3080:
3304:
are written in Ito calculus form. It is possible to write them in
Stratonovich calculus form, which turns out to be helpful when deriving filtering approximations based on differential geometry, as in the projection filters. For example, the Kushner-Stratonovich equation written in Stratonovich
1605:
3613:
2257:
647:
280:
823:
2821:
54:. The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the
1261:
1835:
3291:
2348:
3747:, which is finite dimensional. More generally, the evolution of the filter density occurs in an infinite-dimensional function space, and it has to be approximated via a finite dimensional approximation, as hinted above.
1066:
2664:
443:
1388:
1977:
516:
3908:, Bernard Hanzon and François LeGland, A Differential Geometric approach to nonlinear filtering: the Projection Filter, I.E.E.E. Transactions on Automatic Control Vol. 43, 2 (1998), pp 247--252.
1481:
62:. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated
1689:
3841:. (1967). Nonlinear filtering: The exact dynamical equations satisfied by the conditional mode. Automatic Control, IEEE Transactions on Volume 12, Issue 3, Jun 1967 Page(s): 262 - 267
3917:
Damiano Brigo, Bernard Hanzon and François Le Gland, Approximate
Nonlinear Filtering by Projection on Exponential Manifolds of Densities, Bernoulli, Vol. 5, N. 3 (1999), pp. 495--534
3311:
2777:
3119:
2813:
3741:
3155:
2126:
2090:
2016:
3697:
3677:
2378:
2146:
536:
169:
3075:{\displaystyle {\cal {L}}_{t}^{*}f(t,y)=-\sum _{i}{\frac {\partial }{\partial y_{i}}}+{\frac {1}{2}}\sum _{i,j}{\frac {\partial ^{2}}{\partial y_{i}\partial y_{j}}}}
2050:
2698:
2431:
705:
2401:
1099:
35:
set of observations. While originally motivated by problems in engineering, filtering found applications in many fields from signal processing to finance.
4042:
1700:
3163:
2263:
2019:
969:
3896:
Maybeck, Peter S., Stochastic models, estimation, and control, Volume 141, Series
Mathematics in Science and Engineering, 1979, Academic Press
3887:
Mireille
Chaleyat-Maurel and Dominique Michel. Des resultats de non existence de filtre de dimension finie. Stochastics, 13(1+2):83-102, 1984.
2439:
352:
1302:
4012:
3990:
2132:. These equations can be formulated for the above system, but to simplify the exposition one can assume that the unobserved signal
94:
160:
3643:, so that the densities give complete knowledge of the filter. Under the particular linear-constant assumptions with respect to
1884:
454:
1600:{\displaystyle P_{K}:L^{2}(\Omega ,\Sigma ,\mathbf {P} ;\mathbf {R} ^{n})\to L^{2}(\Omega ,F,\mathbf {P} ;\mathbf {R} ^{n})}
78:
are another option to attack the infinite dimensional filtering problem and are based on sequential Monte Carlo methods.
3743:
is
Gaussian and it can be characterized by its mean and variance-covariance matrix, whose evolution is described by the
3771:, a well-known filtering algorithm for linear systems, related both to the filtering problem and the smoothing problem
4037:
3763:
1620:
32:
3608:{\displaystyle dp_{t}={\cal {L}}_{t}^{\ast }\,p_{t}\,dt-{\frac {1}{2}}\,p_{t}\,\,dt+p_{t}\,^{T}\circ dZ_{t}\ .}
1840:
This elementary result is the basis for the general
Fujisaki-Kallianpur-Kunita equation of filtering theory.
4032:
2707:
1464:
3774:
3088:
67:
39:
3929:"Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems"
50:'s, who introduced a simplified dynamics for the unnormalized conditional law of the filter known as the
3814:
Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise
2782:
1412:
82:
20:
3964:
Bain, A., and Crisan, D. (2009). Fundamentals of
Stochastic Filtering. Springer-Verlag, New York,
3809:
895:
3939:(2) (Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996) ed.): 438â495.
3928:
3785:
71:
59:
2252:{\displaystyle \mathrm {d} Y_{t}=b(t,Y_{t})\,\mathrm {d} t+\sigma (t,Y_{t})\,\mathrm {d} B_{t},}
642:{\displaystyle \mathrm {d} Z_{t}=c(t,Y_{t})\,\mathrm {d} t+\gamma (t,Y_{t})\,\mathrm {d} W_{t},}
275:{\displaystyle \mathrm {d} Y_{t}=b(t,Y_{t})\,\mathrm {d} t+\sigma (t,Y_{t})\,\mathrm {d} B_{t},}
4008:
4000:
3986:
3838:
3757:
3710:
3124:
2095:
2059:
1985:
818:{\displaystyle {\big |}c(t,x){\big |}+{\big |}\gamma (t,x){\big |}\leq C{\big (}1+|x|{\big )}}
106:
43:
3682:
3662:
2363:
38:
The problem of optimal non-linear filtering (even for the non-stationary case) was solved by
3940:
3869:
3861:
2053:
2025:
63:
28:
2676:
2409:
3865:
3857:
3854:
3790:
1428:
661:
294:
141:
130:
86:
75:
70:
or the assumed density filters, or more methodologically oriented such as for example the
2383:
2129:
157:
51:
74:, some sub-families of which are shown to coincide with the Assumed Density Filters.
4026:
3905:
3853:(1969), On the optimal filtering of diffusion processes. Zeit. Wahrsch. 11 230â243.
3768:
3744:
1284:
1256:{\displaystyle \mathbf {E} \left=\inf _{Y\in K}\mathbf {E} \left.\qquad {\mbox{(M)}}}
899:
90:
55:
1830:{\displaystyle {\hat {Y}}_{t}=P_{K(Z,t)}{\big (}Y_{t}{\big )}=\mathbf {E} {\big }.}
521:
this gives the following stochastic integral representation for the observations
3850:
2353:
In other terms, the system is simplified by assuming that the observation noise
47:
3945:
3286:{\displaystyle \mathrm {d} q_{t}={\cal {L}}_{t}^{*}q_{t}\ dt+q_{t}^{T}dZ_{t}.}
2343:{\displaystyle \mathrm {d} Z_{t}=c(t,Y_{t})\,\mathrm {d} t+\mathrm {d} W_{t}.}
3965:
3780:
1061:{\displaystyle K=K(Z,t)=L^{2}(\Omega ,G_{t},\mathbf {P} ;\mathbf {R} ^{n}).}
127:
448:
Adopting the ItÅ interpretation of the stochastic differential and setting
2406:
For this particular system, the
Kushner-Stratonovich SPDE for the density
3873:
2659:{\displaystyle \mathrm {d} p_{t}={\cal {L}}_{t}^{*}p_{t}\ dt+p_{t}^{T}}
438:{\displaystyle H_{t}=c(t,Y_{t})+\gamma (t,Y_{t})\cdot {\mbox{noise}}.}
1287:, and the general theory of Hilbert spaces implies that the solution
4005:
Stochastic
Differential Equations: An Introduction with Applications
85:
applies, then filtering also arises as part of the solution of an
3827:
Application of the Markov processes theory to optimal filtering
1383:{\displaystyle {\hat {Y}}_{t}=P_{K(Z,t)}{\big (}Y_{t}{\big )},}
93:
is the estimation part of the optimal control solution to the
3334:
3188:
2828:
2789:
2464:
3829:. Radio Engineering and Electronic Physics, 5:11, pp.1-19.
1972:{\displaystyle p_{t}(y)\ dy={\bf {P}}(Y_{t}\in dy|G_{t}),}
3635:
conditional on the sigma-field generated by observations
3777:, an extension of the Kalman filter to nonlinear systems
1878:. If this probability law admits a density, informally
511:{\displaystyle Z_{t}=\int _{0}^{t}H_{s}\,\mathrm {d} s,}
321:
is the diffusion field. It is assumed that observations
2403:
but we assume this has been taken out by re-scaling.
1247:
426:
3713:
3685:
3665:
3314:
3166:
3127:
3091:
2824:
2785:
2710:
2679:
2442:
2412:
2386:
2366:
2266:
2149:
2098:
2062:
2028:
1988:
1887:
1703:
1623:
1484:
1305:
1102:
972:
708:
539:
457:
355:
342:
may, in general, be unequal) are taken for each time
172:
2700:
denotes the expectation with respect to the density
1852:
would be given by the probability law of the signal
1982:then under some regularity assumptions the density
3735:
3691:
3671:
3607:
3285:
3149:
3113:
3074:
2807:
2771:
2692:
2658:
2425:
2395:
2372:
2342:
2251:
2120:
2084:
2044:
2010:
1971:
1829:
1683:
1599:
1382:
1255:
1060:
885:By "based on those observations" it is meant that
817:
641:
510:
437:
274:
1610:is exactly the conditional expectation operator
1175:
3626:one can calculate all statistics of the signal
1848:The complete knowledge of the filter at a time
31:of a system from an incomplete and potentially
2360:One might keep a deterministic time dependent
1844:More advanced result: nonlinear filtering SPDE
3960:
3958:
3956:
3707:being Gaussian or deterministic, the density
1819:
1802:
1785:
1770:
1753:
1684:{\displaystyle P_{K}(X)=\mathbf {E} {\big }.}
1673:
1663:
1653:
1475:-algebra of Σ then the orthogonal projection
1372:
1355:
1226:
1202:
1155:
1115:
810:
784:
771:
746:
736:
711:
66:may be more based on heuristics, such as the
8:
3703:, with the initial condition for the signal
3472:
3450:
3418:
3396:
1296:of the minimization problem (M) is given by
3157:, the Zakai SPDE for the same system reads
1463:). Furthermore, it is a general fact about
1080:minimizes the mean-square distance between
882:of the system based on those observations?
691: : [0, +â) ×
679: : [0, +â) ×
313: : [0, +â) ×
301: : [0, +â) ×
3760:, closely related to the filtering problem
3983:Stochastic Processes and Filtering Theory
3966:https://doi.org/10.1007/978-0-387-76896-0
3944:
3718:
3712:
3684:
3664:
3593:
3577:
3541:
3536:
3507:
3501:
3487:
3475:
3439:
3434:
3421:
3392:
3386:
3381:
3371:
3361:
3355:
3350:
3344:
3339:
3333:
3332:
3322:
3313:
3274:
3261:
3230:
3208:
3198:
3193:
3187:
3186:
3176:
3167:
3165:
3132:
3126:
3105:
3090:
3027:
3011:
2998:
2984:
2978:
2966:
2952:
2907:
2891:
2878:
2872:
2838:
2833:
2827:
2826:
2823:
2799:
2794:
2788:
2787:
2784:
2715:
2709:
2684:
2678:
2615:
2610:
2597:
2581:
2545:
2540:
2506:
2484:
2474:
2469:
2463:
2462:
2452:
2443:
2441:
2417:
2411:
2385:
2365:
2331:
2322:
2311:
2310:
2301:
2276:
2267:
2265:
2240:
2231:
2230:
2221:
2194:
2193:
2184:
2159:
2150:
2148:
2103:
2097:
2067:
2061:
2036:
2027:
1993:
1987:
1957:
1948:
1933:
1920:
1919:
1892:
1886:
1818:
1817:
1811:
1801:
1800:
1794:
1784:
1783:
1778:
1769:
1768:
1762:
1752:
1751:
1730:
1717:
1706:
1705:
1702:
1672:
1671:
1662:
1661:
1652:
1651:
1646:
1628:
1622:
1588:
1583:
1574:
1553:
1537:
1532:
1523:
1502:
1489:
1483:
1371:
1370:
1364:
1354:
1353:
1332:
1319:
1308:
1307:
1304:
1246:
1231:
1225:
1224:
1211:
1201:
1200:
1190:
1178:
1160:
1154:
1153:
1146:
1135:
1134:
1124:
1114:
1113:
1103:
1101:
1046:
1041:
1032:
1023:
1004:
971:
809:
808:
803:
795:
783:
782:
770:
769:
745:
744:
735:
734:
710:
709:
707:
630:
621:
620:
611:
584:
583:
574:
549:
540:
538:
497:
496:
490:
480:
475:
462:
456:
425:
413:
385:
360:
354:
263:
254:
253:
244:
217:
216:
207:
182:
173:
171:
27:describes the problem of determining the
3883:
3881:
3121:. If we choose the unnormalized density
2136:and the partially observed noisy signal
2020:stochastic partial differential equation
3802:
16:Mathematical model for state estimation
2772:{\displaystyle E_{p}=\int f(y)p(y)dy,}
113:) and suppose that the (random) state
1071:By "best estimate", it is meant that
847:is the following: given observations
7:
4007:(Sixth ed.). Berlin: Springer.
3114:{\displaystyle a=\sigma \sigma ^{T}}
2779:and the forward diffusion operator
1267:Basic result: orthogonal projection
3168:
3004:
2991:
2981:
2884:
2880:
2808:{\displaystyle {\cal {L}}_{t}^{*}}
2444:
2323:
2312:
2268:
2232:
2195:
2151:
1562:
1517:
1511:
1013:
622:
585:
541:
498:
255:
218:
174:
14:
4043:Stochastic differential equations
3647:, where the systems coefficients
95:linear-quadratic-Gaussian control
3816:. Radiofizika, 2:6, pp. 892-901.
1921:
1779:
1647:
1584:
1575:
1533:
1524:
1191:
1104:
1042:
1033:
161:stochastic differential equation
136:of a system of interest at time
2128:satisfies a linear SPDE called
1861:conditional on the sigma-field
1245:
954:that are square-integrable and
3730:
3724:
3574:
3570:
3567:
3555:
3549:
3526:
3514:
3508:
3484:
3481:
3468:
3456:
3447:
3414:
3402:
3393:
3258:
3254:
3242:
3236:
3144:
3138:
3069:
3066:
3054:
3048:
3036:
3020:
2946:
2943:
2931:
2925:
2913:
2900:
2859:
2847:
2757:
2751:
2745:
2739:
2727:
2721:
2653:
2644:
2641:
2629:
2623:
2587:
2578:
2574:
2571:
2559:
2553:
2530:
2518:
2512:
2307:
2288:
2227:
2208:
2190:
2171:
2115:
2109:
2079:
2073:
2005:
1999:
1963:
1949:
1926:
1904:
1898:
1746:
1734:
1711:
1640:
1634:
1594:
1559:
1546:
1543:
1508:
1348:
1336:
1313:
1140:
1052:
1010:
994:
982:
913:generated by the observations
804:
796:
766:
754:
731:
719:
617:
598:
580:
561:
419:
400:
391:
372:
250:
231:
213:
194:
1:
3981:Jazwinski, Andrew H. (1970).
3933:Annals of Applied Probability
2054:Kushner-Stratonovich equation
3985:. New York: Academic Press.
2056:, or a unnormalized version
864:, what is the best estimate
156:given by the solution to an
3825:Stratonovich, R.L. (1960).
152: : Ω â
4059:
3927:Del Moral, Pierre (1998).
3764:Filter (signal processing)
3618:From any of the densities
1870:generated by observations
668:and the initial condition
101:The mathematical formalism
89:problem. For example, the
2357:is not state dependent.
950:-valued random variables
3736:{\displaystyle p_{t}(y)}
3655:are linear functions of
3150:{\displaystyle q_{t}(y)}
2121:{\displaystyle p_{t}(y)}
2085:{\displaystyle q_{t}(y)}
2011:{\displaystyle p_{t}(y)}
1465:conditional expectations
946:) the collection of all
309:is the drift field, and
3946:10.1214/aoap/1028903535
3692:{\displaystyle \gamma }
3672:{\displaystyle \sigma }
2673:denotes transposition,
2373:{\displaystyle \gamma }
2140:satisfy the equations
2018:satisfies a non-linear
42:(1959, 1960), see also
3775:Extended Kalman filter
3737:
3693:
3673:
3609:
3287:
3151:
3115:
3076:
2809:
2773:
2694:
2660:
2427:
2397:
2374:
2344:
2253:
2122:
2086:
2046:
2045:{\displaystyle dZ_{t}}
2012:
1973:
1831:
1685:
1601:
1384:
1257:
1089:and all candidates in
1062:
819:
643:
512:
439:
276:
68:extended Kalman filter
40:Ruslan L. Stratonovich
3738:
3694:
3674:
3610:
3288:
3152:
3116:
3077:
2810:
2774:
2695:
2693:{\displaystyle E_{p}}
2661:
2428:
2426:{\displaystyle p_{t}}
2398:
2375:
2345:
2254:
2123:
2087:
2047:
2013:
1974:
1832:
1686:
1602:
1413:orthogonal projection
1385:
1283:) of candidates is a
1258:
1063:
820:
644:
513:
440:
277:
3711:
3683:
3663:
3312:
3164:
3125:
3089:
2822:
2783:
2708:
2677:
2440:
2410:
2384:
2364:
2264:
2147:
2096:
2060:
2026:
1986:
1885:
1701:
1621:
1482:
1303:
1100:
970:
898:with respect to the
706:
537:
455:
353:
170:
83:separation principle
21:stochastic processes
3810:Stratonovich, R. L.
3349:
3203:
2843:
2804:
2479:
485:
81:In general, if the
4001:Ãksendal, Bernt K.
3874:10.1007/BF00536382
3786:Projection filters
3745:Kalman-Bucy filter
3733:
3689:
3669:
3605:
3331:
3283:
3185:
3147:
3111:
3072:
2977:
2877:
2825:
2805:
2786:
2769:
2690:
2656:
2461:
2423:
2396:{\displaystyle dW}
2393:
2370:
2340:
2249:
2118:
2082:
2042:
2008:
1969:
1827:
1681:
1597:
1380:
1253:
1251:
1189:
1058:
873:of the true state
856:for 0 â€
836:and some constant
815:
639:
508:
471:
435:
430:
272:
72:projection filters
60:Kalman-Bucy filter
4038:Signal estimation
4019:(See Section 6.1)
3758:smoothing problem
3699:do not depend on
3601:
3379:
3216:
3018:
2962:
2960:
2898:
2868:
2492:
2022:(SPDE) driven by
1909:
1714:
1419:(Ω, Σ,
1316:
1250:
1174:
1143:
845:filtering problem
664:, independent of
656:denotes standard
429:
289:denotes standard
109:(Ω, Σ,
107:probability space
44:Harold J. Kushner
19:In the theory of
4050:
4018:
3996:
3968:
3962:
3951:
3950:
3948:
3924:
3918:
3915:
3909:
3903:
3897:
3894:
3888:
3885:
3876:
3848:
3842:
3836:
3830:
3823:
3817:
3807:
3791:Particle filters
3742:
3740:
3739:
3734:
3723:
3722:
3698:
3696:
3695:
3690:
3678:
3676:
3675:
3670:
3614:
3612:
3611:
3606:
3599:
3598:
3597:
3582:
3581:
3548:
3547:
3546:
3545:
3506:
3505:
3480:
3479:
3446:
3445:
3444:
3443:
3426:
3425:
3391:
3390:
3380:
3372:
3360:
3359:
3348:
3343:
3338:
3337:
3327:
3326:
3296:These SPDEs for
3292:
3290:
3289:
3284:
3279:
3278:
3266:
3265:
3235:
3234:
3214:
3213:
3212:
3202:
3197:
3192:
3191:
3181:
3180:
3171:
3156:
3154:
3153:
3148:
3137:
3136:
3120:
3118:
3117:
3112:
3110:
3109:
3081:
3079:
3078:
3073:
3035:
3034:
3019:
3017:
3016:
3015:
3003:
3002:
2989:
2988:
2979:
2976:
2961:
2953:
2912:
2911:
2899:
2897:
2896:
2895:
2879:
2876:
2842:
2837:
2832:
2831:
2814:
2812:
2811:
2806:
2803:
2798:
2793:
2792:
2778:
2776:
2775:
2770:
2720:
2719:
2699:
2697:
2696:
2691:
2689:
2688:
2665:
2663:
2662:
2657:
2622:
2621:
2620:
2619:
2602:
2601:
2586:
2585:
2552:
2551:
2550:
2549:
2511:
2510:
2490:
2489:
2488:
2478:
2473:
2468:
2467:
2457:
2456:
2447:
2432:
2430:
2429:
2424:
2422:
2421:
2402:
2400:
2399:
2394:
2379:
2377:
2376:
2371:
2349:
2347:
2346:
2341:
2336:
2335:
2326:
2315:
2306:
2305:
2281:
2280:
2271:
2258:
2256:
2255:
2250:
2245:
2244:
2235:
2226:
2225:
2198:
2189:
2188:
2164:
2163:
2154:
2127:
2125:
2124:
2119:
2108:
2107:
2091:
2089:
2088:
2083:
2072:
2071:
2051:
2049:
2048:
2043:
2041:
2040:
2017:
2015:
2014:
2009:
1998:
1997:
1978:
1976:
1975:
1970:
1962:
1961:
1952:
1938:
1937:
1925:
1924:
1907:
1897:
1896:
1836:
1834:
1833:
1828:
1823:
1822:
1816:
1815:
1806:
1805:
1799:
1798:
1789:
1788:
1782:
1774:
1773:
1767:
1766:
1757:
1756:
1750:
1749:
1722:
1721:
1716:
1715:
1707:
1690:
1688:
1687:
1682:
1677:
1676:
1667:
1666:
1657:
1656:
1650:
1633:
1632:
1606:
1604:
1603:
1598:
1593:
1592:
1587:
1578:
1558:
1557:
1542:
1541:
1536:
1527:
1507:
1506:
1494:
1493:
1389:
1387:
1386:
1381:
1376:
1375:
1369:
1368:
1359:
1358:
1352:
1351:
1324:
1323:
1318:
1317:
1309:
1262:
1260:
1259:
1254:
1252:
1248:
1241:
1237:
1236:
1235:
1230:
1229:
1216:
1215:
1206:
1205:
1194:
1188:
1170:
1166:
1165:
1164:
1159:
1158:
1151:
1150:
1145:
1144:
1136:
1129:
1128:
1119:
1118:
1107:
1067:
1065:
1064:
1059:
1051:
1050:
1045:
1036:
1028:
1027:
1009:
1008:
922:, 0 â€
824:
822:
821:
816:
814:
813:
807:
799:
788:
787:
775:
774:
750:
749:
740:
739:
715:
714:
648:
646:
645:
640:
635:
634:
625:
616:
615:
588:
579:
578:
554:
553:
544:
517:
515:
514:
509:
501:
495:
494:
484:
479:
467:
466:
444:
442:
441:
436:
431:
427:
418:
417:
390:
389:
365:
364:
281:
279:
278:
273:
268:
267:
258:
249:
248:
221:
212:
211:
187:
186:
177:
76:Particle filters
64:nonlinear filter
4058:
4057:
4053:
4052:
4051:
4049:
4048:
4047:
4023:
4022:
4015:
3999:
3993:
3980:
3977:
3975:Further reading
3972:
3971:
3963:
3954:
3926:
3925:
3921:
3916:
3912:
3904:
3900:
3895:
3891:
3886:
3879:
3849:
3845:
3839:Kushner, Harold
3837:
3833:
3824:
3820:
3808:
3804:
3799:
3753:
3714:
3709:
3708:
3681:
3680:
3661:
3660:
3634:
3589:
3573:
3537:
3532:
3497:
3471:
3435:
3430:
3417:
3382:
3351:
3318:
3310:
3309:
3305:calculus reads
3270:
3257:
3226:
3204:
3172:
3162:
3161:
3128:
3123:
3122:
3101:
3087:
3086:
3023:
3007:
2994:
2990:
2980:
2903:
2887:
2883:
2820:
2819:
2781:
2780:
2711:
2706:
2705:
2680:
2675:
2674:
2611:
2606:
2593:
2577:
2541:
2536:
2502:
2480:
2448:
2438:
2437:
2413:
2408:
2407:
2382:
2381:
2362:
2361:
2327:
2297:
2272:
2262:
2261:
2236:
2217:
2180:
2155:
2145:
2144:
2099:
2094:
2093:
2092:of the density
2063:
2058:
2057:
2032:
2024:
2023:
1989:
1984:
1983:
1953:
1929:
1888:
1883:
1882:
1869:
1860:
1846:
1807:
1790:
1758:
1726:
1704:
1699:
1698:
1624:
1619:
1618:
1582:
1549:
1531:
1498:
1485:
1480:
1479:
1454:
1429:linear subspace
1410:
1360:
1328:
1306:
1301:
1300:
1295:
1269:
1223:
1207:
1199:
1195:
1152:
1133:
1120:
1112:
1108:
1098:
1097:
1088:
1079:
1040:
1019:
1000:
968:
967:
962:
921:
912:
893:
881:
872:
855:
704:
703:
674:
662:Brownian motion
626:
607:
570:
545:
535:
534:
529:
486:
458:
453:
452:
409:
381:
356:
351:
350:
329:
295:Brownian motion
259:
240:
203:
178:
168:
167:
151:
142:random variable
131:Euclidean space
121:
103:
87:optimal control
17:
12:
11:
5:
4056:
4054:
4046:
4045:
4040:
4035:
4033:Control theory
4025:
4024:
4021:
4020:
4013:
3997:
3991:
3976:
3973:
3970:
3969:
3952:
3919:
3910:
3898:
3889:
3877:
3843:
3831:
3818:
3801:
3800:
3798:
3795:
3794:
3793:
3788:
3783:
3778:
3772:
3766:
3761:
3752:
3749:
3732:
3729:
3726:
3721:
3717:
3688:
3668:
3630:
3616:
3615:
3604:
3596:
3592:
3588:
3585:
3580:
3576:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3544:
3540:
3535:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3504:
3500:
3496:
3493:
3490:
3486:
3483:
3478:
3474:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3442:
3438:
3433:
3429:
3424:
3420:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3389:
3385:
3378:
3375:
3370:
3367:
3364:
3358:
3354:
3347:
3342:
3336:
3330:
3325:
3321:
3317:
3294:
3293:
3282:
3277:
3273:
3269:
3264:
3260:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3233:
3229:
3225:
3222:
3219:
3211:
3207:
3201:
3196:
3190:
3184:
3179:
3175:
3170:
3146:
3143:
3140:
3135:
3131:
3108:
3104:
3100:
3097:
3094:
3083:
3082:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3033:
3030:
3026:
3022:
3014:
3010:
3006:
3001:
2997:
2993:
2987:
2983:
2975:
2972:
2969:
2965:
2959:
2956:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2910:
2906:
2902:
2894:
2890:
2886:
2882:
2875:
2871:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2841:
2836:
2830:
2802:
2797:
2791:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2718:
2714:
2687:
2683:
2667:
2666:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2618:
2614:
2609:
2605:
2600:
2596:
2592:
2589:
2584:
2580:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2548:
2544:
2539:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2509:
2505:
2501:
2498:
2495:
2487:
2483:
2477:
2472:
2466:
2460:
2455:
2451:
2446:
2420:
2416:
2392:
2389:
2369:
2351:
2350:
2339:
2334:
2330:
2325:
2321:
2318:
2314:
2309:
2304:
2300:
2296:
2293:
2290:
2287:
2284:
2279:
2275:
2270:
2259:
2248:
2243:
2239:
2234:
2229:
2224:
2220:
2216:
2213:
2210:
2207:
2204:
2201:
2197:
2192:
2187:
2183:
2179:
2176:
2173:
2170:
2167:
2162:
2158:
2153:
2130:Zakai equation
2117:
2114:
2111:
2106:
2102:
2081:
2078:
2075:
2070:
2066:
2039:
2035:
2031:
2007:
2004:
2001:
1996:
1992:
1980:
1979:
1968:
1965:
1960:
1956:
1951:
1947:
1944:
1941:
1936:
1932:
1928:
1923:
1918:
1915:
1912:
1906:
1903:
1900:
1895:
1891:
1865:
1856:
1845:
1842:
1838:
1837:
1826:
1821:
1814:
1810:
1804:
1797:
1793:
1787:
1781:
1777:
1772:
1765:
1761:
1755:
1748:
1745:
1742:
1739:
1736:
1733:
1729:
1725:
1720:
1713:
1710:
1692:
1691:
1680:
1675:
1670:
1665:
1660:
1655:
1649:
1645:
1642:
1639:
1636:
1631:
1627:
1608:
1607:
1596:
1591:
1586:
1581:
1577:
1573:
1570:
1567:
1564:
1561:
1556:
1552:
1548:
1545:
1540:
1535:
1530:
1526:
1522:
1519:
1516:
1513:
1510:
1505:
1501:
1497:
1492:
1488:
1450:
1442:) =
1397:
1391:
1390:
1379:
1374:
1367:
1363:
1357:
1350:
1347:
1344:
1341:
1338:
1335:
1331:
1327:
1322:
1315:
1312:
1291:
1268:
1265:
1264:
1263:
1244:
1240:
1234:
1228:
1222:
1219:
1214:
1210:
1204:
1198:
1193:
1187:
1184:
1181:
1177:
1173:
1169:
1163:
1157:
1149:
1142:
1139:
1132:
1127:
1123:
1117:
1111:
1106:
1084:
1075:
1069:
1068:
1057:
1054:
1049:
1044:
1039:
1035:
1031:
1026:
1022:
1018:
1015:
1012:
1007:
1003:
999:
996:
993:
990:
987:
984:
981:
978:
975:
958:
917:
908:
889:
877:
868:
851:
826:
825:
812:
806:
802:
798:
794:
791:
786:
781:
778:
773:
768:
765:
762:
759:
756:
753:
748:
743:
738:
733:
730:
727:
724:
721:
718:
713:
672:
650:
649:
638:
633:
629:
624:
619:
614:
610:
606:
603:
600:
597:
594:
591:
587:
582:
577:
573:
569:
566:
563:
560:
557:
552:
548:
543:
525:
519:
518:
507:
504:
500:
493:
489:
483:
478:
474:
470:
465:
461:
446:
445:
434:
424:
421:
416:
412:
408:
405:
402:
399:
396:
393:
388:
384:
380:
377:
374:
371:
368:
363:
359:
325:
283:
282:
271:
266:
262:
257:
252:
247:
243:
239:
236:
233:
230:
227:
224:
220:
215:
210:
206:
202:
199:
196:
193:
190:
185:
181:
176:
147:
117:
102:
99:
52:Zakai equation
15:
13:
10:
9:
6:
4:
3:
2:
4055:
4044:
4041:
4039:
4036:
4034:
4031:
4030:
4028:
4016:
4014:3-540-04758-1
4010:
4006:
4002:
3998:
3994:
3992:0-12-381550-9
3988:
3984:
3979:
3978:
3974:
3967:
3961:
3959:
3957:
3953:
3947:
3942:
3938:
3934:
3930:
3923:
3920:
3914:
3911:
3907:
3906:Damiano Brigo
3902:
3899:
3893:
3890:
3884:
3882:
3878:
3875:
3871:
3867:
3863:
3859:
3856:
3852:
3847:
3844:
3840:
3835:
3832:
3828:
3822:
3819:
3815:
3811:
3806:
3803:
3796:
3792:
3789:
3787:
3784:
3782:
3779:
3776:
3773:
3770:
3769:Kalman filter
3767:
3765:
3762:
3759:
3755:
3754:
3750:
3748:
3746:
3727:
3719:
3715:
3706:
3702:
3686:
3666:
3658:
3654:
3650:
3646:
3642:
3638:
3633:
3629:
3625:
3621:
3602:
3594:
3590:
3586:
3583:
3578:
3564:
3561:
3558:
3552:
3542:
3538:
3533:
3529:
3523:
3520:
3517:
3511:
3502:
3498:
3494:
3491:
3488:
3476:
3465:
3462:
3459:
3453:
3440:
3436:
3431:
3427:
3422:
3411:
3408:
3405:
3399:
3387:
3383:
3376:
3373:
3368:
3365:
3362:
3356:
3352:
3345:
3340:
3328:
3323:
3319:
3315:
3308:
3307:
3306:
3303:
3299:
3280:
3275:
3271:
3267:
3262:
3251:
3248:
3245:
3239:
3231:
3227:
3223:
3220:
3217:
3209:
3205:
3199:
3194:
3182:
3177:
3173:
3160:
3159:
3158:
3141:
3133:
3129:
3106:
3102:
3098:
3095:
3092:
3063:
3060:
3057:
3051:
3045:
3042:
3039:
3031:
3028:
3024:
3012:
3008:
2999:
2995:
2985:
2973:
2970:
2967:
2963:
2957:
2954:
2949:
2940:
2937:
2934:
2928:
2922:
2919:
2916:
2908:
2904:
2892:
2888:
2873:
2869:
2865:
2862:
2856:
2853:
2850:
2844:
2839:
2834:
2818:
2817:
2816:
2800:
2795:
2766:
2763:
2760:
2754:
2748:
2742:
2736:
2733:
2730:
2724:
2716:
2712:
2703:
2685:
2681:
2672:
2650:
2647:
2638:
2635:
2632:
2626:
2616:
2612:
2607:
2603:
2598:
2594:
2590:
2582:
2568:
2565:
2562:
2556:
2546:
2542:
2537:
2533:
2527:
2524:
2521:
2515:
2507:
2503:
2499:
2496:
2493:
2485:
2481:
2475:
2470:
2458:
2453:
2449:
2436:
2435:
2434:
2418:
2414:
2404:
2390:
2387:
2367:
2358:
2356:
2337:
2332:
2328:
2319:
2316:
2302:
2298:
2294:
2291:
2285:
2282:
2277:
2273:
2260:
2246:
2241:
2237:
2222:
2218:
2214:
2211:
2205:
2202:
2199:
2185:
2181:
2177:
2174:
2168:
2165:
2160:
2156:
2143:
2142:
2141:
2139:
2135:
2131:
2112:
2104:
2100:
2076:
2068:
2064:
2055:
2037:
2033:
2029:
2021:
2002:
1994:
1990:
1966:
1958:
1954:
1945:
1942:
1939:
1934:
1930:
1916:
1913:
1910:
1901:
1893:
1889:
1881:
1880:
1879:
1877:
1873:
1868:
1864:
1859:
1855:
1851:
1843:
1841:
1824:
1812:
1808:
1795:
1791:
1775:
1763:
1759:
1743:
1740:
1737:
1731:
1727:
1723:
1718:
1708:
1697:
1696:
1695:
1678:
1668:
1658:
1643:
1637:
1629:
1625:
1617:
1616:
1615:
1613:
1589:
1579:
1571:
1568:
1565:
1554:
1550:
1538:
1528:
1520:
1514:
1503:
1499:
1495:
1490:
1486:
1478:
1477:
1476:
1474:
1470:
1466:
1462:
1458:
1453:
1449:
1445:
1441:
1437:
1433:
1430:
1426:
1422:
1418:
1414:
1408:
1404:
1400:
1396:
1377:
1365:
1361:
1345:
1342:
1339:
1333:
1329:
1325:
1320:
1310:
1299:
1298:
1297:
1294:
1290:
1286:
1285:Hilbert space
1282:
1278:
1274:
1266:
1242:
1238:
1232:
1220:
1217:
1212:
1208:
1196:
1185:
1182:
1179:
1171:
1167:
1161:
1147:
1137:
1130:
1125:
1121:
1109:
1096:
1095:
1094:
1092:
1087:
1083:
1078:
1074:
1055:
1047:
1037:
1029:
1024:
1020:
1016:
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91:Kalman filter
88:
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79:
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57:
56:Wiener filter
53:
49:
46:'s work and
45:
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3892:
3851:Zakai, Moshe
3846:
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2380:in front of
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1411:denotes the
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163:of the form
153:
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3639:up to time
2052:and called
1874:up to time
1471:is any sub-
1427:) onto the
334:(note that
128:dimensional
105:Consider a
48:Moshe Zakai
4027:Categories
3866:0164.19201
3797:References
3659:and where
1271:The space
896:measurable
3781:Smoothing
3687:γ
3667:σ
3584:∘
3559:⋅
3530:−
3518:⋅
3460:⋅
3428:−
3406:⋅
3369:−
3346:∗
3252:⋅
3200:∗
3103:σ
3099:σ
3005:∂
2992:∂
2982:∂
2964:∑
2885:∂
2881:∂
2870:∑
2866:−
2840:∗
2801:∗
2734:∫
2639:⋅
2604:−
2569:⋅
2534:−
2528:⋅
2476:∗
2368:γ
2206:σ
1940:∈
1712:^
1563:Ω
1547:→
1518:Σ
1512:Ω
1446:(Ω,
1314:^
1218:−
1183:∈
1141:^
1131:−
1014:Ω
777:≤
752:γ
596:γ
473:∫
423:⋅
398:γ
229:σ
97:problem.
25:filtering
4003:(2003).
3812:(1959).
3751:See also
1614:, i.e.,
1467:that if
903:-algebra
828:for all
699:satisfy
58:and the
1694:Hence,
1459:;
1455:,
1438:,
1423:;
1279:,
942:,
4011:
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3864:
3858:242552
3600:
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3085:where
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2669:where
2491:
2433:reads
1908:
1393:where
901:σ
675:, and
652:where
285:where
2704:,
428:noise
140:is a
33:noisy
29:state
4009:ISBN
3987:ISBN
3756:The
3679:and
3651:and
3622:and
3300:and
843:The
832:and
687:and
338:and
3941:doi
3870:doi
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