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We consider several problems where the approach developed in this work can be useful for their solutions. The first one is related to localizing a source on a plane using observations from K detectors. It is assumed that the detectors observe Gaussian signals from the source in the presence of white Gaussian noise. The unknown parameters are the coordinates of the source. Several variations of the problem are studied. First, we assume that the source is fixed and starts emitting at some known moment ( \(t=0\) ). The position of the source is determined by the times required for the signals to reach the detectors. The properties of the estimators of the position depend significantly on the form of the signal front (smooth function, function with cusp-type singularity, or discontinuous function), and we describe the properties of the MLE and BE estimators for such models in the asymptotic of small noise in both equations. Then, we consider a similar problem with a moving source and discuss other possible generalizations. In the next model the deterministic signal depending on some unknown finite dimensional parameter is present in the hidden equation, and we study the problem of its estimation. The partially observed system related to the security prices process depending on unknown parameters is considered in the Sect. 7.3. The equations of adaptive filtering and adaptive extrapolation are proposed. Further the change point problems for partially observed linear system are discussed in three situations: change point in the observation equations, in the state equation, and in both equations. The properties of different estimators and the possibilities of adaptive filtering are discussed. In the last section the problem concerns the approximation of the solution of backward stochastic differential equation (BSDE) in the situation where the solution of the forward equation with unknown volatility is observed in WGN. The proposed approximation algorithm is based on the One-step MLE-process. The question of asymptotic optimality of such approximation is discussed.

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  • 发表日期：

    2025年

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  • Yury A. Kutoyants

摘要

We consider several problems where the approach developed in this work can be useful for their solutions. The first one is related to localizing a source on a plane using observations from K detectors. It is assumed that the detectors observe Gaussian signals from the source in the presence of white Gaussian noise. The unknown parameters are the coordinates of the source. Several variations of the problem are studied. First, we assume that the source is fixed and starts emitting at some known moment ( \(t=0\) ). The position of the source is determined by the times required for the signals to reach the detectors. The properties of the estimators of the position depend significantly on the form of the signal front (smooth function, function with cusp-type singularity, or discontinuous function), and we describe the properties of the MLE and BE estimators for such models in the asymptotic of small noise in both equations. Then, we consider a similar problem with a moving source and discuss other possible generalizations. In the next model the deterministic signal depending on some unknown finite dimensional parameter is present in the hidden equation, and we study the problem of its estimation. The partially observed system related to the security prices process depending on unknown parameters is considered in the Sect. 7.3. The equations of adaptive filtering and adaptive extrapolation are proposed. Further the change point problems for partially observed linear system are discussed in three situations: change point in the observation equations, in the state equation, and in both equations. The properties of different estimators and the possibilities of adaptive filtering are discussed. In the last section the problem concerns the approximation of the solution of backward stochastic differential equation (BSDE) in the situation where the solution of the forward equation with unknown volatility is observed in WGN. The proposed approximation algorithm is based on the One-step MLE-process. The question of asymptotic optimality of such approximation is discussed.