# 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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### Hunting an invisible target

An invisible target on the integer line starts at $0$. On each round it either stays put, moves to the left or moves to the right by $1$ with probability $\frac{1}{3}$ each. You are then asked to ...

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### How to optimally bet on a biased coin?

A number $p$ is drawn uniformly at random from $[0, 1]$. You are then given a biased coin that turns up heads with probability $p$, but the number $p$ is not known to you.
You start with a total ...

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### Filtering where the observed process has noise coefficient depending on state

In continuous-time, the state process is assumed to follow
$$dX_t = f(X_t, t)dt + g(X_t, t)dW^1_t$$
and the observation process is usually assumed to follow something like
$$dZ_t = h(X_t, t)dt + dW^...

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### Lipschitzness of conditional law of a stochastic filtering problem wrt the Wasserstein distance

Let $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be a pair of stochastic processes taking values in $\mathbb{R}^n$ and in $\mathbb{R}^m$; defined on a filtered probability spaces $(\Omega,\mathcal{F},(\...

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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 ...

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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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### What are the optimal times to sample a process?

Let $X$ be a one dimensional Ito diffusion given by
$$X_t = b \,W_t$$
where $b$ is a constant, and $W$ is a standard Brownian motion.
Let $B$ be another Brownian motion independent of $W$, and define ...

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### Filtering the period and amplitude of a sine wave corrupted by noise

Let $W$ be a standard Brownian motion and $\mathcal F_t$ its natural filtration.
Suppose $\theta, A$ are positive $L^1$ random variables independent of $\mathcal F_t$.
Let $Y_t$ be the process
$$Y_t :=...

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### When enlarging a filtration makes a stochastic processes into a solution to an SDE

Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...

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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 ...

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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 ...

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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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### Parseval's equivalent of Norm that includes a Projection matrix

I need to optimize the norm, ${\bf x}^H {\bf P}_{\bf B} {\bf x} $, where, ${\bf P}_{\bf B} = {\bf B}^H({\bf B} {\bf B}^H)^{-1} {\bf B}$, ${\bf B}$ is a known $M \times N$ matrix, with $M < N$ and $...

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### Kernel of the adjoint of the infinitesimal generator of Levy SDE

Consider S.D.Es driven by a combination of Brownian and non-Brownian Levy noise (like say Gamma). Then we know that the flow of the density of the S.D.E variable is given by the adjoint of the ...

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### When does the predictable $\sigma$-algebra $\mathcal{P}$ coincide with the optional $\sigma$-algebra $\mathcal{O}$?

The setup of my question is the following: Suppose that we have a measurable space $(\Omega,\mathcal{F})$ and a filtration $\mathbf{F} = (\mathcal{F}_t)_{t \geq 0}$ on it. Let $\mathcal{P}(\mathbf{F})$...

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### Continuous version of conditional probability distributions $( \mathcal{L}(X_t | \mathcal{G}) )_{t \geq 0}$ if $(X_t)_{t \geq 0}$ is continuous?

Let me first explain the setup:
Let $(X_t)_{t \geq 0}$ be a stochastic process on some probability space $(\Omega,\mathcal{F},P)$ with values in a complete and separable metric space $E$ (e.g. $E = \...

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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 ...

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### Extended Kalman Filter and its State Transition Matrix

Sorry for what might be a long post, I want to give background.
Initially I had regular Kalman filter, and the state model was defined by Newtonian kinematics, with initial position 0 and speed of 2. ...

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### Filtration exercise

I am struggling with 1.7 exercise from the Karatzas, Shreve "Brownian motion and stoch. calulus".
Denote by $\mathcal{F}^X_{t_0}$ the natural filtration corresponding to a process $X:[0,\infty)\times ...

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### Ergodicity of differentiated processes

Let $S$ be a vector space, and $X$ a jointly-measurable random process/field with two parameters:
$$ X: [0,\infty)\times\mathbb{R}\times\Omega\to S,$$
i.e. $X_{t,\theta}:\Omega\to S$ are random ...

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### A conjecture in rate distortion theory and stochastic filtering

Let $(X_t)_{t\in T}$ be a stationary random process with known and fixed law $P_X$ describing a dynamic source.
This source is to be encoded real-time by an encoder $e$ into an encoded message $E_t$ ...

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### Why would one work with Kushner-FKK equation over Zakai equation?

In stochastic filtering you are interested in a process called the optimal filter $\pi_t$ which is a probability measure(d stochastic process). You can consider the unnormalized version $V_t$.
The ...

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### Filtering Mixed Discrete and Continous

Suppose I have signal process $\lambda_t$ following the dynamics
\begin{equation}
\begin{aligned}
\zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\
\xi_t&=\mu^{\xi}(...