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

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

Zbl 0655.60035
Full Text:

### References:

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