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Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approach

  • Yacine Ait-Sahalia

When a continuous-time diffusion is observed only at discrete dates, not necessarily close together, the likelihood function of the observations is in most cases not explicitly computable. Researchers have relied on simulations of sample paths in between the observations points, or numerical solutions of partial differential equations, to obtain estimates of the function to be maximized. By contrast, we construct a sequence of fully explicit functions which we show converge under very general conditions, including non-ergodicity, to the true (but unknown) likelihood function of the discretely-sampled diffusion. We document that the rate of convergence of the sequence is extremely fast for a number of examples relevant in finance. We then show that maximizing the sequence instead of the true function results in an estimator which converges to the true maximum-likelihood estimator and shares its asymptotic properties of consistency, asymptotic normality and efficiency. Applications to the valuation of derivative securities are also discussed.

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Paper provided by National Bureau of Economic Research, Inc in its series NBER Technical Working Papers with number 0222.

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Date of creation: Feb 1998
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Publication status: published as "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approach", Econometrica, Vol. 70, pp. 223-262, (2002).
Handle: RePEc:nbr:nberte:0222
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  1. Lo, Andrew W., 1988. "Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data," Econometric Theory, Cambridge University Press, vol. 4(02), pages 231-247, August.
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