PrO-C*-代数的顺从性和核性

研究了Pro—C^*-代数的顺从性和核性．主要证明了(1)顺从Pro—C^*代数的闭理想是顺从的；(2)核Pro—C^*代数类对归纳极限封闭；(3)交换σ-C^*-代数和核C^*-代数都是核，σ-C^*-代数并且核σ-C^*-代数类对于商运算、张量积运算和可数逆向极限封闭．进一步得到核，σ-C^*-代数的扩张保持核性的条件。     

Stochastic realization of a Gaussian stochastic control system

The stochastic realization problem is considered of representing a stationary Gaussian process as the observation process of a Gaussian stochastic control system. The problem formulation includes that the lastm components of the observation process form the Gaussian white noise input process to the system. Identifiability of this class of systems motivates the problem. The results include a necessary and sufficient condition for the existence of a stochastic realization. A subclass of Gaussian stochastic control systems is defined that is almost a canonical form for this stochastic realization problem. For a structured Gaussian stochastic control system an equivalent condition for identifiability of the parametrization is stated.The research of this paper is supported in part by the Commission of the European Communities through the SCIENCE Program by the projectSystem Identification with contract number SC1-CT92-0779.     

Optimal transmission of Gaussian signals through a feedback channel

Using the methodology and results of the theory of filtering of conditionally Gaussian processes, the optimal schemes of transmission of Gaussian signals through the noisy feedback channel are constructed under the new power conditions.     

Fuzzy σ-algebras and fuzzy measurable functions

In this paper we first give an axiomatic definition of a fuzzy σ-algebra which is a generalisation of the family of fuzzy events considered by L.A. Zadeh [13]. The relationship between classical and fuzzy σ-algebras and between topologies and σ-algebras, in both cases, classical and fuzzy, is established in the notation of commutative diagrams. For fuzzy topologies we refer to the work of R. Lowen [8]. After a comparison between classical measurability of functions and fuzzy measurability defined in this paper we finally introduce the product of fuzzy σ-algebras.     

On killed processes and stopped filtrations

This note considers three σ-algebras that contain information about what happens to a stochastic process prior to a random time T. It is shown that the three σ-algebras coincide in a canonical setting but not in general.     

Permanental vectors

A permanental vector is a generalization of a vector with components that are squares of the components of a Gaussian vector, in the sense that the matrix that appears in the Laplace transform of the vector of Gaussian squares is not required to be either symmetric or positive definite. In addition, the power of the determinant in the Laplace transform of the vector of Gaussian squares, which is −1/2, is allowed to be any number less than zero.     

Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay

To the best of the authors’ knowledge, there are no results based on the so-called Razumikhin technique via a general decay stability, for any type of stochastic differential equations. In the present paper, the Razumikhin approach is applied to the study of both pth moment and almost sure stability on a general decay for stochastic functional differential equations with infinite delay. The obtained results are extended to stochastic differential equations with infinite delay and distributed infinite delay. Some comments on how the considered approach could be extended to stochastic functional differential equations with finite delay are also given. An example is presented to illustrate the usefulness of the theory.     

Construction of Fuzzy σ-algebras using triangular norms

Generalizing the definitions given by the author [Fuzzy Sets and Systems4 (1980), 83–93] we introduce and study T-fuzzy σ-algebras, T being any triangular norm. The main result is that for a large class of triangular norms each T-fuzzy σ-algebra is generated, i.e., consists of all functions μ:X → [0, 1] being measurable with respect to some σ-algebra on X.     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

On the Banach space isomorphism type of AF C*-algebras and their triangular subalgebras

It is shown that all the approximately finite dimensional C^*-algebras which are not of Type I are isomorphic as Banach spaces. This generalizes the matroid case given previously by Arazy. Analogous results are obtained for various families of triangular subalgebras of AF C^*-algebras. In addition the classification of various continua of Type I AF C^*-algebras is discussed.     

Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). The financial position is given by an RCLL adapted process. We first state some properties of RBSDEs with jumps when the obstacle process is RCLL only. We then prove that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of optimal stopping times is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, we investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/optimal stopping game problems. We prove that, under some hypothesis, the value function is equal to the solution of an RBSDE. We then study the existence of saddle points when the obstacle is left-upper semi-continuous along stopping times.     

De Finetti’s optimal dividends problem with an affine penalty function at ruin

In a Lévy insurance risk model, under the assumption that the tail of the Lévy measure is log-convex, we show that either a horizontal barrier strategy or the take-the-money-and-run strategy maximizes, among all admissible strategies, the dividend payments subject to an affine penalty function at ruin. As a key step for the proof, we prove that, under the aforementioned condition on the jump measure, the scale function of the spectrally negative Lévy process has a log-convex derivative.     

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.     

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.Partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Generalized joint spectral radius and stability of switching systems

This paper extends the notion of generalized joint spectral radius with exponents, originally defined for a finite set of matrices, to probability distributions. We show that, under a certain invariance condition, the radius is calculated as the spectral radius of a matrix that can be easily computed, extending the classical counterpart. Using this result we investigate the mean stability of switching systems. In particular we establish the equivalence of mean square stability, simultaneous contractibility in square mean, and the existence of a quadratic Lyapunov function. Also the stabilization of positive switching systems is studied. Numerical examples are given to illustrate the results.     

Optimal stopping for partially observed piecewise-deterministic Markov processes

This paper deals with the optimal stopping problem under partial observation for piecewise-deterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an ?$?$-optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence.     

Some notes on extremal discriminant analysis

Classical discriminant analysis focusses on Gaussian and nonparametric models where in the second case the unknown densities are replaced by kernel densities based on the training sample. In the present article we assume that it suffices to base the classification on exceedances above higher thresholds, which can be interpreted as observations in a conditional framework. Therefore, the statistical modeling of truncated distributions is merely required. In this context, a nonparametric modeling is not adequate because the kernel method is inaccurate in the upper tail region. Yet one may deal with truncated parametric distributions like the Gaussian ones. Our primary aim is to replace truncated Gaussian distributions by appropriate generalized Pareto distributions and to explore properties and the relationship of discriminant functions in both models.     

Correspondence between fuzzy measures and classical measures

In this paper we study the question whether, given a fuzzy measure (as defined in [3] and [4]). there exists a classical measure such that the fuzzy measure of a measurable fuzzy set μ equals the classical measure of the area below the membership function of μ. The results are that in the case of finite additivity there is a one-to-one correspondence between classical measures and fuzzy measures, whereas in the case of countable additivity this result only holds for generated fuzzy σ-algebras. Finally, some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy σ-algebra σ to the generated fuzzy σ-algebra $\text{σ}$ are discussed.     

Fuzzy probability spaces defined by means of the fuzzy relation ‘less than’

In this paper a fuzzy relation ‘less than’ and fuzzy probability spaces on the real line are defined. Fuzzy σ-algebras of events are introduced in an analogous way to the classical theory of probability by means of the above relation. Furthermore, specific properties of the fuzzy probability measures presented here are given.     

The probabilistic structure of controlled diffusion processes

This paper surveys those aspects of controlled diffusion processes wherein the control problem is treated as an optimization problem on a set of probability measures on the path space. This includes: (i) existence results for optimal admissible or Markov controls (both in nondegenerate and degenerate cases), (ii) a probabilistic treatment of the dynamic programming principle, (iii) the corresponding results for control under partial observations, (iv) a probabilistic approach to the ergodic control problem. The paper is expository in nature and aims at giving a unified treatment of several old and new results that evolve around certain central ideas.