Exit problems for the difference of a compound Poisson process and a compound renewal process

In this paper we solve a two-sided exit problem for a difference of a compound Poisson process and a compound renewal process. More specifically, we determine the Laplace transforms of the joint distribution of the first exit time, the value of the overshoot and the value of a linear component at this time instant. The results obtained are applied to solve the two-sided exit problem for a particular class of stochastic processes, i.e. the difference of the compound Poisson process and the renewal process whose jumps are exponentially distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process. We determine the Laplace transforms of the busy period of the systems M ^? |G ^δ |1|B, G ^δ |M ^? |1|B in case when δ～exp?(λ). Additionally, we prove the weak convergence of the two-boundary characteristics of the process to the corresponding functionals of the standard Wiener process.     

Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

In this paper, we consider a product of a symmetric stable process in ? ^d and a one-dimensional Brownian motion in ?^?+?. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally Hölder continuous. We also argue a result on Littlewood–Paley functions which are obtained by the α-harmonic extension of an L ^p (? ^d ) function.     

Representation of functions of Markov processes as solutions of stochastic equations

Let X_t be a homogeneous Markov process generated by the weak infinitesimal operator A. Let $\text{H}$ be the class of functions f such that f, f²?D_A, the domain of A. The main result of this paper states that for ? ∈ $\text{H}$ can be represented by a stochastic integral and other terms. If the process is generated by a second order differential operator (with ‘poor’ coefficients possibly) on C₀²(R^d) then the process itself can be represented as the solution of an Itô stochastic differential equation.     

Construction and Strong Feller Property of Distorted Elliptic Diffusion with Reflecting Boundary

Using Dirichlet form techniques we prove existence of a diffusion process with singular drift on an open domain Ω???? ^d , d?∈??, d?≥?2, and generalized reflection at the boundary. The boundary is assumed to be C ²-smooth except for a sufficiently small set. We prove an elliptic regularity result which gives L ^p -strong Feller property for the semigroup and resolvent for p?≥?2 with $p > \frac{d}{2}$ . This result allows us to construct the process pointwisely except for an explicitly known set. For starting points outside this known set the process solves the corresponding martingale problem. The results are applied to prove the existence of the stochastic dynamics of a finite interacting particle system and for the Ginzburg-Landau interface model with a hard wall.     

Dirichlet Heat Kernel Estimates for Stable Processes with Singular Drift In Unbounded C 1,1 Open Sets

Suppose d ≥ 2 and α ∈ (1, 2). Let D be a (not necessarily bounded) C ^1,1 open set in ? ^d and μ = (μ ¹, . . . , μ ^d ) where each μ ^j is a signed measure on ? ^d belonging to a certain Kato class of the rotationally symmetric α-stable process X. Let X ^μ be an α-stable process with drift μ in ? ^d and let X ^μ,D be the subprocess of X ^μ in D. In this paper, we derive sharp two-sided estimates for the transition density of X ^μ,D .     

Non-linear functionals of the Brownian bridge and some applications

Let {b^F(t),t∈[0,1]} be an F-Brownian bridge process. We study the asymptotic behaviour of non-linear functionals of regularizations by convolution of this process and apply these results to the estimation of the variance of a non-homogeneous diffusion and to the convergence of the number of crossings of a level by the regularized process to a modification of the local time of the Brownian bridge as the regularization parameter goes to 0.     

Toward a stochastic calculus for several Markov processes

In this paper we investigate a class of harmonic functions associated with a pair xt = (xt11, xt22) of strong Markov processes. In the case where both processes are Brownian motions, a smooth function f is harmonic if Δx1Δx2f(x1,x2) = 0. For these harmonic functions we investigate a certain boundary value problem which is analogous to the Dirichlet problem associated with a single process. One basic tool for this study is a generalization of Dynkin's formula, which can be thought of as a kind of stochastic Green's formula. Another important tool is the use of Markov processes xti?i obtained from xtii by certain random time changes. We call such a process a stochastic wave since it propogates deterministically through a certain family of sets; however its position on a given set is random.     

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C^1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.     

Hotelling’s T charts with variable sample size and control limit

The ideas of variable sampling interval (VSI), variable sample size (VSS), variable sample size and sampling interval (VSSI), and variable parameters (VP) in the univariate case have been successfully applied to the multivariate case to improve the efficiency of Hotelling’s T² chart with fixed sampling rate (FSR) in detecting small process shifts. However, the main disadvantage in using most of these control schemes is an increasing in the complexity due to the adaptive changes in sampling intervals. In this paper, retaining the lengths of sampling intervals constant, a variable sample size and control limit (VSSC) T² chart is proposed and described. The statistical efficiency of the VSSC T² chart in terms of the average time to signal a shift in process mean vector is compared with that of the VP, VSSI, VSS, VSI, and FSR T² charts. From the results of comparison, it shows that the VSSC T² chart for a (very) small shift in the process mean vector gives a better performance than the VSSI, VSS, VSI, and FSR T² charts; meanwhile, it presents a similar performance to the VP T² chart. Furthermore, from the viewpoint of practicability, it is more convenient for administrating the control chart than the VSI, VSSI, and VP T² chart. Thus, it may provide a good option for quick response to small shifts in a multivariate process.     

Optimal replacement policy for a deteriorating system with increasing repair times

This paper considers an optimal maintenance policy for a practical and reparable deteriorating system subject to random shocks. Modeling the repair time by a geometric process and the failure mechanism by a generalized δ-shock process, we develop an explicit expression of the long-term average cost per time unit for the system under a threshold-type replacement policy. Based on this average cost function, we propose a finite search algorithm to locate the optimal replacement policy N^∗ to minimize the average cost rate. We further prove that the optimal policy N^∗ is unique and present some numerical examples. Many practical systems fit the model developed in this paper.     

A branching particle system approximation for nonlinear stochastic filtering

The optimal filter π = {π t,t ∈ [0,T ]} of a stochastic signal is approximated by a sequence {π n t } of measure-valued processes defined by branching particle systems in a random environment(given by the observation process).The location and weight of each particle are governed by stochastic differential equations driven by the observation process,which is common for all particles,as well as by an individual Brownian motion,which applies to this specific particle only.The branching mechanism of each particle depends on the observation process and the path of this particle itself during its short lifetime δ = n 2α,where n is the number of initial particles and α is a fixed parameter to be optimized.As n →∞,we prove the convergence of π n t to π t uniformly for t ∈ [0,T ].Compared with the available results in the literature,the main contribution of this article is that the approximation is free of any stochastic integral which makes the numerical implementation readily available.     

A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue

We consider a model in which the production of new molecules in a chemical reaction network occurs in a seemingly stochastic fashion, and can be modeled as a Poisson process with a varying arrival rate: the rate is λ _i when an external Markov process J(?) is in state i. It is assumed that molecules decay after an exponential time with mean μ ^?1. The goal of this work is to analyze the distributional properties of the number of molecules in the system, under a specific time-scaling. In this scaling, the background process is sped up by a factor N ^α , for some α>0, whereas the arrival rates become N λ _i , for N large. The main result of this paper is a functional central limit theorem (F-CLT) for the number of molecules, in that, after centering and scaling, it converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i) if α > 1 the background process jumps faster than the arrival process, and consequently the arrival process behaves essentially as a (homogeneous) Poisson process, so that the scaling in the F-CLT is the usual \(\sqrt {N}\), whereas (ii) for α≤1 the background process is relatively slow, and the scaling in the F-CLT is N ^1?α/2. In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process J(?).     

Existence of a double random integral with respect to stable measures

In this paper, the authors study a double random integral of the form ∫₀¹∫₀¹f(s,t) M(ds) M(dt), where M(0,t) is a stable process with independent increments. Basically, the Wiener approach is used, and the existence of the above integral is established for a wide class of functions f.     

Well-posedness and asymptotic behavior a multidimensional model of morphogen transport

Morphogen transport is a biological process, occurring in the tissue of living organisms, which is a determining step in cell differentiation. We present rigorous analysis of a simple model of this process, which is a system coupling parabolic PDE with ODE. We prove existence and uniqueness of solutions for both stationary and evolution problems. Moreover, we show that the solution converges exponentially to the equilibrium in C ^1,?? ×?C ^0,?? topology. We prove all results for arbitrary dimension of the domain. Our results improve significantly previously known results for the same model in the case of one-dimensional domain.     

Franchissement et temps local pour l'oscillateur harmonique

We consider the second order Stochastic Differential Equation dP_t^β = V_t^β dt with P₀^β = p₀, dV_t^β = βV_t^βdt − βω²P_t^β + βdW_t with V₀^β = v₀, where W stands for a standard Wiener process and where ω is a real constant. It is well-known that P^β converges, as β goes to infinity, to an Ornstein-Uhlenbeck process P. In this Note, we study the convergence of the crossings of P^β at level u during the time interval [0, t] · (N_t^Pβ (u)) to the local time of P(L_t^P (u)).     

Distribution and characteristic functions for correlated complex Wishart matrices

Let A(t) be a complex Wishart process defined in terms of the M×N complex Gaussian matrix X(t) by A(t)=X(t)X(t)^H. The covariance matrix of the columns of X(t) is Σ. If X(t), the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t₁, t₂, where t₁<t₂. In particular, we derive the following results: the joint density of the elements of A(t₁), A(t₂), the joint density of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂), the characteristic function of the elements of A(t₁), A(t₂), the characteristic function of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂). In addition, we give the characteristic functions of the eigenvalues of a central and non-central complex Wishart, and some applications of the results in statistics, engineering and information theory are outlined.     

On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L^p (0 < p ≤ 2). We use the theory of L^p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L^q into real or complex L^p for 0 < p < q ≤ 2.     

A note on iterative process for G-multi degree reduction of Bézier curves

In the paper [A. Rababah, S. Mann, Iterative process for G²-multi degree reduction of Bézier curves, Applied Mathematics and Computation 217 (2011) 8126-8133], Rababah and Mann proposed an iterative method for multi-degree reduction of Bézier curves with C¹ and G²-continuity at the endpoints. In this paper, we provide a theoretical proof for the existence of the unique solution in the first step of the iterative process, while the proof in their paper applies only in some special cases. Also, we give a complete convergence proof for the iterative method. We solve the problem by using convex quadratic optimization.     

Differentiation formulas for stochastic integrals in the plane

For a one-parameter process of the form X_t=X₀+∫^t₀φ_sdW_s+∫^t₀ψ_sds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(X_t) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process.Let {W₂, z∈R²₊} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals ∫ φdWand ∫ψdWdW, as well as mixed integrals ∫h dz dW and ∫gdW dz. Now let X_z be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(X_z) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(X_z) on increasing paths in $\text{R}{}^2{}_{\mathrm{+}}$.     

Systems of p-Laplacian differential inclusions with large diffusion

In this paper we consider coupled systems of p-Laplacian differential inclusions and we prove, under suitable conditions, that a homogenization process occurs when diffusion parameters become arbitrarily large. In fact we obtain that the attractors are continuous at infinity on L²(Ω)×L²(Ω) topology, with respect to the diffusion coefficients, and the limit set is the attractor of an ordinary differential problem.