Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =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.     

The Social Worker as 'Bad Object'

Correspondence to Dr Marguerite Valentine, 56 Osbaldeston Road, London N16 7DR Summary A ‘bad object’ has been defined as an ‘objectwhom the subject hates or fears, who is experienced as malevolent.A bad object may be either an internal or an external object’(Rycroft, 1988, p. 100). This paper explores the circumstancesin which public fears and anxieties over child abuse becomeprojected into the social worker. It discusses how social workersintroject these feelings, become a ‘bad object’,and how social services departments have developed a numberof strategies which defend against feelings of blame and responsibility.The exploration is derived from my own experiences and observations,and applies Kleinian psychoanalytic theory to studies of institutionaldefences.