Stochastic Differential Equations

Mao, Xuerong

doi:10.1533/9780857099402.47

Cited by 813 publications

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“…This point is discussed in Ref. [13] for instance. It is also illustrated in the paper [6] for the case of disease-free equilibrium of an SIS model and in Ref.…”

Section: Introductionmentioning

confidence: 89%

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Stability of an SEIR epidemic model with independent stochastic perturbations

Witbooi

2013

Physica A: Statistical Mechanics and its Applications

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Stability of an SEIR epidemic model with independent stochastic perturbationsPeter J. Witbooi AbstractFor an epidemic model of the type mentioned, we prove a theorem on almost sure exponential stability of the disease-free equilibrium. For small values of the diffusion parameter, σ , we describe the stability of the disease free equilibrium point in terms of an appropriate analogue, R σ , of the basic reproduction number R 0 of the deterministic special case. Whenever σ > 0 then R σ < R 0 . For small values of σ , the stability theorem guarantees almost sure exponential stability whenever R σ < 1. We also discuss the effect of increasing σ .

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“…This point is discussed in Ref. [13] for instance. It is also illustrated in the paper [6] for the case of disease-free equilibrium of an SIS model and in Ref.…”

Section: Introductionmentioning

confidence: 89%

“…In fact, these stability studies may well be the most active research theme in epidemiological modeling. There are various notions of stability for systems of sde's, see the book [13] of Mao. Stochastic perturbation is capable of improving stability, or to stabilize an otherwise unstable equilibrium solution of a system.…”

Section: Introductionmentioning

confidence: 99%

Stability of an SEIR epidemic model with independent stochastic perturbations

Witbooi

2013

Physica A: Statistical Mechanics and its Applications

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“…There are numerous examples in the literature where authors discretize SDDEs, typically with an Euler-type scheme (see e.g. [20][21][22]) so as to study their properties. Moreover, one can observe immediately that such an approach will allow the use of Monte Carlo simulations so as to compute the expected value of either a function of S(T ) or a functional of {S(t) : 0 ≤ t ≤ T }, and thus obtain the expected payoff of an option (see e.g.…”

Section: Euler-maruyama Approximationmentioning

confidence: 99%

“…This idea was developed in [20] and used recently by [3]. However, we will need the local Lipschitz condition in the next sections when we study the sensitivity of the time lag.…”

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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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“…We denote by R n + the positive cone in R n , that is R n + = {x ∈ R n : x i ≥ 0 for all 1 ≤ i ≤ n}. In general, we consider a d-dimensional stochastic differential equation [13] …”

Section: Introductionmentioning

confidence: 99%

Stationary distribution and pathwise estimation of n-species mutualism system with stochastic perturbation

Fang¹,

Han²,

Wen³

et al. 2016

J. Nonlinear Sci. Appl.

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In this paper, we develop a new stochastic mutualism population modelBy constructing suitable Lyapunov functions, we show the system has a stationary distribution. We also discuss the pathwise behaviour of the solution. The conclusions of this paper is very powerful since they are independent both of the system parameters and of the initial value. It is also independent of the noise intensity as long as the noise intensity σ 2 i > 0. Computer simulations are used to illustrated our results.

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Stochastic Differential Equations

Cited by 813 publications

References 0 publications

Stability of an SEIR epidemic model with independent stochastic perturbations

Stability of an SEIR epidemic model with independent stochastic perturbations

Delay geometric Brownian motion in financial option valuation

Stationary distribution and pathwise estimation of n-species mutualism system with stochastic perturbation

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