A new technique for simulating the likelihood of stochastic differential equations

Abstract

This article presents a new simulation-based technique for estimating the likelihood of stochastic differential equations. This technique is based on a result of DacunhaCastelle and Florens-Zmirou. These authors proved that the transition densities of a nonlinear diffusion process with a constant diffusion coefficient can be written in a closed form involving a stochastic integral. We show that this stochastic integral can be easily estimated through simulations and we prove a convergence result. This simulator for the transition density is used to obtain the simulated maximum likelihood (SML) estimator. We show through some Monte Carlo experiments that our technique is highly computationally efficient and the SML estimator converges rapidly to the maximum likelihood estimator.

Simulated Likelihood Estimation of Diffusions with an Application to Exchange Rate Dynamics in Incomplete Markets

“…The need for this transformation limits the transparency and adaptability of the method. Finally, Ogawa (1995), Hurn and Lindsay (1999), and Nicolau (2000) apply nonparametric density estimation to simulated data from the Euler discretizations to approximate the transition densities of the diffusion. This approach suffers from the usual problems with nonparametric density estimation: a slower convergence rate (in the number of simulations) and the curse of dimensionality.…”

Maximum likelihood estimation of drift and diffusion functions

“…For the case of Markovian processes, an integral form of the estimator has been derived. A slight simplification of this estimator, equation (13), is purely based on two point conditional pdfs, that can be calculated numerically from the Fokker-Planck equation in case of drift and diffusion processes. The integral form of the estimator allows for the reduction of huge datasets to their conditional transition pdfs prior to the iterative analysis procedure.…”

“…On the other hand, the analysis of discrete stochastic processes by means of maximum likelihood -methods has made great progress in recent years: Since it has become evident, that the maximisation of the likelihood function is a powerful tool for the analysis of Markovian time series [11], several methods have been proposed to optimise the calculation of the required conditional transition pdfs [12,13]. For a recent study on the preferences of current methods we refer to [14].…”

Parametric inference for stochastic differential equations with random effects in the drift coefficient

“…A natural approach would be likelihood inference, but the transition densities of the process are rarely known, and thus it is usually not possible to write the likelihood function explicitly. Many references proposed approximations for the unknown likelihood function, for general mixed SDEs an approximations of the likelihood have been proposed (Picchini and Ditlevsen, (2011) [16]), linearization (Beal and Sheiner (1982) [17])), approximate the transition density (Pedersen (1995) [18] Brandt and Santa-Clara (2002) [19] Nicolau (2002) [20], Hurn and Lindsay (1999) [21]), by solving numerically the Kolmogorov partial differential equations satisfied by the transition density (Lo [22] (1988)) or approximating the conditional transition density of the diffusion process given the random effects by a Hermit expansion, (Aït-Sahalia [23] (2002)). Delattre [24] studied the maximum likelihood estimator for random effects in more generally for fixed T and n tending to infinity and found an explicit expression for likelihood function and exact likelihood estimator by investigate the linear random effect in the drift (multiple case) together with a specific distribution for the random effect.…”