Random matrix models of stochastic integral type for free infinitely divisible distributions

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ?? ₀ ^?? e ^?1 d?? _t ^d , d ?? 1, where ?? _t ^d is a d × d matrix-valued Lévy process satisfying an I _log condition.     

Diffusions conditionnelles. II. Générateur conditionnel. Application au filtrage

This paper is a continuation of “Diffusions conditionelles, I.” If (x_t, z_t) is a two-component diffusion process, it is shown that under appropriate conditions, the process x_t (t ? T), given (z_s, s ? T) is a nonhomogeneous strong Markov process, whose generator is explicitly found by using the theory of stochastic flows. The filtering equation is reduced to an ordinary partial differential equation.     

Convergence in probability and almost sure with applications

A necessary and sufficient condition is given for the convergence in probability of a stochastic process {X_t}. Moreover, as a byproduct, an almost sure convergent stochastic process {Y_t} with the same limit as {X_t} is identified. In a number of cases {Y_t} reduces to {X_t} thereby proving a.s. convergence. In other cases it leads to a different sequence but, under further assumptions, it may be shown that {X_t} and {Y_t} are a.s. equivalent, implying that {X_t} is a.s. convergent. The method applies to a number of old and new cases of branching processes providing an unified approach. New results are derived for supercritical branching random walks and multitype branching processes in varying environment.     

Crossing points of distributions and a theorem that relates them to second order stochastic dominance

We state formal definitions for crossing points in pairs of distributions and give a detailed proof of a theorem that relates those points to the second order stochastic dominance (SSD). The theorem states that the fulfillment of the area balance conditions for SSD at the t values that correspond to crossing points, and at the limit t→∞, is a necessary and sufficient condition for its fulfillment at all t: {−∞<t<∞}, as required for the existence of SSD. We provide examples for the application of the theorem in the case of continuous distributions, including a continuous counter example to prove that the Mean-Variance criterion is not sufficient to state preferences under risk aversion.     

Stationary-excess operator and convex stochastic orders

The present paper aims to point out how the stationary-excess operator and its iterates transform s-convex stochastic orders and the associated moment spaces. This allows us to propose a new unified method on constructing s-convex extrema for distributions that are known to be t-monotone. Both discrete and continuous cases are investigated. Several extremal distributions under monotonicity conditions are derived. They are illustrated with some applications in insurance.     

Multivariate operator-self-similar random fields

Multivariate random fields whose distributions are invariant under operator-scalings in both the time domain and the state space are studied. Such random fields are called operator-self-similar random fields and their scaling operators are characterized. Two classes of operator-self-similar stable random fields X={X(t),t∈R^d} with values in R^m are constructed by utilizing homogeneous functions and stochastic integral representations.     

A trivariate version of ‘Lévy's equivalence’

It is shown that the trivariate stochastic processes {(M_t−W_t, M_t, Θ_t), t ≥ 0} and {(|W_t|, L_t, T_t), t ≥ 0} have the same distributions when: W = {W_t, t ≥ 0} is a Wiener process, M_t is the maximum value attained by W over the time interval [0, t], Θ_t is the time the maximum value is attained, L_t is the local time of W at level zero and time t, and T_t is the last time W is zero in the time interval [0, t]. A straightforward proof, based on ‘Tanaka's formula, establishes this result by deriving an almost sure version of the equivalence.     

Open-loop control of stochastic fluid systems and applications

We consider a stochastic process Q(t) satisfying a linear differential equation, driven by a jump process which is modulated by a binary sequence. We prove multimodular properties related to the process Q(t) and apply those results to optimal strategies in storage systems models and in ruin problems.     

Stochastic mechanics of systems with zero potential

Nelson's stochastic mechanics is studied for a system with zero potential. It is shown that, with probability one, the sample paths of this process behave asymptotically like paths in classical mechanics in the limit t → + ∞.     

SDDP for some interstage dependent risk-averse problems and application to hydro-thermal planning

We consider interstage dependent stochastic linear programs where both the random right-hand side and the model of the underlying stochastic process have a special structure. Namely, for equality constraints (resp. inequality constraints) the right-hand side is an affine function (resp. a given function b _t ) of the process value for the current time step t. As for m-th component of the process at time step t, it depends on previous values of the process through a function h _tm . For this type of problem, to obtain an approximate policy under some assumptions for functions b _t and h _tm , we detail a stochastic dual dynamic programming algorithm. Our analysis includes some enhancements of this algorithm such as the definition of a state vector of minimal size, the computation of feasibility cuts without the assumption of relatively complete recourse, as well as efficient formulas for sharing optimality and feasibility cuts between nodes of the same stage. The algorithm is given for both a non-risk-averse and a risk-averse model. We finally provide preliminary results comparing the performances of the recourse functions corresponding to these two models for a real-life application.     

The influence of the nonrecent past in prediction for stochastic processes

Consider the stochastic processes X₁, X₂,… and Λ₁, Λ₂,… where the X process can be thought of as observations on the Λ process. We investigate the asymptotic behavior of the conditional distributions of X_t+v given X₁,…, X_t and Λ_t+v given X₁,…, X_t with regard to their dependency on the “early” part of the X process. These distributions arise in various time series and sequential decision theory problems. The results support the intuitively reasonable and often used (as a basic tenet of model building) assumption that only the more recent past is needed for near optimal prediction.     

On α-symmetric multivariate distributions

A random vector is said to have a 1-symmetric distribution if its characteristic function is of the form φ(|t₁| + … + |t_n|). 1-Symmetric distributions are characterized through representations of the admissible functions φ and through stochastic representations of the radom vectors, and some of their properties are studied.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Correlation between viscosity and positronium life-times for simple molecular liquids

The life-times of positrons inn-octane andn-hexadecane are measured at different temperatures. The viscosities and densities are shown to be correlated with the life-times on the basis of the free volume model. Similar data forn-alkanes and chemical isomers are interpreted in terms of this model.     

PARAMETER ESTIMATION FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SMALL STABLE NOISES FROM DISCRETE OBSERVATIONS

We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small α-stable noises, observed at n regularly spaced time points ti=i/n, i=1,···,n on [0, 1]. Under some regularity conditions, we obtain the consistency and the rate of convergence of the least squares estimator (LSE) when a small dispersion parameter ε→0 and n→∞ simultaneously. The asymptotic distribution of the LSE in our setting is shown to be stable, which is completely different from the classical cases where asymptotic distributions are normal.     

The matrix-t distribution and its applications in predictive inference

The predictive distributions of the future responses and regression matrix under the multivariate elliptically contoured distributions are derived using structural approach. The predictive distributions are obtained as matrix-t which are identical to those obtained under matrix normal and matrix-t distributions. This gives inference robustness with respect to departures from the reference case of independent sampling from the matrix normal or dependent but uncorrelated sampling from matrix-t distributions. Some successful applications of matrix-t distribution in the field of spatial prediction have been addressed.     

Characteristic functions of scale mixtures of multivariate skew-normal distributions

We obtain the characteristic function of scale mixtures of skew-normal distributions both in the univariate and multivariate cases. The derivation uses the simple stochastic relationship between skew-normal distributions and scale mixtures of skew-normal distributions. In particular, we describe the characteristic function of skew-normal, skew-t, and other related distributions.     

Stochastic clearing systems

A stochastic clearing system is characterized by a non-decreasing stochastic input process $\text{{Y(t), t ≧ 0}}$, where Y(t) is the cumulative quantity entering the system in [0, t], and an output mechanism that intermittently and instantaneously clears the system, that is, removes all the quantity currently present. Examples may be found in the theory of queues, inventories, and other stochastic service and storage systems. In this paper we derive an explicit expression for the stationary (in some cases, limiting) distribution of the quantity in the system, under the assumption that the clearing instants are regeneration points and, in particular, first entrance times into sets of the form {y: y>q}. The expression is in terms of the sojourn measure W associated with $\text{{Y(t), t ≧ 0}: W{A} =}\text{E}${time spent in A by Y(t), 0 ≤ t < ∞}. The results are applied to compound input processes and processes with stationary independent increments. In particular, we show that, contrary to a wide-spread belief, the uniform stationary distribution characteristic of deterministic models does not usually carry over to genuinely stochastic models.     

Generalized solutions of a stochastic partial differential equation

We discuss the Cauchy problem of a certain stochastic parabolic partial differential equation arising in the nonlinear filtering theory, where the initial data and the nonhomogeneous noise term of the equation are given by Schwartz distributions. The generalized (distributional) solution is represented by a partial (conditional) generalized expectation ofT(t)° _0,t ^–1 , whereT(t) is a stochastic process with values in distributions and _s,t is a stochastic flow generated by a certain stochastic differential equation. The representation is used for getting estimates of the solution with respect to Sobolev norms.Further, by applying the partial Malliavin calculus of Kusuoka-Stroock, we show that any generalized solution is aC -function under a condition similar to Hörmander's hypoellipticity condition.     

A stochastic-dynamic approach to pension funding

A stochastic-dynamic pension fund model is introduced via a stochastic differential equation in the variable, X_t, representing the fund ratio. The process X_t is analyzed by Lyapunov type methods and also the first and second moments of the process are computed. The advantages of these two methods of analysis are discussed.