We show that in the absence of any information about the ‘hidden’ target in terms of the observed sample path, and irrespectively of the distribution law of the observed process, the ‘median’ rule is optimal in both the space domain and the time domain. While the fact that the median...

We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean–Vlasov evolution equation. Moreover, we show...

Parametric estimation of two-dimensional hypoelliptic diffusions is considered when complete observations–both coordinates discretely observed–or partial observations–only one coordinate observed–are available. Since the volatility matrix is degenerate, Euler contrast estimators cannot...

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance for particle approximations of time-homogeneous Feynman–Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control...

We show an averaging result for a system of stochastic evolution equations of parabolic type with slow and fast time scales. We derive explicit bounds for the approximation error with respect to the small parameter defining the fast time scale. We prove that the slow component of the solution of...

We prove an extension of the Ocone–Karatzas integral representation, valid for all BV functions on the classical Wiener space. We also establish an elementary chain rule formula and combine the two results to compute explicit integral representations for some classes of BV composite random...

Let U be an open set in Rd. We show that under a mild assumption on the richness of the generator, a Feller process in (U,B(U)) with (predictable) killing is a semimartingale. To this end, we generalize the notion of semimartingales in a natural way to those ‘with killing’. Furthermore we...

Particle filters are numerical methods for approximating the solution of the filtering problem which use systems of weighted particles that (typically) evolve according to the law of the signal process. These methods involve a corrective/resampling procedure which eliminates the particles that...

We prove some heavy-traffic limit theorems for processes which encompass the fractionally integrated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a non-Gaussian stable distribution.

We consider filtration consistent nonlinear expectations in probability spaces satisfying only the usual conditions and separability. Under a domination assumption, we demonstrate that these nonlinear expectations can be expressed as the solutions to Backward Stochastic Differential Equations...