Closed-Form Likelihood Expansions for Multivariate Diffusions

By Yacine Aït-Sahalia. 2002

498 citations recorded by the source corpus.

This paper provides closed-form expansions for the transition density and likelihood function of arbitrary multivariate diffusions. The expansions are based on a Hermite series, whose coefficients are calculated explicitly by exploiting the special structure afforded by the diffusion hypothesis. Because the transition function for most diffusion models is not known explicitly, the expansions of this paper can help make maximum-likelihood a practical estimation method for discretely sampled multivariate diffusions. Examples of interest in financial econometrics are included.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6acebaeca6914149b1f4bd3229030dbe21110aeb

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Citations in the corpus

Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case (2005, methodology): “We refer to Bandi and Moloche [2] for the analysis of kernel estimators for the drift vector and the diffusion matrix and to Aı̈t-Sahalia [1] for a recent discussion of applications for multidimensional diffusion processes in econometrics.”
Term Structure Dynamics in Theory and Reality (2003, methodology): “Evidence supporting a nonlinear conditional mean for the short rate is discussed in Ait-Sahalia (1996) and Stanton (1997). In principle, the finding of nonlinear drifts for one-factor models could be a consequence of misspecifying the number of factors. However, the nonparametric analyses in Boudouk”
Stochastic Gradient Descent in Continuous Time (2016, unknown): “Some relevant publications in the ﬁnancial statistics literature include [1], [2], [7], and [14].”
Market price of risk speci-fications for a ne models: theory and evidence (2006, methodology): “We therefore use the approximate maximum likelihood approach of Aı̈t-Sahalia (2001), as implemented in Aı̈t-Sahalia and Kimmel (2005), for these five models.”
Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion) (2006, methodology): “Aït-Sahalia (2002) advocated the use of closed form analytic approximations to the unknown transition density; see Aït-Sahalia (2004) for multidimensional extensions.”
Properties of Foreign Exchange Risk Premiums (2012, methodology): “An alternative likelihood approximation is developed in Aı̈t-Sahalia (2008). It has been used successfully in connection with affine term structure models in Aı̈t-Sahalia and Kimmel (2010) and with affine equity models in Aı̈t-Sahalia and Kimmel (2007) within a maximum likelihood context.”
Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach (2011, methodology): “As noted by Aı̈t-Sahalia and Kimmel (2007), “The volatility can be estimated to an arbitrary degree of precision by sampling frequently enough, but the drift estimate is independent of sampling frequency.””
Hermite Polynomial Based Expansion of European Option Prices (2014, unknown): “It may be possible to further expand the option price around state variables, as developed in Aït-Sahalia (2008) for density expansion.”
Nonparametric Specification Testing for Continuous-Time Models with Applications to Term Structure of Interest Rates (2005, unknown): “See Ait-Sahalia (2002b) for more details. at C ornell U niversity L ibrary on N ovem ber 22, 2013 http://rfs.oxfordjournals.org/ D ow nloaded from 2.2.2.”
Maximum Likelihood Estimation of Latent Affine Processes (2006, unknown): “, Durham and Gallant, 2002), or by good approximation techniques (Aït-Sahalia, 2002).”
Saddlepoint Approximations for Continuous-Time Markov Processes ∗ (2006, unknown): “This has given rise to the development of a method designed to approximate it in closed form: see, Aı¨t-Sahalia (1999, 2001, 2002) for the original work on diffusions.”
Maximum-likelihood estimation for diffusion processes via closed-form density expansions (2013, methodology): “, Aı̈t-Sahalia [3]) labeled as DMROU, whose transition density is bivariate correlated normal:”
Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the Chinese Yuan (2007, unknown): “This expansion is discussed in detail in Aı¨t-Sahalia (2006) (see especially Theorem 2 and the remarks thereafter).”