View source for Duncan-Mortensen-Zakai equation

''DMZ equation''

An equation whose solution is the unnormalized [[Conditional probability|conditional probability]] density function for a non-linear filtering problem. The non-linear filtering problem was motivated by the solution of the linear filtering problem, especially in [[#References|[a3]]], where the signal or state process is modeled by the solution of a linear differential equation with a Gaussian white noise input (the formal derivative of a [[Brownian motion|Brownian motion]] or [[Wiener process|Wiener process]]), so the signal process is a Gauss–Markov process. The differential of the observation process, the process from which an estimate is made of the state process, is a linear transformation of the signal plus Gaussian white noise. This linear filtering problem was motivated by one where the infinite past of the observations [[#References|[a5]]], [[#References|[a7]]] is required. The non-linear filtering problem is described by a signal process that is the solution of a non-linear differential equation with a Gaussian white noise input and an observation process whose differential is a non-linear function of the signal process plus a Gaussian white noise. The precise description of such a filtering problem requires the theory of stochastic differential equations (e.g., [[#References|[a4]]]; see also [[Stochastic differential equation|Stochastic differential equation]]).

Before introducing the Duncan–Mortensen–Zakai equation [[#References|[a1]]], [[#References|[a6]]], [[#References|[a8]]], it is necessary to describe precisely a non-linear filtering problem. A basic filtering problem is described by two stochastic processes (cf. [[Stochastic process|Stochastic process]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203001.png" />, which is called the signal or state process and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203002.png" />, which is called the observation process. These two processes satisfy the following stochastic differential equations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203004.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d1203009.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030015.png" /> are independent standard Brownian motions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030017.png" />, respectively. The stochastic processes are defined on a fixed [[Probability space|probability space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030018.png" /> with a filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030019.png" /> (cf. also [[Stochastic processes, filtering of|Stochastic processes, filtering of]]). Some smoothness and non-degeneracy assumptions are made on the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030023.png" /> to ensure existence and uniqueness of the solutions of the stochastic differential equations (a1), (a2) and to ensure that the partial differential operators in the Duncan–Mortensen–Zakai equation are well defined.

A filtering problem is to estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030025.png" />, based on the observations of (a2) until time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030026.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030027.png" /> which is the sigma-algebra generated by the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030028.png" /> (cf. also [[Optional sigma-algebra|Optional sigma-algebra]]).

The conditional probability density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030029.png" /> given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030030.png" /> represents all of the probabilistic information of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030031.png" /> that is known from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030032.png" />, that is, the observation process. A stochastic partial differential equation can be given for this conditional probability density function, but it satisfies a non-linear stochastic partial differential equation.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030033.png" /> be the process that satisfies the stochastic differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030034.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030035.png" />, which is obtained from (a2) by letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030036.png" />. The measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030038.png" /> for the processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030040.png" /> are mutually absolutely continuous (cf. also [[Absolute continuity|Absolute continuity]]) and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030041.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030042.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030043.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030044.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030045.png" /> is integration on the measure for the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030046.png" />. Using some elementary properties of [[Conditional mathematical expectation|conditional mathematical expectation]] and [[Absolute continuity|absolute continuity]] of measures it follows that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030047.png" /></td> </tr></table>

Thus, the unnormalized conditional probability density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030048.png" /> given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030049.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030050.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030051.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030052.png" /> is the transition probability density (cf. also [[Transition probabilities|Transition probabilities]]) for the [[Markov process|Markov process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030053.png" />.

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030054.png" /> satisfies a linear stochastic partial differential equation that is called the Duncan–Mortensen–Zakai equation [[#References|[a1]]], [[#References|[a6]]], [[#References|[a8]]] and is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030055.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030056.png" /> is the forward differential operator for the Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030057.png" />. The normalization factor for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030058.png" /> to obtain the conditional probability density is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120300/d12030059.png" />. An extensive description of the solution of a non-linear filtering problem can be found in [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.E. Duncan,   "Probability densities for diffusion processes with applications to nonlinear filtering theory and detection theory"  ''PhD Diss. Stanford Univ.''  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Kallianpur,   "Stochastic filtering theory" , Springer  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.E. Kalman,   R.S. Bucy,   "New results in linear filtering and prediction"  ''Trans. ASME Ser. D'' , '''83'''  (1961)  pp. 95–107</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Karatzas,   S.E. Shreve,   "Brownian motion and stochastic calculus" , Springer  (1991)  (Edition: Second)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.N. Kolmogorov,   "Sur l'interpolation et extrapolation des suites stationnaires"  ''C.R. Acad. Sci. Paris'' , '''208'''  (1939)  pp. 2043</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.E. Mortensen,   "Optimal control of continuous-time stochastic systems"  ''PhD Diss. Univ. California, Berkeley''  (1966)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  N. Wiener,   "Extrapolation, interpolation and smoothing of stationary time series with engineering applications" , Technol. Press&amp;Wiley  (1949)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Zakai,   "On the optimal filtering of diffusion processes"  ''ZWvG'' , '''11'''  (1969)  pp. 230–243</TD></TR></table>