About Gaussian filtering problems with general exponential quadratic criteria

Filtering problems with exponential criteria for general Gaussian signals

The explicit solution of the discrete time filtering problems with exponential criteria for a general Gaussian signal is obtained through an approach based on a conditional Cameron-Martin type formula. This key formula is derived for conditional expectations of exponentials of some quadratic forms of Gaussian sequences. The formula involves conditional expectations and conditional covariances in some auxiliary optimal risk-neutral filtering problem which is used in the proof. Closed form recursions of Volterra type for these ingredients are provided. Particular cases for which the results can …

Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering

We take advantage of recent and new results on optimal quantization theory to improve the quadratic optimal quantization error bounds for backward stochastic differential equations (BSDE) and nonlinear filtering problems. For both problems, a first improvement relies on a Pythagoras like Theorem for quantized conditional expectation. While allowing for some locally Lipschitz functions conditional densities in nonlinear filtering, the analysis of the error brings into playing a new robustness result about optimal quantizers, the so-called distortion mismatch property: $L^r$-quadratic optimal qu…

A Bernstein-type inequality for stochastic processes of quadratic forms of Gaussian variables

We introduce a Bernstein-type inequality which serves to uniformly control quadratic forms of gaussian variables. The latter can for example be used to derive sharp model selection criteria for linear estimation in linear regression and linear inverse problems via penalization, and we do not exclude that its scope of application can be made even broader.

离散和连续时间下部分观测多项式过程的最优线性滤波

Optimal linear filtering of partially observed polynomial processes in discrete and continuous time

针对部分观测的多项式过程，利用其与高斯过程在二阶矩上的不可区分性，构造高斯等价过程并显式计算最优线性滤波器、预测器和平滑器。

The reproducing kernel Hilbert spaces underlying linear SDE Estimation, Kalman filtering and their relation to optimal control

It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories, allowing for a convenient handling of state constraints and meeting points. We now extend this viewpoint to estimation problems where it is known that kernels are the covariances of stochastic processes. Here, the Markovian Gaussian processes stem from the linear stochastic differential equations describing the continuous-time dynamics and observations. Taking…

Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in [4], we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new, fully non linear …