An inequality for the multivariate normal distribution

Herman Chernoff used Hermite polynomials to prove an inequality for the normal distribution. This inequality is useful in solving a variation of the classical isoperimetric problem which, in turn, is relevant to data compression in the theory of element identification. As the inequality is of interest in itself, we prove a multivariate generalization of it using a different argument.     

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.     

Superconvergence to the central limit and failure of the Cramér theorem for free random variables

Summary We show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected. Thus, the distributions which tend to the semicircle become absolutely continuous in finite time, and the densities converge in a very strong sense. We also show that the semicircle law is the free convolution of laws which are not semicircular, thus proving that Cramér's classical result for the normal distribution does not have a free counterpart. The authors were partially supported by grants from the National Science Foundation     

Cramér large deviation expansions for martingales under Bernstein’s condition

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.     

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein–Uhlenbeck process associated with Bertoin’s tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Lévy processes.     

Infinite-dimensional Dirichlet operators I: Essential selfadjointness and associated elliptic equations

We give sufficient conditions for essential selfadjointness of operators associated with classical Dirichlet forms on Hilbert spaces and of potential perturbations of Dirichlet operators. We also study the smoothness of generalized solutions of elliptic equations corresponding to the Dirichlet operators.Supported by the Alexander von Humboldt Foundation.     

Statistical theory of shape under elliptical models and singular value decompositions

The size-and-shape and shape distributions based on non-central and non-isotropic elliptical distributions are derived in this paper by using the singular value decomposition (SVD). The general densities require the computation of new integrals involving zonal polynomials. The invariance of the central shape distribution is also proved. Finally, some particular densities are applied in a classical data of Biology, and the inference based on exact distributions is performed after choosing the best model by using a modified BIC^∗ criterion.     

Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which is determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with a finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration.     

Multivariate quadratic forms of random vectors

We obtain the distribution of the sum of n random vectors and the distribution of their quadratic forms: their densities are expanded in series of Hermite and Laguerre polynomials. We do not suppose that these vectors are independent. In particular, we apply these results to multivariate quadratic forms of Gaussian vectors. We obtain also their densities expanded in Mac Laurin series or in the form of an integral. By this last result, we introduce a new method of computation which can be much simpler than the previously known techniques. In particular, we introduce a new method in the very classical univariate case. We remark that we do not assume the independence of normal variables.     

Bivariate distributions characterized by one family of conditionals and conditional percentile or mode functions

It is well known that full knowledge of all conditional distributions will typically serve to completely characterize a bivariate distribution. Partial knowledge will often suffice. For example, knowledge of the conditional distribution of X given Y and the conditional mean of Y given X is often adequate to determine the joint distribution of X and Y. In this paper, we investigate the extent to which a conditional percentile function or a conditional mode function (of Y given X), together with knowledge of the conditional distribution of X given Y will determine the joint distribution. Finally, using this methodology a new characterization of the classical bivariate normal distribution is given.     

Remarks on t-transformations of measures and convolutions

A family of transformations of probability measures is constructed, and used to define transformations of convolutions. The relations between moments and cumulants of a measure and its transformation are presented. For transformed classical and free convolutions the central limit measures and the Poisson type limit measures are computed. Families of non-commutative random variables are constructed, which are associated to these central limit measures. They provide examples of “position operators” which act on the Interacting Fock Spaces.     

Infinite products of relations, set-valued series and uniform openness of multifunctions

Using the notations of convergent series of sets and convergent products of relations, general open mapping theorems are presented which encompass classical results of Banach, Ptak, Khanh, and others.     

Filtering on a Partially Observed Ultra-High-Frequency Data Model

A model for intraday stock price movements is considered. The jump-intensity of the logreturn process is a function of the whole history of a hidden marked point process. The aim is to find the conditional law of such intensity given the history of the logreturn process. Under a Markovianity assumption, related with the weak form of market efficiency, classical filtering techniques are used. The law of the jump-intensity, given the history of the logreturn price, is evaluated and a discussion on a particular case is performed.     

A new kind of modified transportation cost inequalities and polynomial concentration inequalities

This paper introduces a new kind of modified transportation cost inequalities and presents some sufficient and necessary conditions for it. Using these conditions, this paper provides some applications of polynomial concentration inequalities to probability measures with no exponential moments. The tensorization property is also provided.     

Bounds for functions of multivariate risks

Li et al. [Distributions with Fixed Marginals and Related Topics, vol. 28, Institute of Mathematics and Statistics, Hayward, CA, 1996, pp. 198-212] provide bounds on the distribution and on the tail for functions of dependent random vectors having fixed multivariate marginals. In this paper, we correct a result stated in the above article and we give improved bounds in the case of the sum of identically distributed random vectors. Moreover, we provide the dependence structures meeting the bounds when the fixed marginals are uniformly distributed on the k-dimensional hypercube. Finally, a definition of a multivariate risk measure is given along with actuarial/financial applications.     

Beyond the Poisson renewal process: A tutorial survey

After sketching the basic principles of renewal theory and recalling the classical Poisson process, we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of Mittag–Leffler and of Wright type. We compare these three processes with each other.     

Analogues of Jacobi's inversion formula for the incomplete elliptic integral of the first kind

In this article, we revisit Ramanujan's cubic analogue of Jacobi's inversion formula for the classical elliptic integral of the first kind. Our work is motivated by the recent work of Milne (Ramanujan J. 6(1) (2002) 7-149), Chan and Chua (Ramanujan J., to appear) on the representations of integers as sums of even squares.     

Long time asymptotics for constrained diffusions in polyhedral domains

We study long time asymptotic properties of constrained diffusions that arise in the heavy traffic analysis of multiclass queueing networks. We first consider the classical diffusion model with constant coefficients, namely a semimartingale reflecting Brownian motion (SRBM) in a d$d$-dimensional positive orthant. Under a natural stability condition on a related deterministic dynamical system [P. Dupuis, R.J. Williams, Lyapunov functions for semimartingale reflecting brownian motions, Annals of Probability 22 (2) (1994) 680–702] showed that an SRBM is ergodic. We strengthen this result by establishing geometric ergodicity for the process. As consequences of geometric ergodicity we obtain finiteness of the moment generating function of the invariant measure in a neighborhood of zero, uniform time estimates for polynomial moments of all orders, and functional central limit results. Similar long time properties are obtained for a broad family of constrained diffusion models with state dependent coefficients under a natural condition on the drift vector field. Such models arise from heavy traffic analysis of queueing networks with state dependent arrival and service rates.     

Functions operating on multivariate distribution and survival functions—With applications to classical mean-values and to copulas

Functions operating on multivariate distribution and survival functions are characterized, based on a theorem of Morillas, for which a new proof is presented. These results are applied to determine those classical mean values on [0,1]ⁿ which are distribution functions of probability measures on [0,1]ⁿ. As it turns out, the arithmetic mean plays a universal rôle for the characterization of distribution as well as survival functions. Another consequence is a far reaching generalization of Kimberling’s theorem, tightly connected to Archimedean copulas.