Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract

Abstract. We consider N independent stochastic processes (Xi (t), t ∈ [0,Ti]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φi. The distribution of the random effect φi depends on unknown parameters which are to be estimated from the continuous observation of the processes Xi. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φi and φi has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when Ti=T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.

On Bayesian asymptotics in stochastic differential equations with random effects

“…Although a great deal of work on mixed effects models exists in the statistical literature, mixed effects models where within subject variability is modeled via stochastic differential equations (SDE's) are relatively rare. For a relatively short but comprehensive review we refer the reader to Delattre et al (2013), who also undertake theoretical and asymptotic investigation of a class of SDE-based mixed effects models having the following form: for i = 1, . .…”

Estimation of parameters in stochastic differential equations with two random effects

“…As we know, the kullback equal zero if and only if = , then for any ∈> 0 , the set ∈ is non-empty provided that \{ } is non-empty. From the above we see that 1 ( , ) is continuous in (see [6]), and since the parameter space is compact , then from the properties of real analyses it is clear that 1 ( , ) is uniformly continuous on , that is mean , for any ∈> 0 , there exist such that…”

“…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[12]), linearization (Beal and Sheiner (1982) [3]), or approximating the conditional transition density of the diffusion process given the random effects by a Hermit expansion, (Aït-Sahalia (2002) [1]). Maitra et al (2015) [11]) studied consistency and asymptotic normality of the posterior distribution of the parameters in the SDE's with one random effect in the drift term, Delattre et al (2012) [6] and alkreemawi et al (2015) [2] are studied the maximum likelihood estimator for random effects in more generally for fixed T and n tending to infinity (for non i.i.d. sample paths, see Maitra et al (2014) [10]) and they found an explicit expression for likelihood function and exact likelihood estimator by investigate the linear random effect in the drift (multiple and additive case respectively) together with a specific distribution for the random effect.…”

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

“…Proposition4.1 Under H1-H3, Proof: By analogue way of proof of proposition7 in [24]. 18).by H4, we have…”

“…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. In this paper we consider the stochastic differential equation with (i-two random effects, ii-random effect and unknown parameter ,iii-random effect and constant ) in drift coefficient and suppose that the diffusion coefficient without random effect.…”