A classification of regressive transformation semigroups on chains

For each subchain X?? of a chain X, let T _RE (X,X??) denote the semigroup under composition of all full regressive transformations, ??:X??X?? satisfying x????x for all x??X. Necessary and sufficient conditions for T _RE (X,X??) and T _RE (Y,Y??) to be isomorphic are given. This isomorphism theorem is applied to classify the semigroup of regressive transformations T _RE (X,X??) where X is one of several familiar subchains of ?, the chain of real numbers.     

Asymptotic behavior of the convex hull of a stationary Gaussian process?

Let X = {X(t), t ?? T} be a stationary centered Gaussian process with values in ? ^d , where the parameter set T equals ? or ?+. Let ?? _t = Cov(X ₀ ,X _t ) be the covariance function of X, and (??,?, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W _t = conv{X _u , u ?? T ?? [0, t]} as t ?? +?? and show that under the condition ??t ?? 0, t????, the rescaled convex hull (2 ln t) ^?1/2 W _t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ? associated to the covariance matrix ?? ₀. The asymptotic behavior of the mathematical expectations E f(W _t ), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171?C179, 2011].     

A Combinatorial Formula for the Schur Coefficients of the Integral Form of the Macdonald Polynomials in the Two Column and Certain Hook Cases

In this paper, we prove that the integral form of Macdonald polynomials J _?? [X; q, t] has the property that J _?? [X; q, t]/(1?q) ⁿ has Schur expansion with positive polynomial coefficient. Our proof proceeds by constructing constructing combinatorial formula for the Schur coefficients when??? is either a two column shape or a certain type of hook shape.     

Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (X_t)_t∈R+ in terms of the convergence of sequence E(X_Tn) for monotone sequences (T_n) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (X_n)_1≤n≤∞ of integrable random variables, lim X_n exists and is equal to X_∞ and (X_T) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(X_Tn) = E(X_∞) for every increasing sequence (T_n) of extended simple stopping times converging to ∞. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (X_t)_t∈R+ with E{X_T} < ∞ for every bounded stopping time T is right continuous if lim E(X_Tn) = E(X_T) for every bounded stopping time T and every descending sequence (T_n) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.     

Non-parametric estimation of the diffusion coefficient from noisy data

We consider a diffusion process (X _t ) _t????0, with drift b(x) and diffusion coefficient ??(x). At discrete times t _k ?=?k ?? for k from 1 to M, we observe noisy data of the sample path, ${Y_{k\delta}=X_{k\delta}+\varepsilon_{k}}$ . The random variables ${\left(\varepsilon_{k}\right)}$ are i.i.d, centred and independent of (X _t ). The process (X _t ) _t????0 is assumed to be strictly stationary, ??-mixing and ergodic. In order to reduce the noise effect, we split data into groups of equal size p and build empirical means. The group size p is chosen such that ???=?p ?? is small whereas M ?? is large. Then, the diffusion coefficient ?? ² is estimated in a compact set A in a non-parametric way by a penalized least squares approach and the risk of the resulting adaptive estimator is bounded. We provide several examples of diffusions satisfying our assumptions and we carry out various simulations. Our simulation results illustrate the theoretical properties of our estimators.     

The optional stopping theorem for quantum martingales

In classical probability theory, a random time T is a stopping time in a filtration (Ft)_t?0 if and only if the optional sampling holds at T for all bounded martingales. Furthermore, if a process (Xt)_t?0 is progressively measurable with respect to (Ft)_t?0, then XT is FT-measurable. Unfortunately, this is not the case in noncommutative probability with the definition of stopped process used until now. It is shown in this article that we can define the stopping of noncommutative processes in Fock space in such a way that all the bounded martingales can be stopped at any stopping time T, are adapted to the filtration of the past before T and satisfy the optional stopping theorem.     

Accuracy of discrete schemes for a class of abstract evolution inequalities

For a triple of Hilbert spaces {V, H, V*}, we study a discrete and a semidiscrete scheme for an evolution inclusion of the form u′(t) + A(t)u(t) + ??(t, u(t)) ? f(t), u(0) = u ₀, t ∈ (0, T], where the pair {A(t), ?(t, ·)} consists of a family of nonlinear operators from V into V* and a family of proper convex lower semicontinuous functionals with common effective domain D(?) ? V. The discrete scheme is a combination of the Galerkin method with perturbations and the implicit Euler method. Under conditions on the data providing the existence and uniqueness of the solution of the problem in the space H ¹(0, T; V) ∩ W _∞ ¹ (0, T;H), we obtain an abstract estimate for the method error in the energy norm of first-order accuracy with respect to the time increment. By way of application, we consider a problem with an obstacle inside the domain, for which we obtain an optimal estimate of the accuracy of two implicit schemes (standard and new) on the basis of the finite element method.     

Estimation for the change point of volatility in a stochastic differential equation

We consider a multidimensional Itô process Y=(Y_t)_t∈[0,T] with some unknown drift coefficient process b_t and volatility coefficient σ(X_t,θ) with covariate process X=(X_t)_t∈[0,T], the function σ(x,θ) being known up to θ∈Θ. For this model, we consider a change point problem for the parameter θ in the volatility component. The change is supposed to occur at some point t^∗∈(0,T). Given discrete time observations from the process (X,Y), we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit theorems of the asymptotically mixed type.     

The Conjugate Class of a Supercyclic Operator

In this paper, we show that the conjugate set of any supercyclic operator T on a separable, infinite dimensional Banach space X contains a path of supercyclic operators which is dense with the strong operator topology, and the set of common supercyclic vectors for the path is a dense G _?? set if ?? _p (T*) is empty.     

Credit Risk in an Economy with New Firms Arrivals

We propose a dynamic model to analyze the credit quality of firms. In the market in which they operate, the firms are divided into a finite number of classes representing their credit status. The cardinality of the population can increase, since new firms can enter the market and the partition is supposed to change over time, due to defaults and changes in credit quality, following a class of Markov processes. Some conditional probabilities related to default times are investigated and the role of occupation numbers is highlighted in this context. In a partial information setting at discrete time, we present a particle filtering technique to numerically compute by simulation the conditional distribution of the number of firms in the credit classes, given the information up to time t.     

Estimation of the Gauss–Markov process from observation of its sign

Let X(t) be the ergodic Gauss–Markov process with mean zero and covariance function e^?|τ|. Let D(t) be +1, 0 or ?1 according as X(t) is positive, zero or negative. We determine the non-linear estimator of X(t₁) based solely on D(t), ?T ? t ? 0, that has minimal mean–squared error ε²(t₁, T). We present formulae for ε²(t₁, T) and compare it numerically for a range of values of t₁ and T with the best linear estimator of X(t₁) based on the same data.     

A poisson convergence theorem for a particle system with dependent constant velocities

Consider an infinite collection of particles travelling in d-dimensional Euclidean space and let X_n denote the initial position of the n^th particle. Assume that the n^th particle has through all time the random velocity V_n and that {V_n} is a sequence of dependent random variables. Let X_n(t) = X_n + V_nt denote the position of the n^th particle at time t. Conditions are obtained for the convergence of {X_n(t)} to a Poisson process as t→∞. Essentially they require that the dependence in the V_n-sequence decrease with increasing distance between the initial positions and that the conditional distribution of V_n given the initial positions of all the particles and V_nk≠n be absolutely continuous with respect to Lebesgue measure.     

Weak convergence to multifractional Brownian motion of Riemann-Liouville type in Besov spaces

We study the weak convergence of the family of processes {V _n (t)} _n??? defined by $$V_n(t)=\int_{0}^t(t-u)^{H(t)-\frac{1}{2}}\theta_n(u)du,$$ where {?? _n (u)} _n??? is a family of processes converging in law to a Brownian motion, as n????. We consider two cases of {?? _n }. First, we construct ?? _n based on the well-known Donsker??s theorem and show that {V _n (t)} _n??? converges in law to a multifractional Brownian motion of Riemann-Liouville type, as n????. Second, we construct ?? _n based on a Poisson process, and then show that a multifractional Brownian motion of Riemann-Liouville type can be approximated in law by {V _n (t)} _n???.     

On the structure of a semigroup of operators with finite-dimensional ranges

In the present paper, we describe the structure of a strongly continuous operator semigroup T(t) (where T: ?₊ → End X and X is a complex Banach space) for which ImT(t) is a finite-dimensional space for all t > 0. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a C ₀-semigroup.     

Minmax subtree cover problem on cacti

Let G=(V,E) be a connected graph such that edges and vertices are weighted by nonnegative reals. Let p be a positive integer. The minmax subtree cover problem (MSC) asks to find a pair (X,T) of a partition X={X₁,X₂,…,X_p} of V and a set T of p subtrees T₁,T₂,…,T_p, each T_i containing X_i so as to minimize the maximum cost of the subtrees, where the cost of T_i is defined to be the sum of the weights of edges in T_i and the weights of vertices in X_i. In this paper, we propose an O(p²n) time (4-4/(p+1))-approximation algorithm for the MSC when G is a cactus.     

On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

We present an explicit solution to the Skorokhod embedding problem for spectrally negative Lévy processes. Given a process X and a target measure μ satisfying an explicit admissibility condition we define functions φ _± such that the stopping time T=inf?{t>0:X _t ∈{?φ _?(L _t ),φ ₊(L _t )}} induces X _T ～μ, where (L _t ) is the local time in zero of X. We also treat versions of T which take into account the sign of the excursion straddling time t. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]). In particular, we compute explicitly the quantities introduced in Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]) in our setup. Our method relies on some new explicit calculations relating scale functions and the Itô excursion measure of X. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.     

The operator-valued Feynman-Kac formula with noncommutative operators

Let X(t) be a right-continuous Markov process with state space E whose expectation semigroup S(t), given by S(t) φ(x) = E_x[φ(X(t))] for functions φ mapping E into a Banach space L, has the infinitesimal generator A. For each x?E, let V(x) generate a strongly continuous semigroup T_x(t) on L. An operator-valued Feynman-Kac formula is developed and solutions of the initial value problem $\text{?u}\text{?t}\text{= Au + V(x)u, u(0) = φ}$ are obtained. Fewer conditions are assumed than in known results; in particular, the semigroups {T_x(t)} need not commute, nor must they be contractions. Evolution equation theory is used to develop a multiplicative operative functional and the corresponding expectation semigroup has the infinitesimal generator A + V(x) on a restriction of the domain of A.     

The spaces Cp(X): decomposition into a countable union of bounded subspaces and completeness properties

A Tychonoff space X has to be finite if C_p(X) is σ-countably compact [23]. However, this is not true if only σ-pseudocompactness of C_p(X) is assumed. It is proved that C_p(X) is σ-pseudocompact iff X is pseudocompact and b-discrete. The technique developed yields an example showing that the theorem of Grothendieck [7] cannot be extended over the class of pseudocompact spaces. Some generalizations of the results of Lutzer and McCoy [9] are obtained. We establish also that ∏{C_p(X_t):tϵT} is a Baire space in case C_p(X_t) is Baire for each t ∈ T.     

Complete semigroups of binary relations defined by finite XI-semilattices of unions

We give conditions under which the X-semilattice of unions is an XI-semilattice, i.e., let D be a finite X-semilattice of unions. The complete semigroup of binary relation B _X (D) is defined by the XI-semilattice of unions if and only if V (D, ??) = D for some ?? ?? B _X (D).     

Optimal Consumption in a Brownian Model with Absorption and Finite Time Horizon

We construct ?-optimal strategies for the following control problem: Maximize $\mathbb {E}[ \int_{[0,\tau)}e^{-\beta s}\,dC_{s}+e^{-\beta\tau}X_{\tau}]$ , where X _t =x+μt+σW _t ?C _t , τ≡inf{t>0|X _t =0}∧T, T>0 is a fixed finite time horizon, W _t is standard Brownian motion, μ, σ are constants, and C _t describes accumulated consumption until time t. It is shown that ?-optimal strategies are given by barrier strategies with time-dependent barriers.