François Coquet

A general converse comparison theorem for backward stochastic differential equations

Abstract In this Note, we establish a general converse comparison theorem for backward stochastic differential equations (BSDEs).

Stability in D of martingales and backward equations under discretization of filtration

We consider a cadlag process the filtration generated by Y and generated by step processes Yn defined from Y by discretization in time. We study the stability in (with Skorokhod topology) of -martingales and of -solutions of related backward equations, when Yn-->Y. We get this stability (in law) when Y is Markov and (in probability) under stronger assumptions on the coefficients of equations.

On Weak Convergence of Filtrations

A sequence of filtrations \(({\mathcal{F}^n_t})_{t{\leq}T}\) converges wealky to a filtration \(({\mathcal{F}}_t)_{t{\leq}T}\) iff, for all \(B{\in}{\mathcal{F}}_T\), the sequence of processes \((E[1_B\vert{\mathcal{F}}^n_t])_{t{\leq}T}\) converges in probability under the Skorokhod topology to the process \((E[1_B\vert{\mathcal{F}}_t])_{t{\leq}T}\). We give some examples of this kind of convergence; then we study, under the weak convergence of filtrations, the convergence in probability of processes \((E[X_t\vert{\mathcal{F}}^n_t])_{t{\leq}T}\) to X where X is an \({\mathcal{F}}_{t}\)-adapted semimartingale.

On the convergence of Dirichlet processes

For a given weakly convergent sequence {Xn} of Dirichlet processes we show weak convergence of the sequence of the corresponding quadratic variation processes as well as stochastic integrals driven by the Xn values provided that the condition UTD (a counterpart to the condition UT for Dirichlet processes) holds true. Moreover, we show that under UTD the limit process of {Xn} is a Dirichlet process, too.

Uniform convergence of the empirical cumulative distribution function under informative selection from a finite population

Consider informative selection of a sample from a finite population. Responses are realized as independent and identically distributed (i.i.d.) random variables with a probability density function (p.d.f.) f, referred to as the superpopulation model. The selection is informative in the sense that the sample responses, given that they were selected, are not i.i.d. f. In general, the informative selection mechanism may induce dependence among the selected observations. The impact of such dependence on the empirical cumulative distribution function (c.d.f.) is studied. An asymptotic framework and weak conditions on the informative selection mechanism are developed under which the (unweighted) empirical c.d.f. converges uniformly, in

On Non-Continuous Dirichlet Processes

L_2

Some examples and counterexamples of convergence of σ-algebras and filtrations

and almost surely, to a weighted version of the superpopulation c.d.f. This yields an analogue of the Glivenko-Cantelli theorem. A series of examples, motivated by real problems in surveys and other observational studies, shows that the conditions are verifiable for specified designs.

Asymptotics for the maximum sample likelihood estimator under informative selection from a finite population

We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C1 transformation.

Filtration Consistent Nonlinear Expectations

We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of σ-algebras to filtrations generated by different discretizations of a single process.

Filtration-consistent nonlinear expectations and related g-expectations

Inference for the parametric distribution of a response given covariates is considered under informative selection of a sample from a finite population. Under this selection, the conditional distribution of a response in the sample, given the covariates and given that it was selected for observation, is not the same as the conditional distribution of the response in the finite population, given only the covariates. It is instead a weighted version of the conditional distribution of interest. Inference must be modified to account for this informative selection. An established approach in this context is maximum “sample likelihood”, developing a weight function that reflects the informative sampling design, then treating the observations as if they were independently distributed according to the weighted distribution. While the sample likelihood methodology has been widely applied, its theoretical foundation has been less developed. A precise asymptotic description of a wide range of informative selection mechanisms is proposed. Under this framework, consistency and asymptotic normality of the maximum sample likelihood estimators are established. The theory allows for the possibility of nuisance parameters that describe the selection mechanism. The proposed regularity conditions are verifiable for various sample schemes, motivated by real problems in surveys. Simulation results for these examples illustrate the quality of the asymptotic approximations, and demonstrate a practical approach to variance estimation that combines aspects of model-based information theory and design-based variance estimation.