A Simple Proof of Uniqueness for Kushner and Zakai Equations

Rozovskiĭ, B. L.

doi:10.1016/b978-0-12-481005-1.50029-6

Cited by 24 publications

(18 citation statements)

References 7 publications

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“…The goal of this paper is to use a unique solvability result from the theory of parabolic integro-differential equation to prove the uniqueness of the solution to the Zakai equation. The result is an extension of the result obtained by Rozovskii [16] to signal process of Itô-Lévy type with small and large jump noises.…”

Section: Introductionmentioning

confidence: 80%

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A stochastic Lagrangian particle model and nonlinear filtering for three dimensional Euler flow with jumps

Sritharan¹,

Xu²

2011

COSA

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Section: Introductionmentioning

confidence: 80%

“…In this section we prove the uniqueness of measure-valued solutions for the FKK and Zakai equations adapting an idea due to Rozovskii [16]. This method of proving existence and uniqueness is also applied in infinite dimensional context [19], [9].…”

Section: Uniqueness Of Solution To Zakai Equationmentioning

confidence: 99%

A stochastic Lagrangian particle model and nonlinear filtering for three dimensional Euler flow with jumps

Sritharan¹,

Xu²

2011

COSA

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“…For all f ∈ cyl and 0 ≤ t ≤ T the weak FKK equation holds: Let us now discuss existence and uniqueness of measure-valued solutions for the FKK and Zakai equations (see [3,30,32]). Proof.…”

Section: Existence and Uniqueness Of Nonlinear Filtering Equationsmentioning

confidence: 99%

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

Fernando

Sritharan

2013

Stochastic Analysis and Applications

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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.

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“…Before making the filter computation, it is important to make sure the FKK and DMZ equations admit a unique solution. The interested readers may refer to Rozovskii (1991) for the results on uniqueness.…”

Section: Stochastic Filtering Equationmentioning

confidence: 99%

A recursive robust Bayesian estimation in partially observed financial market

Huang

2007

Appl. Math. (Warsaw)

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A RECURSIVE ROBUST BAYESIAN ESTIMATION IN PARTIALLY OBSERVED FINANCIAL MARKETAbstract. I propose a nonlinear Bayesian methodology to estimate the latent states which are partially observed in financial market. The distinguishable character of my methodology is that the recursive Bayesian estimation can be represented by some deterministic partial differential equation (PDE) (or evolution equation in the general case) parameterized by the underlying observation path. Unlike the traditional stochastic filtering equation, this dynamical representation is continuously dependent on the underlying observation path and thus it is robust to the modeling errors. Moreover, its advantages in financial econometrics are also discussed.1. Introduction. The cutting-edge works of Merton (1969Merton ( , 1971 and Black and Scholes (1973) open some frontiers of stochastic finance which models the financial states by stochastic processes such as Markov processes, or more specially, the Itô diffusion processes. Among them, the most canonical example is the Black-Scholes formula where the stock price is assumed to be some geometric Brownian motion (GBM). Through judicious choices of its parameters, this model provides enough flexibility to accommodate a wide range of dynamics of financial variables we are interested in. The motivation of my work stems from the real-world phenomenon that in many applications, the financial states of interest are often unobservable directly from the market and corrupted with some noise. Such states bear the name of "latent states". For this reason, the parametric specification of the underlying model is just the preliminary step while in the next step, it is of 2000 Mathematics Subject Classification: 60G35, 60J60, 62M05, 62P05, 91B28, 91B70.

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A Simple Proof of Uniqueness for Kushner and Zakai Equations

Cited by 24 publications

References 7 publications

A stochastic Lagrangian particle model and nonlinear filtering for three dimensional Euler flow with jumps

A stochastic Lagrangian particle model and nonlinear filtering for three dimensional Euler flow with jumps

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

A recursive robust Bayesian estimation in partially observed financial market

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