On the existence of smooth densities for jump processes

Summary We consider a Lévy processX _t and the solutionY _t of a stochastic differential equation driven byX _t; we suppose thatX _t has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the existence of a smooth density forY _t: these conditions are similar to those of the classical Malliavin calculus for continuous diffusions. More generally, we study the smoothness of the law of variablesF defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function ofF.     

Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims

We will study the following problem.Let X_t,t∈[0,T],be an R~d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before t.Thus at time T,the random value of Y(ω) will become known to this agent.The question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε_t[Y] to define the value of Y given by this agent at time t.Thisoperator ε_t[·] assigns an (X_s)0(?)s(?)T-dependent random variable Y to an (X_s)0(?)s(?)t-dependent random variableε_t[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σcontaining an unknown parameter θ=θ_t.We then consider theso called super evaluation when the agent is a seller of the asset Y.We will prove that such super evaluation is afiltration consistent nonlinear expectation.In some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a g-evaluation.We also consider the correspondingnonlinear Markovian situation.     

Robust Approximation in a Filtering Problem with Real State Space and Counting Observations

Let (X _t ,Y _t ) be a pure jump Markov process, where X _t takes values in \bf R and Y _t is a counting process. We compare the filter of this system and a filter of a suitably modified system. We compute an explicit bound for the distance in the so-called bounded Lipschitz metric between the two filters. Finally we show how to use this bound to construct a discrete space approximation of the filter. Accepted 7 December 1999     

On the transition from a Markov chain to a continuous time process

Starting from a real-valued Markov chain X₀,X₁,…,X_n with stationary transition probabilities, a random element {Y(t);t[0, 1]} of the function space D[0, 1] is constructed by letting Y(k/n)=X_k, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t[0,1]};_n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes.     

Time Difference on the Brownian Curve

Let X _t and Y _t be respectively the locations of the maximum and minimum, over [0, t], of a real-valued Wiener process. We establish limsup and liminf iterated logarithm laws for , the time difference between the maximum and the minimum, as well as for max(X _t, Y _t) and min(X _t, Y _t).     

Shy couplings

A pair (X, Y) of Markov processes on a metric space is called a Markov coupling if X and Y have the same transition probabilities and (X, Y) is a Markov process. We say that a coupling is “shy” if inf _{t ≥ 0} dist(X _t , Y _t ) >  0 with positive probability. We investigate whether shy couplings exist for several classes of Markov processes.     

Controlled partially observed diffusions with correlated noise

This work is concerned with separated control problems for optimal stochastic controls under partial observations. Continuity properties of the unnormalized conditional distribution measure are found, and the Nisio nonlinear semigroup is formed in the case when a functionh(X _t ,Y _t ,U _t ) of stateX _t observationY _t , and controlU _t plus correlated additive white noise is observed.     

Oscillatory behavior of linear matrix Hamiltonian systems

We establish some new oscillation criteria for the matrix linear Hamiltonian system X ′ = A (t)X + B (t)Y, Y ′ = C (t)X –A *(t)Y by using a new function class X and monotone functionals on a suitable matrix space. In doing so, many existing results are generalized and improved. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularity of the density for the total weighted occupation measure of super-Brownain motion

Let (X _t ) be a super-Brownian motion in a bounded domain D in ℝ ^d . The random measure Y ^D (·) = ∫₀^∞ X _t (·)dt is called the total weighted occupation time of (X _t ). We consider the regularity properties for the densities of a class of Y ^D . When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y ^D (dx)-measure.     

Asymptotic expansions in the central limit theorem for compound and Markov processes

Summary For independent identically distributed bivariate random vectors (X ₁, Y ₁), (X ₂, Y ₂), ... and for large t the distribution of X ₁ +...+ X _N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y ₁, Y ₂,.... Similar expansions are derived for multivariate X ₁. Furthermore, local asymptotic expansions are valid for the distribution of f(X ₁)+ ...+ f(X _N ) when N is large and nonrandom, and X _i , i=1, 2,..., is a discrete strongly mixing Markov chain.     

Comparaison des deux composantes d''un subordinateur bivarié, puis étude de l''enveloppe supérieure d''un processus de Lévy

Soit (Y,Z) un subordinateur bivarié. Nous donnons une condition suffisante pour que Y_t/Z_t converge vers zéro quand t tend vers 0 ou +∞. Ceci généralise partiellement des résultats de Bertoin et de Kesten–Erickson.Soit X un processus de Lévy et S_t=sup{X_s: st}. Soit f une fonction sous-additive. En appliquant le résultat précédent au subordinateur bivarié d'échelle, nous donnons des conditions nécéssaires et suffisantes pour que et égalent 0 ou +∞.Let (Y,Z) be a bivariate subordinator. Generalizing theorems of Bertoin and Kesten–Erickson, we give a sufficient condition for Y_t/Z_t to converge to 0 when t tends either to 0 or +∞.Let X be a Lévy process. Denote by S_t=sup{X_s: st} and let f be any sub-additive function. Applying our first result to the bivariate ladder process, we give necessary and sufficient conditions for and to be either 0 or +∞.     

The Local Bootstrap for Kernel Estimators under General Dependence Conditions

We consider the problem of estimating the distribution of a nonparametric (kernel) estimator of the conditional expectation g(x; ) = E((X _t+1) | Y _t,m = x) of a strictly stationary stochastic process {X _t , t 1}. In this notation (·) is a real-valued Borel function and Y _t,m a segment of lagged values, i.e., Y_t,m=(X_{t-i 1},X_{t-i 2},...,X_{t-i m}), where the integers i _i , satisfy 0 i₁2...m>. We show that under a fairly weak set of conditions on {X _t , t 1}, an appropriately designed and simple bootstrap procedure that correctly imitates the conditional distribution of X _t+1 given the selective past Y _t,m , approximates correctly the distribution of the class of nonparametric estimators considered. The procedure proposed is entirely nonparametric, its main dependence assumption refers to a strongly mixing process with a polynomial decrease of the mixing rate and it is not based on any particular assumptions on the model structure generating the observations.     

Separately Continuous Functions on Products and Their Dependence on ℵ Coordinates

We investigate necessary and sufficient conditions on topological products X = ∏_{s ∈ s} X _s and Y = ∏_{t ∈ T} Y _t for every separately continuous function f: X × Y → ℝ to be dependent on at most ℵ coordinates with respect to a certain coordinate.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1357 – 1368, October, 2004.     

Logarithmic Estimates for Submartingales and Their Differential Subordinates

In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then

Some results on the residual life at random time

In this paper, we consider the residual life at random time, i.e.X _Y =X−Y\X>Y, whereX andY are non-negative random variables. We establish a number of stochastic comparison properties forX _Y under various assumptions ofX andY. Under the assumption thatY has decreasing reverse hazard rate (DRHR), we show that ifX is in any one of the classes IFR, DFR, DMRL or IMRL thenX _Y is in the same class asX. We also obtain some useful bounds for the distribution and the moment ofX _Y . Because the idle time in classicalGI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system. This work is supported by the National Natural Science Foundation of China.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

A note on a strong germ-Markov property

Summary In a seminal paper on Markovian germ fields, Knight [2] proposes five distinct definitions of the infinitesimal present and shows that these alternative formulations all lead to the same class of germ-Markov processes, but that they lead to different classes of strong germ-Markov processes. The same paper asserts that every germ-Markov process is strongly Markovian relative to a certain right germ field. This note will show that not every germ-Markov process (in fact, not every right continuous Markov process) is strongly germ-Markovian in such a sense. But if X=(X _t) _t0 is germ-Markovian, then the process Y defined by Y _t=(X _t, t), t0, is strongly germ-Markovian in a suitable sense. An analogous argument will settle a conjecture in [2] concerning left continuous processes and a germ-Markov property at stopping times of their natural histories not rendered right continuous.This research was supported in part by the Air Force Office of Scientific Research through its Grant No. AFOSR-80-0252.     

Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

Optimal stopping times for solutions of nonlinear stochastic differential equations and their application to one problem of financial mathematics

We solve the problem of finding the optimal switching time for two alternative strategies at the financial market in the case where a random processX _t ,t ∈ [0, T], describing an investor's assets satisfies a nonlinear stochastic differential equation. We determine this switching time τ∈[0,T] as the optimal stopping time for a certain processY _t generated by the processX _t so that the average investor's assets are maximized at the final time, i.e.,EX _T . Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 804–809, June, 1999.     

A characterization of certain diffusions on the Sierpinski gasket by the exit distributions

LetK be the Sierpinski gasket with verticesa ₁,a ₂,a ₃ forming an equilateral triangle. Suppose that (X _t ,P ^x ) and (Y _t ,Q ^x ) are diffusions onK with the same hitting probabilities to verticesa ₁,a ₂,a ₃. We show that ifX is an asymptotically one-dimensional diffusion or ap-stream diffusion, thenY is a time-change ofX.Part of this work was done while the author was an Alexander von Humboldt fellow at the Universität des Saarlandes in Saarbrücken, Germany.