Estimation for nonlinear stochastic differential equations by a local linearization method<sup>1</sup>

Shoji, Isao; Ozaki, Tohru

doi:10.1080/07362999808809559

Cited by 108 publications

(72 citation statements)

References 10 publications

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“…, M . Durham and Gallant (2002) suggest to apply to the transformed SDE the Shoji and Ozaki (1998) local linearization method, which approximates (2) with an Ornstein-Uhlenbeck process, characterized by a readily available Gaussian transition density:…”

Section: Approximations Based On Importance Sampling Integrationmentioning

confidence: 99%

“…They discuss a large number of available alternatives, among which the most effective is the Shoji and Ozaki (1998) local linearization approach applied to the Lamperti transform of the original diffusion. Anyway, all but one of them are basically Gaussian transition densities.…”

Section: Eis Implementation With Gaussian Subdensitiesmentioning

confidence: 99%

“…95%) percentile range, with a horizontal segment denoting the median, over 1,000 replications. In all cases: ∆ = 1/12; both methods use the same set of normalized random numbers, the Shoji and Ozaki (1998) subdensity and no extrapolation. Figure 2: Dispersion of EIS and MBB estimates of the log transition density for processes for which the latter is unknown.…”

Section: Loglikelihood Approximationmentioning

confidence: 99%

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Efficient importance sampling maximum likelihood estimation of stochastic differential equations

Pastorello

Rossi

2010

Computational Statistics & Data Analysis

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This paper considers ML estimation of a diffusion process observed discretely. Since the exact loglikelihood is generally not available, it must be approximated. We review the most efficient approaches in the literature, and point to some drawbacks. We propose to approximate the loglikelihood using the EIS strategy (Richard and Zhang, 1998), and detail its implementation for univariate homogeneous processes. Some Monte Carlo experiments evaluate its performance against an alternative IS strategy (Durham and Gallant, 2002), showing that EIS is at least equivalent, if not superior, while allowing a greater flexibility needed when examining more complicated models.JEL codes: C13, C15, C22

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Section: Approximations Based On Importance Sampling Integrationmentioning

confidence: 99%

Section: Eis Implementation With Gaussian Subdensitiesmentioning

confidence: 99%

Section: Loglikelihood Approximationmentioning

confidence: 99%

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Efficient importance sampling maximum likelihood estimation of stochastic differential equations

Pastorello

Rossi

2010

Computational Statistics & Data Analysis

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“…Parameter estimation for discretely observed diffusion processes is non-trivial, because of theoretical and implementation difficulties, and during the past decades there has been a great deal of research effort in this area (e.g. Aït-Sahalia (2001, 2002b, Beskos et al (2006), Bibby et al (2004), Bibby and Sørensen (1995), Brandt and Santa-Clara (2002), Dacunha-Castelle and Florens-Zmirnou (1986), Ditlevsen and De Gaetano (2005), Durham and Gallant (2002), Elerian et al (2001), Gallant and Long (1997), Gouriéroux et al (1993), Hurn and Lindsay (1999), Kutoyants (2004), Nicolau (2002), Pedersen (1995), Prakasa Rao (1999), Shoji and Ozaki (1998), Sørensen (2000)). …”

Section: Introductionmentioning

confidence: 99%

Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics

Picchini

Ditlevsen

Gaetano

2008

Mathematical Medicine and Biology

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Stochastic differential equations (SDEs) are assuming an important role in the definition of dynamical models allowing for explanation of internal variability (stochastic noise).SDE models are well-established in many fields, such as investment finance, population dynamics, polymer dynamics, hydrology and neuronal models. The metabolism of glucose and insulin has not yet received much attention from SDE modellers, except from a few recent contributions, because of methodological and implementation difficulties in estimating SDE parameters. Objectives: here we propose a new SDE model for the dynamics of glycemia during a euglycemic hyperinsulinemic clamp experiment, introducing system noise in tissue glucose uptake, and apply for its estimation a closed-form Hermite expansion of the transition densities of the solution process. Results: the present work estimates the new model parameters using a computationally efficient approximate maximum likelihood approach. By comparison with other currently used methods, the estimation process is very fast, obviating the need to use clusters or expensive mainframes to obtain the quick answers needed for everyday iterative modeling. Furthermore, it can introduce the demonstrably essential concept of system noise in this branch of physiological modeling. Conclusions: SDE modeling for metabolic processes is physiologically pertinent and computationally feasible using commonly available resources.

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“…Ozaki [8] and Shoji and Ozaki [10] propose alternative methods, called a local linearization method, which are used for both estimation and approximation purposes. In particular, the method proposed by Shoji and Ozaki [10] shows better performance in estimation through numerical experiments than the method proposed by Ozaki [8] and the Euler method.…”

Section: Introductionmentioning

confidence: 99%

Approximation of continuous time stochastic processes by a local linearization method

Shoji

1998

Math. Comp.

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Abstract. This paper investigates the rate of convergence of an alternative approximation method for stochastic differential equations. The rates of convergence of the one-step and multi-step approximation errors are proved to be O((∆t) 2 ) and O(∆t) in the Lp sense respectively, where ∆t is discrete time interval. The rate of convergence of the one-step approximation error is improved as compared with methods assuming the value of Brownian motion to be known only at discrete time. Through numerical experiments, the rate of convergence of the multi-step approximation error is seen to be much faster than in the conventional method.

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Estimation for nonlinear stochastic differential equations by a local linearization method¹

Cited by 108 publications

References 10 publications

Efficient importance sampling maximum likelihood estimation of stochastic differential equations

Efficient importance sampling maximum likelihood estimation of stochastic differential equations

Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics

Approximation of continuous time stochastic processes by a local linearization method

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Estimation for nonlinear stochastic differential equations by a local linearization method1

Cited by 108 publications

References 10 publications

Efficient importance sampling maximum likelihood estimation of stochastic differential equations

Efficient importance sampling maximum likelihood estimation of stochastic differential equations

Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics

Approximation of continuous time stochastic processes by a local linearization method

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Estimation for nonlinear stochastic differential equations by a local linearization method¹