A partially observed control problem for Markov chains

A finite state, continuous time Markov chain is considered and the solution to the filtering problem given when the observation process counts the total number of jumps. The Zakai equation for the unnormalized conditional distribution is obtained and the control problem discussed in separated form with this as the state. A new feature is that, because of the correlation between the state and observation process, the control parameter appears in the diffusion coefficient which multiplies the Poisson noise in the Zakai equation. By introducing a Gâteaux derivative the minimum principle, satisfied by an optimal control, is derived. If the optimal control is Markov, a stochastic integrand can be obtained more explicitly and new forward and backward equations satisfied by the adjoint process are obtained.This research was partially supported by NSERC Grant A7964, the Air Force Office of Scientific Research, United States Air Force, under Contract AFOSR-86-0332, and the U.S. Army Research Office under Contract DAAL03-87-0102.     

An Approximation for the Zakai Equation

In this article we consider a polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space. An estimate of the convergence rate based on a distance which is equivalent to the weak convergence topology is derived. We also study the density of the unnormalized conditional measure, which is the solution of the Zakai stochastic partial differential equation. An estimate of the convergence rate is also given in this case. 60F25, 60H10.} Accepted 23 April 2001. Online publication 14 August 2001.     

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

The Zakai equation for the unnormalized conditional density is derived as a mild stochastic bilinear differential equation on a suitableL ² space. It is assumed that the Markov semigroup corresponding to the state process isC ₀ on such space. This allows the establishment of the existence and uniqueness of the solution by means of general theorems on stochastic differential equations in Hilbert space. Moreover, an easy treatment of convergence conditions can be given for a general class of finite-dimensional approximations, including Galerkin schemes. This is done by using a general continuity result for the solution of a mild stochastic bilinear differential equation on a Hilbert space with respect to the semigroup, the forcing operator, and the initial state, within a suitable topology.     

The Zakai Equation of Nonlinear Filtering for Jump-Diffusion Observations: Existence and Uniqueness

In this paper we study a nonlinear filtering problem for a general Markovian partially observed system (X,Y), whose dynamics is modeled by correlated jump-diffusions having common jump times. At any time t∈[0,T], the σ-algebra $\mathcal{F}^{Y}_{t}:= \sigma\{ Y_{s}: s\leq t\}$ provides all the available information about the signal X _t . The central goal of stochastic filtering is to characterize the filter, π _t , which is the conditional distribution of X _t , given the observed data $\mathcal{F}^{Y}_{t}$ . In Ceci and Colaneri (Adv. Appl. Probab. 44(3):678–701, 2012) it is proved that π is the unique probability measure-valued process satisfying a nonlinear stochastic equation, the so-called Kushner-Stratonovich equation (in short KS equation). In this paper the aim is to improve the hypotheses to obtain the KS equation and describe the filter π in terms of the unnormalized filter ?, which is solution of a linear stochastic differential equation, the so-called Zakai equation. We prove the equivalence between strong uniqueness of the solution of the KS equation and strong uniqueness of the solution of the Zakai one and, as a consequence, we deduce pathwise uniqueness of the solution of the Zakai equation by applying the Filtered Martingale Problem approach (Kurtz and Ocone in Ann. Probab. 16:80–107, 1988; Ceci and Colaneri in Adv. Appl. Probab. 44(3):678–701, 2012). To conclude, we discuss some particular models.     

Nonlinear Filtering with Fractional Brownian Motion

Our objective is to study a nonlinear filtering problem for the observation process perturbed by a Fractional Brownian Motion (FBM) with Hurst index 1/2 相似文献   

基于分数布朗运动的非线性滤波

本文研究一类由Host指数为1/2相似文献   

Nonlinear Filtering of Stochastic Navier-Stokes Equation with Itô-Lévy Noise

In this article, we study the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Itô-Lévy noise. Nonlinear filtering problem is formulated for the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data. The corresponding Fujisaki-Kallianpur-Kunita and Zakai equations describing the time evolution of the nonlinear filter are derived. Existence and uniqueness of measure-valued solutions are proven for these filtering equations.     

Stochastic partial differential equations and filtering of diffusion processes

We establish basic results on existence and uniqueness for the solution of stochastic PDE's. We express the solution of a backward linear stochastic PDE in terms of the conditional law of a partially observed Markov diffusion process. It then follows that the adjoint forward stochastic PDE governs the evolution of the “unnormalized conditional density”     

Filltering of a partially observed process in the case of a high signal –to–noise ratio for correlated systems

This paper concerns a nonlinear filtering problem with correlated noises in the case of a high signal–to–noise ratio, when only one component of the signal is observed. We compute an approximate filter for the unnormalized filter associated to the system and derive both a Zakai and a Kushner-Stratonovitch type equation for the approximate filter     

Nonlinear Filtering in Rieznnnnifln Manifolds

A general formulation of the nonlinear filtering problem inRiemanman manifolds is given by use of the strong solutionsof the stochastic differential equations for the state and observationprocesses in the orthonormal frame bundles of the state andobservation process manifolds, respectively. A general Bayesformula for the conditional expectation of smooth functionsof the state process is given. This is used to give a directderivation of the Zakai equation for the general problem underconsideration. An example is presented.     

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained. Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G0008.     

Wong–Zakai approximations and attractors for stochastic reaction–diffusion equations on unbounded domains

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction–diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong–Zakai approximations of stochastic reaction–diffusion equation. Then, we show that the attractors of Wong–Zakai approximations converges to the attractor of the stochastic reaction–diffusion equation for both additive and multiplicative noise.     

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.     

Filtering with Marked Point Process Observations via Poisson Chaos Expansion

We study a general filtering problem with marked point process observations. The motivation comes from modeling financial ultra-high frequency data. First, we rigorously derive the unnormalized filtering equation with marked point process observations under mild assumptions, especially relaxing the bounded condition of stochastic intensity. Then, we derive the Poisson chaos expansion for the unnormalized filter. Based on the chaos expansion, we establish the uniqueness of solutions of the unnormalized filtering equation. Moreover, we derive the Poisson chaos expansion for the unnormalized filter density under additional conditions. To explore the computational advantage, we further construct a new consistent recursive numerical scheme based on the truncation of the chaos density expansion for a simple case. The new algorithm divides the computations into those containing solely system coefficients and those including the observations, and assign the former off-line.     

Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in Bandini et al. (2018), we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton–Jacobi–Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear–quadratic model.     

Inverse Problem for a Mathematical Model of Adsorption Dynamics

A mixed problem is considered for a system of partial differential equations modeling the process of adsorption dynamics. An existence and uniqueness theorem is proved for this problem, and the solution properties are investigated. The inverse problem is posed, involving the determination of the system coefficient given additional information about the solution. A uniqueness theorem is proved for the solution of the inverse problem.__________Translated from Prikladnaya Matematika i Informatika, No. 16, pp. 5 – 14, 2004.     

Optimal inventory control with shrinkage and observed sales

We study a discrete-time periodic-review inventory system where the unmet demand is lost, and the excess inventory is subject to shrinkage. We first derive the state evolution and then introduce unnormalized conditional probabilities to transform the nonlinear state evolution into a linear one. We then prove the existence and uniqueness of the solution for the Bellman equation in the case of unbounded costs and show that the solution yields the value function.     

An alternative approach to nonlinear filtering

The structure of a nonlinear filter with observation process having continuous and discontinuous components is considered. The approach is based on the so-called “Bayes” formula for conditional expectations. “Fubini” type theorems for stochastic integrals are given and used to obtain the representations of an optimal estimate and of the conditional likelihood ratio. A linear unnormalized filtering equation for controlled system process is derived.     

Determination of the coefficient in the stationary nonlinear equation of heat conduction

The article considers the problem of determining the solution-dependent coefficient of heat conductivity in a stationary nonlinear equation of heat conduction containing a parameter. Additional information for the determination of heat conductivity is provided by a function dependent on a parameter, which is obtained by solving a boundary-value problem. A uniqueness theorem is proved for the inverse problem.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 13–17, 1993.