# Questions tagged [stochastic-filtering]

Stochastic filtering deals with the problem of finding the best estimate for a signal, given a noisy or incomplete observation.

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### Is the Kalman Filter computationally optimal for Kalman filtering?

Kalman filtering is known to be a recursive process that minimizes mean square error in linear problems. My question is: has anybody shown that this algorithm is computationally optimal, i.e. that you ...
1 vote
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### What's the autocovariance of 2 concurrent hidden states of a stochastic gaussian walk, using the Kalman filter?

A latent variable evolves as $x_t = x_{t-1} + e_t$ where $e_t$ has a gaussian distribution with 0 mean and a variance which defines the volatility of the overall process. Let $o_t$ be the noisy ...
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### Estimator for last value of an Ornstein-Uhlenbeck process given noisy measurement

I have a (discrete) Ornstein Uhlenbeck process $B_{n+1} = aB_n + X_n$, of which I have a noisy measurement $M_n = B_n + Y_n$ (where $Y$ is a white noise), of which I know the parameters. I am looking ...
1 vote
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### Conditions ensuring that conditional law of a process belongs to a given exponential family

Let $(X_t,Y_t)_{t\geq 0}$ be a pair of $\mathbb{R}^n$-(resp. $\mathbb{R}^m$)-valued stochastic processes on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$, ...
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### Preservation of the Markov Property under Conditioning

Let $(X_t,Z_t)_t$ be an $\mathbb{R}^{n}\times \mathbb{R}^m$-valued time-homogeneous Markov process on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$ with transition ...
1 vote
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### If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?

Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
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### The optimality of Kalman filtering

It is known that the Kalman filter estimates the state of the following system recursively. $$x_{k+1}=Ax_k+w_k, \ \ w_k \sim \mathcal{N}(0,Q)$$ $$y_k=Cx_k+v_k, \ \ v_k \sim \mathcal{N}(0,W)$$ In the ...
1 vote
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### Onsager--Machlup functional as the density across a mesh of discrete points

It is known that the ratio of the probability of infinitesimal tubes around paths of Itō diffusion processes converges to the Onsager--Machlup (OM) functional. I wonder whether the ratio of the joint ...
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### Kalman filter distribution of observation process

Let $(X_t,Y_t)$ be a pair of stochastic processes such that \begin{aligned} dX_t =& A_t X_t dt + C_t dW_t,\\ dY_t = & H_t X_t dt + K_tdB_t \end{aligned} for some non-random matrix-valued ...