Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes

In this paper, a new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Lévy process and the integral with respect to an adapted continuous increasing process is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of stochastic partial differential integral equations (PDIEs in short) with a nonlinear Neumann boundary condition is given.     

Modelo de un proceso de cristalización continua de un sorbete por medio de la metodología de momentos

Freezing is an important step in the manufacturing process of ice-cream and sorbet, since the operating conditions have a strong influence on the micro-structure, and consequently on the sensorial attributes of the final product. This steep of freezing is carried out by a scraped surface heat exchanger (SSHE) where the product quality is conditioned by process conditions as the evaporation temperature of a refrigerant fluid, the mix flow rate, the dasher speed and the cylinder pressure due to the air introduction. In order to study the relevance of a control system based on the influence of process variables on product quality, this paper presents a model for a continuous crystallization of a sorbet using the method of moments, which is validated by experimental data.The model created by this methodology has been able to represent the influence of the process conditions during the crystallization of the sorbet on the final product characteristics such as crystal size and the draw temperature in the outlet of the SSHE in absence of air. The model based in moments is studied as a reduced model of the population balance equation and includes the phenomena of heterogeneous nucleation and growth. This model developed represents minimal computational requirements and is highly adapted for optimization and/or process control tasks.     

The hu-meyer formula for non deterministic kernels

In this paper we prove the analogue of the Hu-Meyer formula for random kernels. More precisely, using a suitable notion of trace we give the relation between the multiple Stratonovich integral of a non adapted process and the multiple Skorohod integral     

Numerical Solution of non-Homogeneous Semi-Markov Processes in Transient Case*

In this article a numerical solution for the evolution equation of a continuous time non-homogeneous semi-Markov process (NHSMP) is obtained using a quadrature method. The paper, after a short introduction to continuous time NHSMP, presents the numerical solution of the process evolution equation with a general quadrature method. Furthermore, the paper gives results that justify this approach, proving that the numerical solution tends to the evolution equation of the continuous time NHSMP. Moreover, the formulae related to some specific quadrature methods are given and a method for obtaining the discrete time NHSMP by applying a very particular quadrature formula for the discretization is shown. In this way the relation between the continuous and discrete time NHSMP is proved. Then, the problem of obtaining the continuous time NHSMP from the discrete one is considered. This problem is solved showing that the discrete process converges in law to the continuous one if the discretized time interval tends to zero. In addition, the discrete time NHSMP in matrix form is presented, and the fact that the solution to this process always exists is proved. Finally, an algorithm for solving the discrete time NHSMP is given. To illustrate the use of this algorithm for a discrete NHSMP, an example in the area of finance is presented.     

A stable numerical method for Volterra integral equations with discontinuous kernel

Numerical methods for Volterra integral equations with discontinuous kernel need to be tuned to their peculiar form. Here we propose a version of the trapezoidal direct quadrature method adapted to such a type of equations. In order to delineate its stability properties, we first investigate about the behavior of the solution of a suitable (basic) test equation and then we find out under which hypotheses the trapezoidal direct quadrature method provides numerical solutions which inherit the properties of the continuous problem.     

Stochastic integrals and BDG’s inequalities in Orlicz-type spaces

In this paper we extend an inequality of Lenglart et al. (1980, Lemma 1.1) to general continuous adapted stochastic processes with values in topological spaces. Using this inequality we prove Burkholder–Davies–Gundy’s inequality for stochastic integrals in Orlicz-type spaces (a class of quasi-Banach spaces) with respect to cylindrical Brownian motions. As an application, we show the well-posedness of stochastic heat equations in Orlicz spaces.     

The Wiener integral with respect to second order processes with stationary increments

We prove that for any second order stochastic process X with stationary increments with continuous paths and continuous variance function, there exists a tempered measure μ (for which we give an explicit expression) related with the domain of the Wiener integral with respect to X as follows: the space of tempered distributions f such that the Fourier transform of f is square integrable with respect to μ is always a dense subset of the domain of the Wiener integral. Moreover, we provide sufficient conditions on μ in order that the domain of the integral is exactly this space of distributions. We apply our results to the fractional Brownian motion. In particular, it is proved that the domain of the Wiener integral with respect to the fractional Brownian motion with Hurst parameter H>1/2 contains distributions that are not given by locally integrable functions, this fact was suggested by Pipiras and Taqqu (2000) in [5]. We have also considered the example of the process given by Ornstein and Uhlenbeck as a model for the position of a Brownian particle.     

Decision-making in testing process performance with fuzzy data

Over the years, numerous process capability indices (PCIs) have been proposed to the manufacturing industry to provide numerical measures of process performance. Most research efforts have focused on developing and investigating PCIs that assess process capability by precise measurements of output quality. However, real observations of continuous quantities are not precise numbers; in practice, they are more or less imprecise. Since observations of continuous random variables are imprecise the values of related test statistics become imprecise. Therefore, decision rules for statistical tests have to be adapted to this situation. This article presents a set of confidence intervals that produces triangular fuzzy numbers for the estimation of C_pk index using Buckley’s approach with some modification. Additionally, a three-decision testing rule and step-by-step procedure are developed to assess process performance based on fuzzy critical values and fuzzy p-values. This concept is also illustrated with an example for testing process performance.     

Circular spline fitting using an evolution process

We propose a new method to approximate a given set of ordered data points by a spatial circular spline curve. At first an initial circular spline curve is generated by biarc interpolation. Then an evolution process based on a least-squares approximation is applied to the curve. During the evolution process, the circular spline curve converges dynamically to a stable shape. Our method does not need any tangent information. During the evolution process, the number of arcs is automatically adapted to the data such that the final curve contains as few arc arcs as possible. We prove that the evolution process is equivalent to a Gauss-Newton-type method.     

Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach

This paper presents the numerical solution of the process evolution equation of a homogeneous semi-Markov process (HSMP) with a general quadrature method. Furthermore, results that justify this approach proving that the numerical solution tends to the evolution equation of the continuous time HSMP are given. The results obtained generalize classical results on integral equation numerical solutions applying them to particular kinds of integral equation systems. A method for obtaining the discrete time HSMP is shown by applying a very particular quadrature formula for the discretization. Following that, the problem of obtaining the continuous time HSMP from the discrete one is considered. In addition, the discrete time HSMP in matrix form is presented and the fact that the solution of the evolution equation of this process always exists is proved. Afterwards, an algorithm for solving the discrete time HSMP is given. Finally, a simple application of the HSMP is given for a real data social security example.     

Relaxation-based algorithms for minimax optimization problems with resource allocation applications

We consider minimax optimization problems where each term in the objective function is a continuous, strictly decreasing function of a single variable and the constraints are linear. We develop relaxation-based algorithms to solve such problems. At each iteration, a relaxed minimax problem is solved, providing either an optimal solution or a better lower bound. We develop a general methodology for such relaxation schemes for the minimax optimization problem. The feasibility tests and formulation of subsequent relaxed problems can be done by using Phase I of the Simplex method and the Farkas multipliers provided by the final Simplex tableau when the corresponding problem is infeasible. Such relaxation-based algorithms are particularly attractive when the minimax optimization problem exhibits additional structure. We explore special structures for which the relaxed problem is formulated as a minimax problem with knapsack type constraints; efficient algorithms exist to solve such problems. The relaxation schemes are also adapted to solve certain resource allocation problems with substitutable resources. There, instead of Phase I of the Simplex method, a max-flow algorithm is used to test feasibility and formulate new relaxed problems.Corresponding author.Work was partially done while visiting AT&T Bell Laboratories.     

Properties of a wiener process with coalescence

A Wiener process with coalescence and its analog are discussed. We prove the existence of an initial distribution with preset final probabilities for this analog and investigate the problem of the existence of such distributions concentrated at a single point or absolutely continuous with respect to the Lebesgue measure. The behavior of a semigroup of a Wiener process with coalescence in the two-dimensional case and properties of a Wiener flow with coalescence are studied. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 489–504, April, 2006.     

Anticipated BSDEs driven by a single jump process

In this paper, we discuss a class of anticipated backward stochastic differential equations related to a finite continuous time single jump process. We prove the existence and uniqueness of the adapted solution. Moreover, a comparison theorem for the solutions is also established.     

Separation principle for impulse control with partial information

This article is concerned with the impulse control of stochastic systems, the internal state of which is only known by a noisy observation with both continuous and discontinuous components. It is shown that among the controls depending on the observation there exists an optimal one, that can be recursively constructed as a function of the filter associated with the system.     

A self-adaptive global best harmony search algorithm for continuous optimization problems

This paper presents a self-adaptive global best harmony search (SGHS) algorithm for solving continuous optimization problems. In the proposed SGHS algorithm, a new improvisation scheme is developed so that the good information captured in the current global best solution can be well utilized to generate new harmonies. The harmony memory consideration rate (HMCR) and pitch adjustment rate (PAR) are dynamically adapted by the learning mechanisms proposed. The distance bandwidth (BW) is dynamically adjusted to favor exploration in the early stages and exploitation during the final stages of the search process. Extensive computational simulations and comparisons are carried out by employing a set of 16 benchmark problems from literature. The computational results show that the proposed SGHS algorithm is more effective in finding better solutions than the state-of-the-art harmony search (HS) variants.     

Supercritical Markov branching processes with general set of types

This paper concerns the asymptotic behaviour of normalized averaging processes associated with a supercritical, indecomposable Markov branching process. Results wellknown in case of a finite set of types are extended to processes with an arbitrary set of types. The process parameter is allowed to be discrete or continuous.Convergence in the quadratic mean is proved on the basis of a weak form of positive regularity. In this setting, limit variables corresponding to different averaging functions are proportional almost everywhere. The rate of convergence is such that process skeletons, defined by uniform partitions of the parameter set, converge with probability 1. Starting from the almost sure convergence of skeletons, we obtain almost sure convergence of the processes themselves. The final sections deals with properties of the limiting distribution functions, in particular with the possible jump at the origin and the existence of a continuous density everywhere else.Other investigations of supercritical Markov branching processes with an infinite or arbitrary set of types are to be found in [2, 3, 4, 6, 7, 13, 15, 19, 21, 23], and [24].The author was supported by DFG grant HE 678/1.     

On a nonlinear heat equation with functional dependence

We reduce the Cauchy problem for a heat equation with the nonlinear right-hand side which depends on some functionals to an equivalent integral equation. Considering mainly Banach spaces of continuous, bounded and exponentially bounded functions, we give some natural sufficient conditions for the existence and uniqueness of solutions to these equations. We give a counterexample which shows that the Lipschitz condition is, in general, insufficient for the Cauchy problem with unbounded data and with functional dependence to guarantee an existence result     

Sample path properties of stochastic integrals,and stochastic differentiation

Let B be a Brownian motion, and X = H.B be a stochastic integral of B. We give conditions on the smoothness of the process H which imply that if Ms a singular point of the sample path of B (ω) (such as a local maximum, a slow point, or a fast point) then t is also a singular point of X (ω). In the final section we give an application to stochastic differential equations     

Nonlinear flows of stochastic linear delay equations

The sample non-linearity of the classical Wiener integral as a function of continuous integrands is pointed out. As an application it is shown that the solution of a linear stochastic delay equation is an almost surely non-linear function of the initial trajectory segment.     

Regularity of Backward Stochastic Volterra Integral Equations in Hilbert Spaces

This article investigates backward stochastic Volterra integral equations in Hilbert spaces. The existence and uniqueness of their adapted solutions is reviewed. We establish the regularity of the adapted solutions to such equations by means of Malliavin calculus. For an application, we study an optimal control problem for a stochastic Volterra integral equation driven by a Hilbert space-valued fractional Brownian motion. A Pontryagin-type maximum principle is formulated for the problem and an example is presented.