On the unnormalized solution of the filtering problem with counting process observations

Kliemann, Wolfgang; Koch, G.; Marchetti, Federico

doi:10.1109/18.59936

Cited by 69 publications

(55 citation statements)

References 12 publications

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“…Calzolari et al We stress that in this model the state process is not necessarily Markovian, while the overall system is Markovian. The same holds for the model studied in (Kliemann et al, 1990), where the state is a jump-diffusion process and the observation is a counting process. Recently, nonlinear filtering has been applied in financial problems in the framework of Bayesian analysis.…”

mentioning

confidence: 64%

“…For the diffusion case this problem, initiated in (Clark, 1978;Davis, 1982) when considering feasible filters continuous with respect to the trajectory of the observation process (i.e., robust filters), has been studied by many authors in various frameworks. When dealing with counting observations this problem was studied in (Brémaud, 1981) for the doubly stochastic case, and in (Kliemann et al, 1990) for more general systems.…”

Section: Lemma 1 Assume That the Function A(·) Is A Continuous Delaymentioning

confidence: 99%

See 1 more Smart Citation

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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mentioning

confidence: 64%

Section: Lemma 1 Assume That the Function A(·) Is A Continuous Delaymentioning

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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“…In Theorem 2.2 of Kliemann et al (1990) strong existence and uniqueness is proved under growth conditions on b(x, y), y)) and under the additional assumption that for every fixed y the SDE dX t = b(X t , y)ds + σ(X t , y)dW s has a unique weak solution which is moreover a Feller process. Alternatively, one can impose growth and Lipschitz conditions on the data of the problem; see for instance Appendix 1, Section 4 of Ceci & Gerardi (2006).…”

Section: Remark 21 (Sufficient Conditions For A1)mentioning

confidence: 99%

“…First results can be found in Grigelionis (1972); the papers Kliemann, Koch & Marchetti (1990) and Ceci & Gerardi (2006) are concerned with scalar observations described by a pure jump process. The recent paper Cvitanic, Liptser & Rozovski (2006) on the other hand treats the filtering problem for a very general marked point process model but without common jumps of the state-and the observation process.…”

Section: Introductionmentioning

confidence: 99%

Pricing credit derivatives under incomplete information: a nonlinear-filtering approach

Frey

Runggaldier

2010

Finance Stoch

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This paper considers a general reduced form pricing model for credit derivatives where default intensities are driven by some factor process X. The process X is not directly observable for investors in secondary markets; rather, their information set consists of the default history and of noisy price observation for traded credit products. In this context the pricing of credit derivatives leads to a challenging nonlinear filtering problem. We provide recursive updating rules for the filter, derive a finite dimensional filter for the case where X follows a finite state Markov chain and propose a novel particle filtering algorithm. A numerical case study illustrates the properties of the proposed algorithms.

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“…Subsequently also the case of counting process or marked point process observation has been considered (see Refs. [9][10][11][12][13][14] and reference therein). A more recent literature contains the case of mixed-type observations (marked point processes and diffusions or jump-diffusion processes), see, for, example, 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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On the unnormalized solution of the filtering problem with counting process observations

Cited by 69 publications

References 12 publications

Nonlinear Filtering for Markov Systems with Delayed Observations

Nonlinear Filtering for Markov Systems with Delayed Observations

Pricing credit derivatives under incomplete information: a nonlinear-filtering approach

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

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