Convergence in Nonlinear Filtering for Stochastic Delay Systems

Calzolari, Antonella; Florchinger, Patrick; Nappo, Giovanna

doi:10.1137/050646135

Cited by 10 publications

(11 citation statements)

References 34 publications

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“…The above result generalizes Proposition 4.2 in Calzolari et al [10], since it does not require the boundedness of the coefficients and (there denoted by a and b, respectively). We point out that Proposition 3 requires condition A1 to hold also for k = 2, while k = 1 is sufficient when t is bounded by a deterministic constant.…”

Section: Propositionsupporting

confidence: 49%

“…Proposition 3 can be proved in much the same way as the above quoted Proposition 4.2 from Section 5 of Calzolari et al [10], which deals with the onedimensional case; the boundedness assumption on the coefficients there is made only to get the upper bound (20). In the proof, which we briefly sketch below for the ease of the reader, we also consider the continuous Euler-Maruyama scheme, that is, the processes Z n = Z n t t∈ 0 T defined by Z n t = t for − ≤ t ≤ 0 and, for 0 ≤ t ≤ T , dZ n t = h · s/h h· s/h X n ds + h · s/h h· s/h X n dW s…”

Section: Propositionmentioning

confidence: 77%

“…This problem is actually the motivation of our interest in the modulus of continuity, the aim being to generalise some strong convergence results obtained in Calzolari et al [10].…”

Section: Introductionmentioning

confidence: 96%

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On the Moments of the Modulus of Continuity of Itô Processes

Fischer

Nappo

2009

Stochastic Analysis and Applications

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The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Itô processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Itô processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm

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Section: Propositionsupporting

confidence: 49%

Section: Propositionmentioning

confidence: 77%

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On the Moments of the Modulus of Continuity of Itô Processes

Fischer

Nappo

2009

Stochastic Analysis and Applications

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“…As an example we can consider the simple stochastic delay system of Remark 1.3. Taking into account this remark, the computation of the filter is equivalent to the computation of the filter of the stochastic delay system (13) and (14), for which the convergence result obtained in [5] can be applied. To our knowledge there are only two other papers dealing with the approximation of nonlinear filtering for delayed diffusion systems: Chang [10], and Twardowska, Marnik and Pas lawaska-Po luniak [22].…”

Section: Remark 13 the Case When The State Process Solves The Stochmentioning

confidence: 99%

Approximation of Nonlinear Filters for Markov Systems with Delayed Observations

Calzolari¹,

Florchinger²,

Nappo³

2006

SIAM J. Control Optim.

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The aim of this paper is to give some approximation results for a class of nonlinear filtering problems with delay in the observation. First, we point out some general results on the approximation problem for the filter in nonlinear filtering. In particular we give a general procedure to obtain some upper bounds for the different approximations we consider. This procedure is then applied in the case of nonlinear filtering problems with delay (X, Y ), which can be represented by means of a Markov system (X,Ŷ ), in the sense that Yt =Ŷ a(t) . Finally these upper bounds are computed explicitly in the particular case of Markov jump process with counting observations.

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“…This inspiring example can also be viewed as a particular case of the delay systems considered in (Calzolari et al, 2007).…”

mentioning

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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Convergence in Nonlinear Filtering for Stochastic Delay Systems

Cited by 10 publications

References 34 publications

On the Moments of the Modulus of Continuity of Itô Processes

On the Moments of the Modulus of Continuity of Itô Processes

Approximation of Nonlinear Filters for Markov Systems with Delayed Observations

Nonlinear Filtering for Markov Systems with Delayed Observations

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