Mirror and synchronous couplings of geometric Brownian motions

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.     

Measurability of semimartingale characteristics with respect to the probability law

Given a càdlàg process X$X$ on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let _Psem${\mathfrak{P}}_{sem}$ be the set of all probability measures P$P$ under which X$X$ is a semimartingale. We construct processes (B^P,C,ν^P)$(B^P,C,{\nu }^P)$ which are jointly measurable in time, space, and the probability law P$P$, and are versions of the semimartingale characteristics of X$X$ under P$P$ for each P∈_Psem$P\in {\mathfrak{P}}_{sem}$. This result gives a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.     

A variational approach to second-order approximability of sliding mode control systems

We consider robustness properties of second-order methods for the sliding mode control of nonlinear ordinary differential equations. A new approach is presented based on the theory of well-posed optimization problems. It is shown that the convergence of the real states of the control system to the ideal one is intimately related to Tykhonov well-posedness of suitably defined dynamic optimization problems.     

ON THE ADJOINT SITUATIONS BETWEEN BEHAVIOUR AND REALIZATION

ABSTRACT

Following the Arbib-Manes approach to machines in categories, the categories Mach(C) of machines and TBeh(C) of total behaviours are constructed, where c is a class of state-behaviour processes. It is shown that the total external behaviour functor E: Mach(C) → TBeh(C) has a left adjoint and that the free realization of any total behaviour is reachable. Furthermore, the restriction of E to the full subcategory of Mach(C) with all reachable machines as objects has as right adjoint the minimal realization functor.     

A NOTE ON SUBCATEGORIES OF REACHABLE MACHINES

Abstract

Given a category K. a state-behaviour process X: K → K and an image factorization system (E,M) for K, it is shown that under mild conditions the full subcategory with objects all reachable machines is coreflective. A corresponding result is obtained for coequalizer-reachable machines if X is only an input process.     

A central limit theorem in nonlinear filtering

It is shown that the probability law of a diffusion process conditioned on weakly corrupted observations is asymptotically Gaussian when properly scaled. The method of proof involves Fisher information matrices and a Cramér-Rao inequality.     

Errata: On semidefinite linear complementarity problems

We correct an error in the statement of Theorem 8 in [1]. Received: January 3, 2001 / Accepted: February 26, 2001?Published online May 18, 2001     

The Kalman-Yakubovich-Popov theorem for stabilizable hyperbolic boundary control systems

In this paper we present a version of the Kalman-Yakubovich-Popov theorem for a class of boundary control systems of hyperbolic type. Unstable, controllable systems are considered and stabilizability withunbounded feedbacks is permitted.Paper partially supported by the Italian MINISTERO DELLA RICERCA SCIENTIFICA E TECNOLOGICA within the program of GNAFA-CNR and by NATO CRG program SA.5-2-05 (CRG940161).