In filtering theory

Filtering problem (stochastic processes)

In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some observations of that system...

 the Kushner equation (after Harold Kushner) is an equation for the conditional probability

Conditional probability

In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

 density

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 of the state of a stochastic

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 non-linear dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

, given noisy measurements of the state. It therefore provides the solution of the nonlinear filtering problem in estimation theory

Estimation theory

Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...

. The equation is sometimes referred to as the Stratonovich–Kushner (or Kushner–Stratonovich) equation.

Overview

Assume the state of the system evolves according to


and a noisy measurement of the system state is available:


where dw, dv are independent Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

es. Then the conditional probability density p(x, t) of the state at time t is given by the Kushner equation:


where is the Kolmogorov Forward operator.

The term is the innovation

Innovation (signal processing)

In time series analysis — as conducted in statistics, signal processing, and many other fields — the innovation is the difference between the observed value of a variable at time t and the optimal forecast of that value based on information available prior to time t...

i.e. the difference between the measurement and its expected value.