Filtering problem (stochastic processes)

Consider a probability space (Ω, Σ, P) and suppose that the (random) state Y_t in n-dimensional Euclidean space Rⁿ of a system of interest at time t is a random variable Y_t : Ω → Rⁿ given by the solution to an Itō stochastic differential equation of the form

${\displaystyle \mathrm {d} Y_{t}=b(t,Y_{t})\,\mathrm {d} t+\sigma (t,Y_{t})\,\mathrm {d} B_{t},}$

where B denotes standard p-dimensional Brownian motion, b : [0, +∞) × Rⁿ → Rⁿ is the drift field, and σ : [0, +∞) × Rⁿ → R^n×p is the diffusion field. It is assumed that observations H_t in R^m (note that m and n may, in general, be unequal) are taken for each time t according to

${\displaystyle H_{t}=c(t,Y_{t})+\gamma (t,Y_{t})\cdot {\mbox{noise}}.}$

Adopting the Itō interpretation of the stochastic differential and setting

${\displaystyle Z_{t}=\int _{0}^{t}H_{s}\,\mathrm {d} s,}$

this gives the following stochastic integral representation for the observations Z_t:

${\displaystyle \mathrm {d} Z_{t}=c(t,Y_{t})\,\mathrm {d} t+\gamma (t,Y_{t})\,\mathrm {d} W_{t},}$

where W denotes standard r-dimensional Brownian motion, independent of B and the initial condition Y₀, and c : [0, +∞) × Rⁿ → Rⁿ and γ : [0, +∞) × Rⁿ → R^n×r satisfy

${\displaystyle {\big |}c(t,x){\big |}+{\big |}\gamma (t,x){\big |}\leq C{\big (}1+|x|{\big )}}$

for all t and x and some constant C.

The filtering problem is the following: given observations Z_s for 0 ≤ s ≤ t, what is the best estimate Ŷ_t of the true state Y_t of the system based on those observations?

By "based on those observations" it is meant that Ŷ_t is measurable with respect to the σ-algebra G_t generated by the observations Z_s, 0 ≤ s ≤ t. Denote by K = K(Z, t) be collection of all Rⁿ-valued random variables Y that are square-integrable and G_t-measurable:

${\displaystyle K=K(Z,t)=L^{2}(\Omega ,G_{t},\mathbf {P}$ ;\mathbf {R} ^{n}).}

By "best estimate", it is meant that Ŷ_t minimizes the mean-square distance between Y_t and all candidates in K:

${\displaystyle \mathbf {E} \left[{\big |}Y_{t}-{\hat {Y}}_{t}{\big |}^{2}\right]=\inf _{Y\in K}\mathbf {E} \left[{\big |}Y_{t}-Y{\big |}^{2}\right].\qquad {\mbox{(M)}}}$

The space K(Z, t) of candidates is a Hilbert space, and the general theory of Hilbert spaces implies that the solution Ŷ_t of the minimization problem (M) is given by

${\displaystyle {\hat {Y}}_{t}=P_{K(Z,t)}{\big (}Y_{t}{\big )},}$

where P_K(Z,t) denotes the orthogonal projection of L²(Ω, Σ, P; Rⁿ) onto the linear subspace K(Z, t) = L²(Ω, G_t, P; Rⁿ). Furthermore, it is a general fact about conditional expectations that if F is any sub-σ-algebra of Σ then the orthogonal projection

${\displaystyle P_{K}:L^{2}(\Omega ,\Sigma ,\mathbf {P}$ ;\mathbf {R} ^{n})\to L^{2}(\Omega ,F,\mathbf {P} ;\mathbf {R} ^{n})}

is exactly the conditional expectation operator E[·|F], i.e.,

${\displaystyle P_{K}(X)=\mathbf {E} {\big [}X{\big |}F{\big ]}.}$

Hence,

${\displaystyle {\hat {Y}}_{t}=P_{K(Z,t)}{\big (}Y_{t}{\big )}=\mathbf {E} {\big [}Y_{t}{\big |}G_{t}{\big ]}.}$

This elementary result is the basis for the general Fujisaki-Kallianpur-Kunita equation of filtering theory.

• The Smoothing problem is closely related to the Filtering problem.
• Filtering (disambiguation)
• Not to be confused with Filter (signal processing)
• Kalman filter most famous filtering algorithm in the sense of 'filtering problem' and 'smoothing problem'.
• Smoothing (not to be confused with the Smoothing problem)
• Smoothing (disambiguation)

• Jazwinski, Andrew H. (1970). Stochastic Processes and Filtering Theory. New York: Academic Press. ISBN 0-12-381550-9.
• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 6.1)