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ABIM
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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 functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear that $X_t$ follows a (multidimensional) Ornstein-Uhlembeck process so is distributed according to this wiki post. However, what is the distribution of $Y_t$? It's not Obviously, it's Gaussian (see standard proofs on Kalman filtering) so the meat of the question is...what is itits mean and co-variance?

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 functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear that $X_t$ follows a (multidimensional) Ornstein-Uhlembeck process so is distributed according to this wiki post. However, what is the distribution of $Y_t$? It's not Gaussian... is it?

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 functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear that $X_t$ follows a (multidimensional) Ornstein-Uhlembeck process so is distributed according to this wiki post. However, what is the distribution of $Y_t$? Obviously, it's Gaussian (see standard proofs on Kalman filtering) so the meat of the question is...what is its mean and co-variance?

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YCor
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Kalman Filter Distributionfilter distribution of Observation Processobservation 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 functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear that $X_t$ follows a (mult-dimensionalmultidimensional) Ornstein-UlembeckUhlembeck process so is distributed according to this wiki post. However, what is the distribution of $Y_t$? It's not Gaussian... is it?

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 functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear that $X_t$ follows a (mult-dimensional) Ornstein-Ulembeck process so is distributed according to this wiki post. However, what is the distribution of $Y_t$? It's not Gaussian... is it?

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 functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear that $X_t$ follows a (multidimensional) Ornstein-Uhlembeck process so is distributed according to this wiki post. However, what is the distribution of $Y_t$? It's not Gaussian... is it?

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ABIM
ABIM
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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 functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear that $X_t$ follows a (mult-dimensional) Ornstein-Ulembeck process so is distributed according to this wiki post. However, what is the distribution of $Y_t$? It's not Gaussian... is it?