Uniqueness and Robustness of Solution of Measure-Valued Equations of Nonlinear Filtering

Bhatt, Abhay G.; Kallianpur, G.; Karandikar, Rajeeva L.

doi:10.1214/aop/1176987808

Cited by 40 publications

(19 citation statements)

References 30 publications

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“…When ξ is a Markov diffusion, the above Zakai equation can also be obtained, under some additional hypotheses, when h is unbounded by means of the same arguments whenσ = 0, see (Baras, Blankenship, and Hopkins, 1983;Hopkins, 1982), and by different techniques whenσ does not depend on y and η is a onedimensional process, see (Florchinger, 1993). Furthermore, note that, under suitable hypotheses, a Zakai equation can be obtained when the state process ξ is a general Markov process (not necessarily given by (2)), and the observation process η is a diffusion process given by (3), with ω independent of ξ, see, e.g., (Bhatt, Kallianpur and Karandikar, 1995).…”

Section: Nonlinear Filtering For Markov Systems With Delayed Observatmentioning

confidence: 99%

Nonlinear Filtering for Markov Systems with Delayed Observations

Calzolari¹,

Florchinger²,

Nappo³

2009

International Journal of Applied Mathematics and Computer Science

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This paper deals with nonlinear filtering problems with delays, i.e., we consider a system (X , Y ), which can be represented by means of a system (X ,Ŷ ), in the sense that Yt =Ŷ a(t) , where a(t) is a delayed time transformation. We start with X being a Markov process, and then study Markovian systems, not necessarily diffusive, with correlated noises. The interest is focused on the existence of explicit representations of the corresponding filters as functionals depending on the observed trajectory. Various assumptions on the function a(t) are considered.

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Section: Nonlinear Filtering For Markov Systems With Delayed Observatmentioning

confidence: 99%

Nonlinear Filtering for Markov Systems with Delayed Observations

Calzolari¹,

Florchinger²,

Nappo³

2009

International Journal of Applied Mathematics and Computer Science

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“…Let the coordinate processes on D IR and C IR be denoted once more by (ξ t (·)) and (β t (·)) respectively. Let P (1) µ be the unique measure on (D IR , B(D IR )) which satisfies for E 1 , · · · , E n ∈ B(IR) and −∞ < t 1 < t 2 · · · < t n < ∞,…”

Section: Proofmentioning

confidence: 99%

“…µ ⊗ Q (1) . In this section, we will consider the coordinate processes (ξ t ), (β t ) to be defined on the product space (Ω 1 , B(Ω 1 ), R (1) µ ).…”

Section: Proofmentioning

confidence: 99%

“…The proof in [9] requires showing that (π (0) t ) and (π (1) t ) are Markov with semigroup T t which is shown in [9] by appealing to the uniqueness of the solution to Kushner-Stratonovich equation. Though we don't need the Markov properties for our proof above, nevertheless using the Feller property of T t , they are seen to hold as argued below.…”

Section: Proofmentioning

confidence: 99%

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Markov Property and Ergodicity of the Nonlinear Filter

Bhatt¹,

Budhiraja²,

Karandikar³

2000

SIAM J. Control Optim.

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In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair: (signal,filter) are Feller-Markov processes.The state space of the signal is allowed to be non locally compact and the observation function: h can be unbounded. Our proofs in contrast to those of Kunita(1971Kunita( ,1991, Stettner(1989) do not depend upon the uniqueness of the solutions to the filtering equations. We then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process. These results extend those of Kunita and Stettner, which hold for locally compact state space and bounded h, to our general framework. Finally we show that the recent results of Ocone-Pardoux [11] on asymptotic stability of the nonlinear filter, which use the Kunita-Stettner setup, hold for the general situation considered in this paper.

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“…Other references in a similar framework are given, for instance, by Refs. [4][5][6][7][8]. Subsequently also the case of counting process or marked point process observation has been considered (see Refs.…”

Section: Introductionmentioning

confidence: 99%

Recent Advances in Nonlinear Filtering with a Financial Application to Derivatives Hedging under Incomplete Information

Ceci¹,

Colaneri²

2017

Bayesian Inference

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In this chapter, we present some recent results about nonlinear filtering for jump diffusion signal and observation driven by correlated Brownian motions having common jump times. We provide the Kushner-Stratonovich and the Zakai equation for the normalized and the unnormalized filter, respectively. Moreover, we give conditions under which pathwise uniqueness for the solutions of both equations holds. Finally, we study an application of nonlinear filtering to the financial problem of derivatives hedging in an incomplete market with partial observation. Precisely, we consider the risk-minimizing hedging approach. In this framework, we compute the optimal hedging strategy for an informed investor and a partially informed one and compare the total expected squared costs of the strategies.

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Uniqueness and Robustness of Solution of Measure-Valued Equations of Nonlinear Filtering

Cited by 40 publications

References 30 publications

Nonlinear Filtering for Markov Systems with Delayed Observations

Nonlinear Filtering for Markov Systems with Delayed Observations

Markov Property and Ergodicity of the Nonlinear Filter

Recent Advances in Nonlinear Filtering with a Financial Application to Derivatives Hedging under Incomplete Information

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