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Beskos, Alexandros; Papaspiliopoulos, Omiros; Roberts, Gareth

Monte Carlo maximum likelihood estimation for discretely observed diffusion processes. (English) Zbl 1169.65004

Ann. Stat. 37, No. 1, 223-245 (2009).
The Monte Carlo method introduced in this paper gives unbiased and a.s. continuous estimates of the likelihood function of the discrete observations of a diffusion process. The likelihood contribution of each time step is estimated through independent copies of a random function. This function is based on a recently developed retrospective rejection sampling algorithm called the Exact Algorithm. There is no discretization error. Under regularity conditions, the Monte Carlo maximum likelihood estimator converges a.s. to the true maximum likelihood estimator of the unknown parameter. When the datasize \(n\) tends to infinity, the optimal number of Monte Carlo iterations should be tuned as \({\mathcal O}(n^{1/2})\) and the resulting Monte Carlo maximal likelihood estimator converges a.s. to the true parameter value. A numerical illustration of the method is given for the estimation of the three parameters of a logistic growth stochastic differential equation.
Reviewer: Dominique Lepingle (Orléans)

Cited in 30 Documents

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
62M05 Markov processes: estimation; hidden Markov models
60J60 Diffusion processes
65C05 Monte Carlo methods
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness

Keywords:

coupling; exact simulation; Brownian bridge; maximum likelihood estimator; strong law of large numbers on Banach space; numerical examples; Monte Carlo method; retrospective rejection sampling algorithm; logistic growth stochastic differential equation
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References:

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