in Random WSNs

Pereira

Pagès-Zamora

2010

IEEE Trans. Signal Process.

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“…As a standing hypothesis, we assume the coefficients are locally Lipschitz continuous (see, e.g., [16,17]). …”

Section: R(t))db(t)mentioning

confidence: 99%

Structured Robust Stability and Boundedness of Nonlinear Hybrid Delay Systems

Fei¹,

Hu²,

Mao³

et al. 2018

SIAM J. Control Optim.

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Abstract. Taking different structures in different modes into account, the paper has developed a new theory on the structured robust stability and boundedness for nonlinear hybrid stochastic differential delay equations (SDDEs) without the linear growth condition. A new Lyapunov function is designed in order to deal with the effects of different structures as well as those of different parameters within the same modes. Moreover, a lot of effort is put into showing the almost sure asymptotic stability in the absence of the linear growth condition. 1. Introduction. Systems in many branches of science and industry not only depend on the present state and the past ones but may also experience abrupt changes in their structures and parameters. Hybrid stochastic differential delay equations (SDDEs; also known as SDDEs with Markovian switching) have been widely used to model these systems (see, e.g., the books [23,24] and the references therein). One of the important issues in the study of hybrid SDDEs is the asymptotic analysis of stability and boundedness (see, e.g., [3,5,13,19]). In asymptotic analysis, robust stability and boundedness have been two of most popular topics. For example, Ackermann [1] gave a nice motivation of robust stability. Hinrichsen and Pritchard [7,8] presented an excellent discussion of the stability radii of linear systems with structured perturbations. Su [26] and Tseng, Fong, and Su [27] discussed robust stability for linear delay equations. In the aspect of robustness of stochastic stability, Haussmann [6] studied robust stability for a linear system and Ichikawa [11] for a semilinear system. Mao, Koroleva, and Rodkina [21] discussed the robust stability of uncertain linear or semilinear stochastic delay systems. Mao [20] investigated the stability of the stochastic delay interval system with Markovian switching. For more information on the stability and boundedness of hybrid SDDEs, please see, e.g., [12,22,23,25]. However, all of the papers, up to 2013, in this area only considered these robust

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“…[18,19,22]). If we would apply this general theory to the DGBM (2.3), we could have been forced to assume that the volatility function V is locally Lipschitz continuous and is bounded.…”

Section: Properties Of the Delay Geometric Brownian Motionmentioning

confidence: 99%

Delay geometric Brownian motion in financial option valuation

Mao

Sabanis

2012

Stochastics

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Motivated by influential work on complete stochastic volatility models, such as Hobson and Rogers (1998), we introduce a model driven by a delay geometric Brownian motion (DGBM) which is described by the stochastic delay differential equationWe show that the equation has a unique positive solution under a very general condition, namely that the volatility function V is a continuous mapping from R + to itself. Moreover, we show that the delay effect is not too sensitive to time lag changes. The desirable robustness of the delay effect is demonstrated on several important financial derivatives as well as on the value process of the underlying asset. Finally, we introduce an Euler-Maruyama numerical scheme for our proposed model and show that this numerical method approximates option prices very well. All these features show that the proposed DGBM serves as a rich alternative in modelling financial instruments in a complete-market framework.

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Stability of Stochastic Differential Equations

Cited by 109 publications

References 0 publications

Mean Square Convergence of Consensus Algorithms in Random WSNs

Mean Square Convergence of Consensus Algorithms in Random WSNs

Structured Robust Stability and Boundedness of Nonlinear Hybrid Delay Systems

Delay geometric Brownian motion in financial option valuation

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