Stochastic Filtering Theory

Kallianpur, G.

doi:10.1007/978-1-4757-6592-2

Cited by 611 publications

(161 citation statements)

References 0 publications

Supporting

Mentioning

161

Contrasting

Order By: Relevance

“…This construction differs from two standard constructions. The first one deals with the case of Gaussian infinite-dimensional basic variables for which mathematical methods used for the construction of Fock spaces are applicable [12,19,11]. The second standard method deals with multidimensional polynomial approximations over product vector spaces.…”

mentioning

confidence: 99%

Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure

Soize¹,

Ghanem

2004

SIAM J. Sci. Comput.

531

410

View full text Add to dashboard Cite

Abstract. The basic random variables on which random uncertainties can in a given model depend can be viewed as defining a measure space with respect to which the solution to the mathematical problem can be defined. This measure space is defined on a product measure associated with the collection of basic random variables. This paper clarifies the mathematical structure of this space and its relationship to the underlying spaces associated with each of the random variables. Cases of both dependent and independent basic random variables are addressed. Bases on the product space are developed that can be viewed as generalizations of the standard polynomial chaos approximation. Moreover, two numerical constructions of approximations in this space are presented along with the associated convergence analysis.

show abstract

mentioning

confidence: 99%

Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure

Soize¹,

Ghanem

2004

SIAM J. Sci. Comput.

531

410

View full text Add to dashboard Cite

show abstract

“…x with probability 1 and where ϕ i (0, x) is the initial conditional density and I i (t) is the Innovations process, which is a Brownian motion, defined above. This can be shown, for instance, by following [17,Theorem 11.2.1]. Based on the consistency based approach to MFG [15], we now provide a decoupling result which demonstrates that the closed loop dynamics of each agent in the infinite population limit is approximated by MV SDEs in the partially observed setup.…”

Section: Introduction For Dynamical Games Of Mean Field Type It Has mentioning

confidence: 79%

Mean field game theory for agents with individual-state partial observations

Sen

Caines

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

Abstract. Subject to reasonable conditions, in large population stochastic dynamics games, where the agents are coupled by the system's mean field (i.e. the state distribution of the generic agent) through their nonlinear dynamics and their nonlinear cost functions, it can be shown that a best response control action for each agent exists which (i) depends only upon the individual agent's state observations and the mean field, and (ii) achieves a ǫ-Nash equilibrium for the system. In this work we formulate a class of problems where each agent has only partial observations on its individual state. We employ nonlinear filtering theory and the Separation Principle in order to analyze the game in the asymptotically infinite population limit. The main result is that the ǫ-Nash equilibrium property holds where the best response control action of each agent depends upon the conditional density of its own state generated by a nonlinear filter, together with the system's mean field. Finally, comparing this MFG problem with state estimation to that found in the literature with a major agent whose partially observed state process is independent of the control action of any individual agent, it is seen that, in contrast, the partially observed state process of any agent in this work depends upon that agent's control action.

show abstract

“…Although global Lipschitzcontinuity is postulated for the coefficients of the state/observation equation (2.7) (see Condition 2.3) it is well to note that the complexity of the drift term {γ 0 t } means that standard results on pathwise-uniqueness for classical Itô SDEs (see e.g. Theorem 5.1.1 of [Kallianpur [11], p.97]) do not apply directly to the SDE with unit covariance and drift {γ 0 t }. In fact, the global Lipschitzcontinuity of Condition 2.3 is essential for securing the representation of the drift {γ 0 t } due to Bhatt and Karandikar [3], but is otherwise used only rather indirectly in the proof of pathwise-uniqueness (at (4.98), (4.102), (4.105) and (4.108)).…”

Section: Remark 44mentioning

confidence: 99%

“…The simplest of these is the linear Gaussian model, for which the innovations conjecture may be established by functional-analytic methods which depend on the underlying linearity (see e.g. Kallianpur [11], Section 10.2). In the genuinely nonlinear and non-Gaussian case the problem of establishing the innovations conjecture with correlated data becomes much more challenging.…”

Section: Introductionmentioning

confidence: 99%