Proceedings of the International Congress of Mathematicians Berkeley, California, USA, 1986

Nonlinear Filtering and Stochastic Flows

M. H. A. DAVIS

1. Introduction. The aim of this article is to describe some recent issues in filtering theory, mainly related to robustness and continuity properties. In general terms, nonlinear filtering refers to the problem of calculating the condi­tional distribution of a "signal" xt given "observations" {Y3,0 < s <t}, where {xa}, {Y3}, s E [0,T], are stochastic processes defined on the same probability space, denoted throughout (fi, T,P). In this generality, very little can be said, and the vast majority of work in this area has concerned the case where {xt} is a Markov process and {Yt} is given by

Yt = h(xt)+nt, (1.1)

where {nt} is some form of "wide band noise." The most familiar model is the "additive white noise" model where we define

Vt = / Y3 ds, Wt=l n3 ds Jo Jo

and take {w®} to be Brownian motion (BM), giving an observation model usually written in differential form as

dyt = h(xt)dt + dw°. (1.2)

The best known result in this area is of course the Kaiman filter, where {xt, yt} satisfy linear stochastic equations

dxt = Axt dt + Cdwt, dyt = Hxtdt + Gdwt (1.3)

where {wt} is a vector BM and it is assumed that GGT > 0 and that the initial state XQ is independent of {wt} with normal distribution N(mo, Po). Then

=the=conditional-distribution=of-^irgiven=-(y^0-^s=^^is==iV=(^t7J2(i)=)=where— {xt} and P(t) satisfy a linear stochastic differential equation and a deterministic Riccati equation respectively; see [6]. The immense success of the Kaiman filter in applications is largely due to its modest computational complexity: P(t) is nonrandom, so the conditional distribution N(xt,P(t)) is parametrized by the low-dimensional sufficient statistic xt.

© 1987 International Congress of Mathematicians 1986

1000

NONLINEAR FILTERING AND STOCHASTIC FLOWS 1001

The extended generator of a homogeneous Markov process on a state space E is an operator (A, D(A)) such that for each / E D(A), the process

G( := f(xt) - /(so) - f Af(x3) ds (1.4) Jo

is a martingale. The standard results in nonlinear filtering concern a situation where the signal is a Markov process with given extended generator and initial distribution 7T°, and the linear observation equation (1.3) of the Kaiman filter is replaced by the nonlinear equation (1.2). We assume that E is a Polish space and that the sample paths of {xt} are in DE[0,T] (the space of right-continuous E-valued functions on [0,T] with left-hand limits). yt and w$ take values in Rm and {w$} is m-dimensional BM; yo = w§ = 0. Let us denote by yt = a{y3,0 < s < t} the natural filtration of the observation process, and by 7T* the conditional distribution of xt given 2/t; we also write irt(f) = fE f(x)irt (dx) for / E B(E). In this case it is generally too much to expect that there will be any low-dimensional sufficient statistic for Trt, which should be thought of as a î/t-adapted process taking values in P(E), the set of probability measures on E. Its evolution is described in two equivalent ways. The first of these involves the innovations process dvt = dyt — iït(h)dt which, as in the Kaiman filter, is a standard BM. The direct nonlinear equivalent of the Kaiman filter is the Kushner-Stratonovich or Fujisaki-Kallianpur-Kunita equation, a nonlinear stochastic differential equation (SDE) satisfied by 7Tj. For / E D(A),

dhrt(f) = n(Af) dt + (*t(hf) - n(h)n(f) + E[a{ | yt\) dvt, Mf) = Mf). (FKK)

Here a{ = d(C^,w°)t/dt where (M,N)t denotes the "joint variation" process for square integrable martingales. In many cases of interest, the state space E is a manifold and a{ is given by

a{ = Zf(xt) (1.5)

where Z is a vector field on E. Then E[a{ \ yt] = nt(Zf). Of course, a? = 0 if {xt} and {w?} are independent, as is commonly the case. Derivations of this equation can be found in full detail in the textbooks [17, 23]. A quick account, also covering the Zakai equation below, is given in [10].

It turns out that the awkwardly nonlinear coefficient of dyt in (FKK) is oc­casioned by the requirement that 7rt E P(E), i.e., 7Tt(l) = 1 (here 1 denotes the function l(x) = 1). For / E D(A) define

0*(/) = n(f) exp f / TT3(h)dVs+2 [ns(h)]2 dsj .

Since 7Tt(l) = 1, the exponential term is equal to ot(l) and t(f) = 0"t(/)/o"t(l). ot is an unnormalized conditioned distribution-, it is a l/t-adapted M_j_( )-valued process, where M+(E) denotes the set of positive measures on E. It satisfies the Zakai equation [10, 28]

dat(f) = at(Af) dt + at(Df) dyt, cr0(f) = TT°(/) (Z)

1002 M. H. A. DAVIS

where Df = Zf-r-hf, assuming ot{ is given by (1.5). When a{ = 0 this reduces to

dvt(f) = °t(Af) dt + at(hf) dyt. (Z')

This is substantially simpler than (FKK) since each coefficient is linear in at, and furthermore, the equation is driven by the observation process {yt} directly rather than by the indirectly-defined innovations process {^}. For these reasons recent work has concentrated almost entirely on analysis of (Z) rather than (FKK).

An integral expression for at can be obtained using the so-called reference probability method. Let us suppose that {xt} and {w®} are independent and that h E B(E), i.e., h is bounded. Then according to Girsanov's theorem, the formula

-^p- = exp f - / h(x3) dw^-- h2(x3) ds J

defines a probability measure on (fi, 7) under which {yt} is a BM and {xt}, {yt} are independent. The inverse Radon-Nikodym derivative is given by

d P A

dïv = Â T = e x p I / h(x3) dys-,} h2(x3) ds J

Let EQ denote expectation with respect to Po- By a standard formula of condi­tional expectations,

Mf)=mxt)\yt] = E o [ ^ t ^ t ] p-™. (1.6)

It is not hard to show that the numerator of this expression is equal to crt(f) (and hence the denominator is o*(l)). Since {xt} is independent of yt under P0, the conditional expectation can be evaluated by integrating with respect to the sample space measure ßt of { a?0 < s < t} on DE[0,ì\:

EQ[f(xt)Kt \yt]= / f(xt)exp ( / h(x3) dy3-- h2(x3) ds) ßt (dx). JDE[o,t] wo * Jo /

(1.7) (1.6), (1.7) are known as the KallianpurStriebel (KS) formula. When a{ ^ 0 the KS formula is substantially more complicated; this will be referred to in §2.

2. Uniqueness. In applications, one is going to compute a solution 7rt or at

to (FKK) or (Z) and then claim that this solution is the conditional distribution „(normalized or not). To substantiate this claim it is of co:urs_e_jie_cemary t_o_showL that the solutions to these equations are unique.

One route to such results is through the theory of stochastic partial differential equations (PDEs) [25]. Suppose that E = Rn and that {xt} is a diffusion process with generator1

1The Einstein summation convention is used here and throughout the rest of this paper.

NONLINEAR FILTERING AND STOCHASTIC FLOWS 1003

If the coefficients a%3, bl are smooth and crt has a smooth density, i.e., there is a random function p(t,x) such that crt(f) = fE f(x)p(t,x)dx, then (Z) can be written in "strong" form as

dp(t, x) = A*p(t, x) dt + D*p(t, x) dyu p(0, x) = p°(x), (1.8)

where A*, D* are the formal adjoints of A, D and p° is the density of the initial measure 7r°. Under the assumption that h and 6 are bounded measurable, a13 is continuous with first derivatives in L°°, and [a13] > 61 for some 8 > 0, p° E L2(R

n), it is shown in [25] that (1.8) has a unique solution

PEL2(nx[o,Ty,H1)nL2(n]c([o,nL2(Rn))),

and that p is the density of vt> Another approach, which treats the problem in much greater generality, is the so-called "filtered martingale problem" idea of Kurtz and Ocone [22]. Recall that in the martingale problem (MP) approach to Markov process theory, initiated by Stroock and Varadhan, the martingale property of C^ in (1.4) is regarded as encapsulating the connection between operator A and process {a:*}. The MP (A,D) is said to be well posed if for each initial measure 7r° there is a unique probability measure Pno on DE(R+)

such that xo has distribution n° and C* is a Pno-martingale for each f E D. If uniqueness holds it is generally not hard to show that xt is a homogeneous Markov process with transition measure Px = Psx (Sx = Dirac measure at x).

Kurtz' and Ocone's approach [21] is to show that if an MP is well posed, then so is an associated family of "filtered" MPs. Note from (FKK) that

Mf)~ f Ks(Af)ds Jo

is a stochastic integral with respect to the innovations process, and hence, it is a local martingale; under suitable conditions it is a martingale. More generally,

nfi'iVt)- / ir3Af(',ya)ds Jo

is a martingale for functions / : E X Rm —• R, where

nf(;y) = / f(x,y)7Tt(dx). JE

Think of {xt,yt} as a joint Markov process with extended generator (A, D(A)) where D(A) C B(ExRm). A process (fi, U) with sample paths in Dp(E)xRm (Ä+) is a solution of the filtered MP for (-4, D) if ß is ^ - a d a p t e d and

Ptf(;Ut)~ f fi8Af(;U3)ds Jo

is an .^-martingale for each / E D(Ä). Here 7^ denotes the natural filtration of {Ut}. Uniqueness holds if any two solutions have the same finite-dimensional distributions. If uniqueness holds and

E[ti0f(;U0)]=E[f(xo,y0)] = f f(x,0)w°(dx), JE

1004 M. H. A. DAVIS

then (7T, Y) has the same distribution as (JJL, U). Since ß is ^-adapted, there is a function H: R+ x GRm(R+) —• P(E) such that p,t = H(t,U) a.s. and hence uniqueness implies 7rt = H(t,y) a.s. Thus any algorithm that solves (FKK) does indeed generate the required conditional distribution. The following result gives uniqueness for the Zakai equation (Z') (the case a{ ^ 0 is also covered but we do not give details here).

THEOREM 2.1 [21]. Suppose (i)Ef*\h(x3)\*ds<oo; (ii) A maps Cb(E) into Cb(E), D(A) is a dense subalgebra ofCi>(E), and the

MP for A is well posed] (iii) f(x)hi(x) E Cb(E) for all f E D(A), i = l,...,m.

If {pt} is a Yt-adapted cadlag M+ -valued process satisfying

Pt(/) = 7T°(/)+ / pa(Af)ds+ f pa(hf)dya, Jo Jo

pt(l) = l+ [ ! Jo

p3(h)dy3,

for all f E D(A), t E T, then pt = at for all t E T, a.s.

Uniqueness holds for (FKK) under the same conditions.

3. Pathwise filtering. For the remainder of the paper we shall consider only signal processes {xt} with continuous sample paths, i.e., sample paths in the space CE[0,T], although many results can be generalized to paths in DE[0,T],

possibly at the expense of some complication. We then have the following simple result.

PROPOSITION 3.1 [16]. Suppose D(A) is an algebra; i.e., f,g E D(A) im­plies fg E P(A) where fg(x) := f(x)g(x). Then (C*,G% = f*&f9Mds where

Af9(x)=A(fg)-fAg-gAf. (3.1)

The idea behind "pathwise filtering" is to recast the equations of nonlinear filtering in a form in which no stochastic integration is involved. Apart from its intrinsic interest, this is important from the point of view of mathematical modelling; see Clark [5] or Davis [7] for discussions of this point.

Let us consider first the independent signal and noise case: {xt} and {w®} are independent. Then irt(f) = at(f)/ct(l), where vt(f) is given by the KS formula (1.8), and we have the following result. For notational simplicity we assume for the moment that m = 1, i.e., yt and w$ are scalar.

THEOREM 3.2 [5]. Suppose that the process t —• h(xt) is a semimartingale and define a function <j> : [0, T] x CR™ [0, T\ x B (E) by

<j>(t,d,f):=EW f(xt)exp(t(t)Hxt))

x e x p f - / Ç(s)dh(x3) - - / h2(x3)ds)

NONLINEAR FILTERING AND STOCHASTIC FLOWS 1005

where E^ denotes integration with respect to the distribution of {xt}. Then (i) For each t, f the function f —* (f>(t, £, / ) is locally Lipschitz continuous with

respect to the uniform norm on Cum [0, T]. (ii) (ß(t,yj) = E0[f(xt) | yt] a.s., i.e., </)(t,y,f) is a version ofat(f).

REMARKS. 1. The functional (j)(t, £, / ) is obtained simply by integrating by parts the stochastic integral in the KS formula (1.8).

2. Property (i) is a "robustness" property of the filter; see [5, 7]. From now on, y(-) will denote an arbitrary but fixed continuous function, and

any integration is always over the distribution of {xt}. We will write vt(f) for a(t,y,f)\ i.e., we always choose the robust version.

Theorem 3.2 provides a pathwise formula in integral form, but we would like to get it in a differential equation form similar to the Zakai equation. The key to this is to notice that the functional

a3(y) := exp ( I y(s) dh(xt) - - / h2(x3) ds)

is a multiplicative functional (m.f.) of {xt} and hence defines a two-parameter semigroup of operators T*t on B (E) by

T*ttf(x)=E3,x[f(xt)ats(v)].

Thus &t(f) can be expressed in the form

*(/) = KtievWf),*0). (3.2)

The following is the main result of pathwise filtering for the independent signal and noise case.

THEOREM 3.3 [8]. Suppose D C D(A) is a set such that h E D and hf E D (A) for all f E D. Then the extended generator A% of the semigroup T^t is given by

A\f(u) = ertOfcto^g-vM*/) _ \h2(x)f(x) (3.3)

= Af(x) - y(t)Ahf(x) + [±y*(t)Ahf(x) - y(t)Ah(x) - \h?(x)} f(x)

where Ahf is given by (3.1).

This is proved by factoring a* (y) into the product of "Girsanov" m.f. and a "potential" m.f.

The significance of the result is that &t(f) can be calculated, in principle (by considering the adjoint semigroup (TJf)* in (3.2)), by the following procedure: let pt be the solution of the "Fokker-Planck" equation

±-tpt = {A\yPu pt = p. (3.4)

Then

°t(f)= f mey{t)h{x)Pt(dx). JE

1006 M. H. A. DAVIS

The exact interpretation of (3.4) depends on the context, but generally it will have the same interpretation as the Fokker-Planck equation for the {xt} process itself since, from (3.3), A% is a "first order" perturbation of A.

EXAMPLE 3.4. Suppose that E is a C°° manifold, that X 0 ,Xi , . . . ,Xn are vector fields, w\,... ,w™ are independent BMs, and {xt} is the solution of the SDE

df(xt) = X0f(xt)dt + Xif(xt) o dw\, f E G°°(E), (3.5)

where "o" denotes the Stratonovich stochastic integral. Then {xt} is a diffusion process with generator A = X0 + \ Y%=1 *ï and Ah*(x) = Y%=1 XihXif. Thus

Aytf(x) = lf^Xff(x)+Xo--y(t)^2xihXif + iP(y(t),x)f(x) 1 l

where

*(V,*) = \y'ìJZ{Xih?-y(xQh + \Y,XÌh) - \h\x).

Thus A% has the same second-order part as A but differs from A in the first and zero-order terms. In particular, pt of (3.4) has a smooth density if A is nondegenerate, i.e., X i , . . . , Xn span TX(E) at each x € E.

All of the above results extend immediately to the case of multidimensional observation n > 1.

4. Pathwise filtering with noise correlation [11]. This cannot be han­dled at the same level of generality as above, and we restrict our attention to the situation considered in Example 3.4 where {xt} is a diffusion on a manifold specified by equation (3.5). It will not be necessary, however, to suppose that {xt} is nondegenerate. We take {yt} to be scalar; in contrast to the situation in §3, this assumption is needed for validity of most results described below. Noise correlation arises when the BMs w% in (3.5) are not independent of w°. Specifically, we assume that w% and w3 are independent for i ^ j ^ 0 but that

(w\w°)t = / a^x^ds. Jo

Referring back to §1, (1.5) then holds with Z = a%(x)X{. What makes the present case more complicated is that when we introduce the measure Po via the Radon-Nikodym derivative (1.7), {yt} becomes a BM but the distributions of {xt} are not preserved. More precisely we have the following results.

THEOREM 4.1 [1Ï]. There are vector fields Yo,Y\,... ,Yn and independent BMs {b\,..., b™} on (fi, J, PQ), independent of {yt} such that {xt} satisfies the following SDE

df(xt) = Y0f(xt) dt + Zf(xt) o dyt + YJ(xt) o db\. (4.1)

In [11] the vector fields Y{ and processes bl are expressed explicitly in terms of the original coefficients and processes. We now decompose (4.1) in a manner

NONLINEAR FILTERING AND STOCHASTIC FLOWS 1007

pioneered by Kunita [19]. Let f (t,x) be the flow of the vector field Z, i.e.,

jtf(<(t,x)) = Zf(ç(t,x)), f&C°°(E),

c(0,x) = x.

We suppose that Z is complete; i.e., f(t, a;) is defined for all t G R. Now define

Çt(x) = i(yt,x). (4.2)

Then £t is a diffeomorphism for all £ > 0 and satisfies

df(Çt(x)) = Zf(tt(x))odyt. (4.3)

Now consider the equation

df(vt) = S^Yofivt) dt + frMfiVt) ° db\ (4.4)

where f 1 is the differential map:

&Y0f{x) = Yü{fo^){it{x)).

This SDE uniquely defines a process r)t, and applying the Kunita-Bismut ex­tended Ito formula [1, 19] we find that

xt(x) = 6 o mi^) = ç{ywnt{x))- (4.5)

Now in (4.2), (4.4), {yt} appears simply as a parameter and the bl are indepen­dent of {yt}- Thus conditioned on yt, rjt is a diffusion process with generator At = Ct*1^ + 2 ZXEt*1^)2 a nd xt is diffeomorphically related to r)t via (4.5). With this information in hand we can derive a pathwise filtering formula by-decomposition of multiplicative functionals, much as before. The result is

THEOREM 4.2 [11]. at(f) = (T^t(ByWf)^°) where T*%t is a two-parame­ter semigroup with extended generator

A\ = eMHvW)A*t exp(-Hyit)) - \%{Zh + A2).

Here Ht(x) := fQ f *h(s) ds and Bt is the group of operators

Btf(x) = it*f(x)exj>([ $h(x)du) Jo

(notation: $f(x) = / o ç(t,x)).

As before, the significance of this result is that at can be computed by solving the Fokker-Planck equation (3.4) and then performing an integration:

*t( / )= / By(t)f(x)nt(dx). JE

All of this is done pathwise, i.e., separately for each sample path y(-, w).

1008 M. H. A. DAVIS

5. Observations on a manifold. Several authors [12, 13, 26] have con­sidered filtering problems where the observations take values in a manifold N (example: measurement of an angle). The observation equation (1.2) is then replaced by an equation of the form

df(yt) = L0f(xt, yt) dt + LJ(yt) o dwf (5.1)

where Li are vector fields on AT, or by the requirement that yt be a nondegenerate diffusion whose generator G has an ^-dependent first-order term. It is shown in [12] that pathwise filtering is possible if Lo in (5.1) takes the form

Lof(x,y) = ^2Lig(x,y)Lif(y) i

for some scalar function g, or, equivalently, if LQ = grad^ g(x,y) (the gradient with respect to the Riemannian metric on N determined by G).

6. Continuity. We have shown in previous sections that when there is no noise correlation it is possible to choose a version of the conditional expectation &t(f) such that the map y —> &t(f)(y) is continuous with respect to the uniform norm on CR™ [0, T] and that this is still possible with noise correlation if the observations are scalar, m = 1. When m > 1, (4.3) is replaced by an equation of the form

d)'(6(a)) = ZiJ•(&(*)) °dyi (6.1)

and the mapping y —> & is no longer continuous unless the vector fields Zi commute [20].

Nonetheless, with smooth coefficients the map x —• &(:r) is almost surely a diffeomorphism, so the decomposition (4.4) and other formulas of §4 are still valid, but only almost surely.

Weaker notions of continuity have been studied by Chaleyat-Maurel and Michel [3, 4]. In [3] it is shown that vt(f)(y) is infinitely differentiate in the sense of Malliavin calculus whereas in [4] a notion of continuity related to the Stroock-Varadhan support theorem [27, 15, §6.8] is introduced. Let at(f) be the solution of the Zakai equation when the signal is a diffusion as in Example 3.4, and let &%($) be the (deterministic) solution when y(t) is replaced by an arbitrary H1 function u(t). Then under smoothness and growth conditions, for each rj > 0,

lim P 0 6->0

s u p | a t ( / ) - < ( / ) | > r / t

sup |y t -« t | <S t

= 0.

7. Existence of conditional densities. Space limitations unfortunately preclude any discussion of this important topic, which is concerned with deter­mining conditions under which the measure at has a (smooth) density when the signal process {xt} is a diffusion. Most of the cases discussed in this paper are best handled by using decompositions similar to (4.5), which reduce the question to one of unconditional diffusions; see [18, 20]. Then "classical" results obtained

NONLINEAR FILTERING AND STOCHASTIC FLOWS 1009

using Hörmander's theorem or Malliavin calculus can be applied. More general cases have been studied by a number of authors using extensions of Malliavin calculus [2, 22, 24].

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[I] J. M. Bismut, A generalized formula of Ito and some other properties of stochastic flows, Z. Wahrsch. Verw. Gebiete 55 (1981), 331-350.

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[3] M. Chaleyat-Maurel, Robustesse du filtre et calcul des variations stochastiquei J. Funct. Anal, (to appear).

[4] M. Chaleyat-Maurel and D. Michel, Une propriété de continuité en filtrage non-linéaire, Stochastics 19 (1986), 11-40.

[5] J. M. C. Clark, The design of robust approximations to the stochastic differential equations of nonlinear filtering, Communication Systems and Random Process Theory (J. K. Skwirzynski, ed.), NATO Advanced Study Institute Series, Sijthoff and Noordhoff, Alphen aan den Rijn, 1978.

[6] M. H. A. Davis, Linear estimation and stochastic control, Chapman and Hall, London; Halsted Press, New York, 1977.

[7] , Pathwise nonlinear filtering, Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Reidel, Dordrecht, 1981, pp. 505-528.

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[12] , Pathwise nonlinear filtering with observations on a manifold, Proc. 25th IEEE Conference on Decision and Control, Athens, 1986, pp. 1337-1338.

[13] T. E. Duncan, Some filtering results in Riemannian manifolds, Inform, and Control 35 (1977), 182-195.

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[15] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland, Amsterdam, 1981.

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[22] S. Kusuoka and D. Stroock, The partial Malliavin calculus with applications to filtering, Stochastics 12 (1984), 83-142.

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[23] R. S. Liptser and A. N. Shiryaev, Statistics of random processes. I, II, Springer-Verlag, Berlin-Heidelberg-New York, 1977, 1978.

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