Numerical solutions of stochastic differential delay equations under local Lipschitz condition

Mao, Xuerong; Sabanis, Sotirios

doi:10.1016/s0377-0427(02)00750-1

Cited by 157 publications

(109 citation statements)

References 8 publications

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“…The consistency analysis in [2] was performed for the Euler-Maruyama method. The latter has also been applied to SFDEs with variable delays and local Lipschitz conditions on the coefficient functions, using an interpolation at non-meshpoints by piecewise constants, in [15]. In [16] …”

Section: A Brief Review Of Methods and Aimsmentioning

confidence: 99%

One-step approximations for stochastic functional differential equations

Buckwar

2006

Applied Numerical Mathematics

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Section: A Brief Review Of Methods and Aimsmentioning

confidence: 99%

One-step approximations for stochastic functional differential equations

Buckwar

2006

Applied Numerical Mathematics

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“…given θ . Markovian systems are more easily real-time implementable and the infinitesimal generator of a Markov process can directly be used to construct its probability distribution by solving the Chapman-Kolmogorov forward equation [20][21][22][23].…”

Section: Time Delay Estimation In Lévy Noisementioning

confidence: 99%

Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

2017

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The stability and convergence of state, disturbance and parametric estimates of a robot have been analyzed using the Lyapunov method in the existing literature. In this paper, we analyze the problem of stochastic stability and also prove some results regarding behavior of statistically averaged Lyapunov energy function in the presence of jerk noise modeled as the sum of independent random variables hitting the robot at Poisson times. This type of noise is also called jerk noise in contrast to white Gaussian noise. Jerk noise is a Lévy process, i.e., a process with stationary independent increments, and is the natural non-Gaussian generalization of white Gaussian noise. Jerk noise can successfully be used to model hand tremor occurring at sporadic time intervals. We also study here the problem of time delay estimation in Lévy noise.

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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%

“…3.1) holds. More importantly, it enable us to define the EM approximate solution to the DGBM (2.3) following the author's earlier paper [21]. Let the time-step size ∆t ∈ (0, 1) be a fraction of τ , that is ∆t = τ /N for some sufficiently large integer N .…”

Section: Euler-maruyama Approximationmentioning

confidence: 99%

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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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Numerical solutions of stochastic differential delay equations under local Lipschitz condition

Cited by 157 publications

References 8 publications

One-step approximations for stochastic functional differential equations

One-step approximations for stochastic functional differential equations

Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

Delay geometric Brownian motion in financial option valuation

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