| Abstract: |
| We focus on the asymptotic behavior of the optimal filter where both signal and observation processes are driven by L\`evy noises. Indeed, we study large deviations for the case where the signal-to-noise ratio is small by considering weak convergence arguments. To that end, we first prove the uniqueness of the solution of the controlled Zakai and Kushner-Stratonovich equations, using a method which transforms the associated equations into SDEs in an appropriate Hilbert space. Taking into account the controlled analogue of Zakai and Kushner-Stratonovich equations, respectively, the large deviation principle follows by employing the existence, uniqueness and tightness of the solutions. |
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