KushnerEquation

Consider a stochastic system given by:

$$ dx(t) = f(x(t), t) dt + G(x(t), t) dw(t), $$

$$ dy(t) = h(x(t), t) dt + v(t), $$

where $x(t)$ is the state vector, $f(x(t), t)$ and $G(x(t), t)$ are system dynamics and noise functions, respectively, $dw(t)$ is the Wiener process, $y(t)$ is the observation vector, $h(x(t), t)$ is the observation function, and $v(t)$ is the observation noise.

The main goal is to estimate the conditional pdf $p(x(t) | y(\tau), \tau \le t)$ for the state $x(t)$ given the noisy observations $y(\tau)$ up to time $t$. The Kushner equation, in this case, is given by:

Consider a stochastic system given by:

$$ dx(t) = f(x(t), t) dt + G(x(t), t) dw(t), $$

$$ dy(t) = h(x(t), t) dt + v(t), $$

where $x(t)$ is the state vector, $f(x(t), t)$ and $G(x(t), t)$ are system dynamics and noise functions, respectively, $dw(t)$ is the Wiener process, $y(t)$ is the observation vector, $h(x(t), t)$ is the observation function, and $v(t)$ is the observation noise.

The main goal is to estimate the conditional pdf $p(x(t) | y(\tau), \tau \le t)$ for the state $x(t)$ given the noisy observations $y(\tau)$ up to time $t$. The Kushner equation, in this case, is given by:

$$ \begin{aligned} dp(x(t)|y(\tau), \tau\leq t) &= \left[ \nabla_x \cdot \left(f(x(t),t) - G(x(t),t) \nabla_x \log(p(x(t)|y(\tau), \tau\leq t))\right) + \frac{1}{2} \operatorname{Tr}\left(G(x(t),t) G'(x(t),t) \nabla_x^2\right) \right] p(x(t)|y(\tau), \tau\leq t) dt \\ &\quad + \left[ h(x(t),t) - \int h(x',t) p(x'(t)|y(\tau), \tau\leq t) dx' \right] p(x(t)|y(\tau), \tau\leq t) dy(t), \end{aligned} $$

where $\nabla_x$ denotes the gradient with respect to $x$, and $\operatorname{Tr}()$ is the trace of a matrix.