The work concerns nonlinear filtering problems of stochastic differential equations with correlated Lévy noises. First, we establish the Kushner–Stratonovich and Zakai equations through martingale representation theorems and the Kallianpur–Striebel formula. Second, we show the pathwise uniqueness and uniqueness in joint law of weak solutions for the Zakai equation. Finally, we investigate the uniqueness in joint law of weak solutions to the Kushner–Stratonovich equation.

中文翻译：

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具有相关Lévy噪声的随机微分方程的非线性滤波

这项工作涉及具有相关Lévy噪声的随机微分方程的非线性滤波问题。首先，我们通过mar表示法和Kallianpur-Striebel公式建立Kushner-Stratonovich和Zakai方程。其次，我们显示了Zakai方程的弱解的联合定律中的路径唯一性和唯一性。最后，我们研究了Kushner–Stratonovich方程的弱解在联合定律中的唯一性。