Nonlinear Filtering Equation of a Jump Process with Counting Observations

We study the filtering problem of an R ^d -valued pure jump process when the observations is a counting process. We assume that the dynamic of the state and the observations may be strongly dependent and that the two processes may jump together. Weak and pathwise uniqueness of solution of the Kushner–Stratonovich equation are discussed.     

Existence of Optimal Controls for Partially Observed Jump Processes

A partially observable control problem for an R ^d -valued jump process with counting observations is studied. The state and the observations may be strongly dependent and, in particular, the two processes may jump together. An equivalent separated problem is introduced and the existence of an optimal control for the separated problem is obtained in the class of relaxed and generalized controls. Equivalence between the initial problem and the relaxed generalized separated control problem is discussed.     

On the optimal control of partially observed inventory systems

This Note introduces recent developments in the analysis of inventory systems with partial observations. The states of these systems are typically conditional distributions, which evolve in infinite dimensional spaces over time. Our analysis involves introducing unnormalized probabilities to transform nonlinear state transition equations to linear ones. With the linear equations, the existence of the optimal feedback policies are proved for two models where demand and inventory are partially observed. In a third model where the current inventory is not observed but a past inventory level is fully observed, a sufficient statistic is provided to serve as a state. The last model serves as an example where a partially observed model has a finite dimensional state. In that model, we also establish the optimality of the basestock policies, hence generalizing the corresponding classical models with full information. To cite this article: A. Bensoussan et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Change-point detection for continuous processes with high-frequency sampling

We consider the detection of a jump in a continuous process over a fixed time interval. We aim to locate the jump position via discrete observations and consider how increasing the frequency of the observations affects the accuracy of the detection process. We show that the classical cumulative-sum estimator fails, and propose a new estimator based on local information that we prove converges exponentially fast. To cite this article: G. Wang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Paley-Wiener theorem for singular support of K-finite distributions on symmetric spaces

We study Fourier transforms of distributions on a symmetric space X. Eguchi et al. [1] characterized the image of $\text{E}$′(X)-distributions of compact support under the Fourier transform. We give a simpler proof of Eguchi's result and characterize the size of the singular support for the K-finite members of E′(X). We apply this Paley-Wiener type theorem to invariant differential equations on X.     

On a robust version of the integral representation formula of nonlinear filtering

The paper is concerned with completing “unfinished business” on a robust representation formula for the conditional expectation operator of nonlinear filtering. Such a formula, robust in the sense that its dependence on the process of observations is continuous, was stated in [2] without proof. The main purpose of this paper is to repair this deficiency.The formula is “almost obvious” as it can be derived at a formal level by a process of integration-by-parts applied to the stochastic integrals that appear in the integral representation formula. However, the rigorous justification of the formula is quite subtle, as it hinges on a measurability argument the necessity of which is easy to miss at first glance. The continuity of the representation (but not its validity) was proved by Kushner [9] for a class of diffusions.Here we follow the definition given in [11].     

Φ-entropy inequality and application for SDEs with jumps

By using the Φ-entropy inequality derived in [16] and [2] for Poisson measures, the same type of inequality is established for a class of stochastic differential equations driven by purely jump Lévy processes. This inequality implies the exponential convergence in Φ-entropy of the associated Markov semigroup. The semigroup Φ-entropy inequality for SDEs driven by Poisson point processes is also considered.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

Optimal Control of Point Processes with Noisy Observations: The Maximum Principle

This paper studies the optimal control problem for point processes with Gaussian white-noised observations. A general maximum principle is proved for the partially observed optimal control of point processes, without using the associated filtering equation . Adjoint flows—the adjoint processes of the stochastic flows of the optimal system—are introduced, and their relations are established. Adjoint vector fields , which are observation-predictable, are introduced as the solutions of associated backward stochastic integral-partial differential equtions driven by the observation process. In a heuristic way, their relations are explained, and the adjoint processes are expressed in terms of the adjoint vector fields, their gradients and Hessians, along the optimal state process. In this way the adjoint processes are naturally connected to the adjoint equation of the associated filtering equation . This shows that the conditional expectation in the maximum condition is computable through filtering the optimal state, as usually expected. Some variants of the partially observed stochastic maximum principle are derived, and the corresponding maximum conditions are quite different from the counterpart for the diffusion case. Finally, as an example, a quadratic optimal control problem with a free Poisson process and a Gaussian white-noised observation is explicitly solved using the partially observed maximum principle. Accepted 8 August 2001. Online publication 17 December, 2001.     

Stability of Merton's portfolio optimization problem for Lévy models

Merton's classical portfolio optimization problem for an investor, who can trade in a risk-free bond and a stock, can be extended to the case where the driving noise of the logreturns is a pure jump process instead of a Brownian motion. Benth et al. [4,5] solved the problem and found the optimal control implicitly given by an integral equation in the hyperbolic absolute risk aversion (HARA) utility case. There are several ways to approximate a Levy process with infinite activity by neglecting the small jumps or approximating them with a Brownian motion, as discussed in Asmussen and Rosinski [1]. In this setting, we study stability of the corresponding optimal investment problems. The optimal controls are solutions of integral equations, for which we study convergence. We are able to characterize the rate of convergence in terms of the variance of the small jumps. Additionally, we prove convergence of the corresponding wealth processes and indirect utilities (value functions).     

Remarks on the optimization method of a manufacturing system with stochastic breakdown and rework process in supply chain management

Chiu et al. (2010) [8] present the proof of convexity of the long-run average cost function E[TCU(t₁)] for a manufacturing system with stochastic breakdown and rework process. This note not only demonstrates that E[TCU(t₁)] is not convex but also adopts the rigorous methods of mathematics to develop the complete solution procedure to find the optimal solution for removing shortcomings of the above paper mentioned.     

Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.     

A note on “Determining the optimal run time for EPQ model with scrap,rework, and stochastic breakdowns”

In a recently published paper by Chiu et al. [Chiu, S.W., Wang, S.-L., Chiu, Y.-S.P., 2007. Determining the optimal run time for EPQ model with scrap, rework and stochastic breakdowns. European Journal of Operational Research 180, 664–676], a theorem on conditional convexity of the integrated total cost function was employed in their solution procedure. We reexamine this theorem and present a direct proof to the convexity of the total cost function. This proof can be used in place of Theorem 1 of Chiu et al.’s paper to enhance quality of their optimization process.     

Numerical analysis for stochastic age-dependent population equations with Poisson jump and phase semi-Markovian switching

In this paper, we shall examine the convergence of semi-implicit Euler approximation for stochastic age-dependent population equations with Poisson jump and phase semi-Markovian switching. Here, the main ideas from the papers Ronghua et al. (2009) [2] and Wang and Wang (2010) [3] are successfully developed to the more general cases. Finally, a numerical example is provided to illustrate the theoretical result of convergence.     

Some refinements of the Hodge decomposition and applications

We extend the nonlinear Hodge decomposition of Iwaniec et al. [1] to other vector fields. We give applications to Leray-Lions type equations with very weak assumptions on data similar to those developed in [1].     

A note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay

In this note, we prove the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay (INSFDEs in short) in which the initial value belongs to the phase space BC((-∞,0]R^d), which denotes the family of bounded continuous R^d-value functions φ defined on (-∞,0] with norm ||φ||=sup_-∞<θ?0|φ(θ)|, under some Carathéodory-type conditions on the coefficients by means of the successive approximation. Especially, we extend the results appeared in Ren et al. [Y. Ren, S. Lu, N. Xia, Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay, J. Comput. Appl. Math. 220 (2008) 364-372], Ren and Xia [Y. Ren, N. Xia, Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput. 210 (2009) 72-79] and Zhou and Xue [S. Zhou, M. Xue, The existence and uniqueness of the solutions for neutral stochastic functional differential equations with infinite delay, Math. Appl. 21 (2008) 75-83].     

The number of vertices whose out-arcs are pancyclic in a 2-strong tournament

An arc going out from a vertex x in a digraph is called an out-arc of x. Yao et al. [Discrete Appl. Math. 99 (2000) 245-249] proved that every strong tournament contains a vertex x such that all out-arcs of x are pancyclic. Recently, Yeo [J. Graph Theory 50 (2005) 212-219] proved that each 3-strong tournament contains two such vertices. In this paper, we confirm that Yeo's result is also true for 2-strong tournaments. Our proof implies a polynomial algorithm to find two such vertices.     

Control problems with time-lag. II

Consider a continuous time Markov chain with stationary transition probabilities. A function of the state is observed. A regular conditional probability distribution for the trajectory of the chain, given observations up to time t, is obtained. This distribution also corresponds to a Markov chain, but the conditional chain has nonstationary transition probabilities. In particular, computation of the conditional distribution of the state at time s is discussed. For s > t, we have prediction (extrapolation), while s < t corresponds to smoothing (interpolation). Equations for the conditional state distribution are given on matrix form and as recursive differential equations with varying s or t. These differential equations are closely related to Kolmogorov's forward and backward equations. Markov chains with one observed and one unobserved component are treated as a special case. In an example, the conditional distribution of the change-point is derived for a Poisson process with a changing intensity, given observations of the Poisson process.     

A Direct Approach to Deriving Filtering Equations for Diffusion Processes

Filtering equations are derived for conditional probability density functions in case of partially observable diffusion processes by using results and methods from the L _p -theory of SPDEs. The method of derivation is new and does not require any knowledge of filtering theory. Accepted 31 July 2000. Online publication 13 November 2000.     

Nonlinear filtering theory for coral/starfish and plant/herbivore interactions †

Several nonlinear filtering problems associated with specific 4 dimensional differential equation models of coral/starfish or chemically mediated plant/herbivore population dynamics are studied. Extensive use is made of H. Kunita's backward Stratonovich calculus and stochastic partial differential equations theory to obtain exact solution measures of the Zakai and Kushner equations. The hypoellipticity problem is solved positively, so that these measures all possess c^∞-densities. Thus, explicit formulas are obtained for the estimation of signal processes conditional on observational data. For example, biomass production/consumption processes are least squares estimated conditional on observations on the population dynamics of the producing and consuming units themselves.