# 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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### A nonlinear PDE on a matrix domain involving a maximum eigenvalue operator

In a research problem in optimal control
$^{\ast}$ I came across the following nonlinear first-order PDE:
$$\frac{\partial V}{\partial t}=\max\text{eig}\left[\sum_{i=1}^m P_i\frac{\partial V}{\partial ...

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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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### Karhunen-Loeve expansion of autoregressive process

I was wondering whether anyone knows of a source for the Karhunen-Loeve expansion (KLE) of an (stable) autoregressive process (ideally of arbitrary order N) driven by iid gaussian noise, i.e. a ...

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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}(...