Nonlinear filtering for jump diffusion observations. (English) Zbl 1251.93123

Ceci, Claudia; Colaneri, Katia

Adv. Appl. Probab. 44, No. 3, 678-701 (2012).

Summary: We deal with the filtering problem of a general jump diffusion process, \(X\), when the observation process, \(Y\), is a correlated jump diffusion process having common jump times with \(X\). In this setting, at any time \(t\) the \(\sigma \)-algebra \(\mathcal F^{Y}_{t}\) provides all the available information about \(X_{t}\), and the central goal is to characterize the filter, \(\pi_{t}\), which is the conditional distribution of \(X_{t}\) given observations \(\mathcal F^{Y}_{t}\). To this end, we prove that \(\pi_{t}\) solves the Kushner-Stratonovich equation and, by applying the filtered martingale problem approach (see T. G. Kurtz, D. L. Ocone [”Unique characterization of conditional distributions in nonlinear filtering”, Ann. Probab. 16, No.1, 80-107 (1988; Zbl 0655.60035)]), that it is the unique weak solution to this equation. Under an additional hypothesis, we also provide a pathwise uniqueness result.

MSC:

93E11	Filtering in stochastic control theory
60J75	Jump processes (MSC2010)
60J60	Diffusion processes

Keywords:

filtering; jump diffusion process

Citations:

Zbl 0655.60035

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