Stochastic Filtering in Manifolds

Copyright © IFAC Cont rol Science a nd T echnology (8th Trienn ial Wo rld Congress) Kyoto. Japa n . 198 1

STOCHASTIC FILTERING IN MANIFOLDS T. E. Duncan Departm ent of Math ematz'cs, University of Kansas, Lawren ce, Kansas, USA

Abstract. A non linear filtering problem is formulated where the state evolves in a compact Riemannian manifold and the observations evolve in a noncompact Riemannian symmetric space. Absolute continuity methods are used to obtain an expression for the conditional probability. A stochastic equation is obtained for the unnormalized conditional probability density. This equation is described in the bundle of orthonormal frames. Keywords. Nonlinear filtering; state estimation; stochastic systema; Markov processes in manifolds; statistics. The Laplacian can be described by various techniques. The most cla~sical description of the Laplacian is in terma of local coordinates. This characterization has the disadvantages that it does not show the intrinsic properties of the Laplacian and the transformation rules between two local coordinate descriptions are cumbersome. The description of the Laplacian in terms of covariant derivatives is intrinsic but it fails to relate the Laplacian to the description of a Riemannian connection in the bundle of orthonormal frames and to the global construction of the Brownian motion via the frame bundle.

INTRODUCTION For the stochastic modelling of many physical phenomena it is necessary to have certain stochastic processes in manifolds. In continuous time an important stochastic process for modelling is Brownian motion. While the construction of Brownian motion in a linear space is relatively easily accomplished. there are complete Riemannian manifolds that do not admit a Brownian motion without finite escape time (e.g. DebiardGaveau-Mazet. 1974). However a Brownian motion without finite escape time does exist in compact Riemannian manifolds (Duncan. 1977. 1979) and in noncompact Riemannian symmetric spaces (Duncan. 1981). These latter manifolds arise as the quotient of a semisimple Lie group by a maximal compact subgroup. Semisimple Lie groups are an important family of Lie groups. The classical matrix groups. SL(n; E). SL(n;~). SO(P.q). SO(n;~). SU(P.q) and Sp(n; lR) are all semisimple Lie group.. These groups appear in many applied mathematical problema. The semisimple Lie group SL(2; E) appears in induced representations of the oscillator representation of the Heisenberg group which is important in stochastic filtering (Brockett. 1981. Mitter. 1981).

A third description of the Laplacian is in terms of the frame bundle. the bundle of orthonormal frames for a Riemannian manifold. A normal coordinate system at x in a Riemannian manifold M is a local coordinate system x1 ••••• xn in a neighborhood of x such that

•.•••_Cl_) ( ....!... Clx 1 axn

forms an orthonormal frame at

x. However. (Cl/ax l ••••• Cl/Clxn ) may not be orthonormal at other points. Let U be a normal coordinate neighborhood of x with a normal coordinate system xl ••••• xn at x. Let u be the orthonormal frame at x given by

PRELIMINARIES

((~,L ,""", C:n) ~

A stochastic filtering problem will be subsequently described where the observations are in a noncompact Riemannian symmetric space. The existence of a Brownian motion without finite escape time in such a space was proved by Duncan (1981). Since an equation for the unnormalized conditional probability density function for a filtering problem will be obtained it is useful to describe certain differential operators on manifolds. in particular the Laplacian.

By the parallel displacement of u along the geodesics through x. an orthonormal frame is assigned to each point in U. This construction gives a cross section CJ : U ... O(M) and is called the cross section adapted to the normal coordinate system xl ••••• xn. Let (Xl ••••• !n) be a local frame field about 553

T. E. Dun can

554

the point p that is adapted to a normal coordinate system at p and let Dxi be the covariant derivative in the direction Xi.

At

the point p of the normal coordinates, the frame field (XI, ••• ,Xn) has the property that DXilp -0 and in particular !>xiXilp .. 0. Now

The infinitesimal generator for this diffusion process can be computed directly from the stochastic differential equation description. The forward and the backward Kolmogorov operators for this M-valued diffusion process expressed in the frame field (XI, ••• ,Xn) evaluated at p are respectively for f t ()O(\f) n DxiDxi fl p - ~ Dxi(Zf)l p io:l i-I n

~

~

Define n

~

Then

The term

Dnxy

2

occurs in DxY because it is ne-

cessary to differentiate the frame. The sign is determined by the natural pairing of one forms and vector fields. The construction of Brownian motion in a manifold M also gives the horizontal lift of this process to the frame bundle O(M) (Duncan, 1977, 1979). This latter process defines parallelism along the paths of Brownian motion. To describe Brownian motion or more generally a process that is obtained from a stochastic differential equation it is convenient to give a frame bundle description which is analogous to the frame bundle description of the Laplacian. Let (XI, ••• ,Xn ) .. X be a local frame field adapted to a normal coordinate system at p. The stochastic differential of Brownian motion in this frame field will be denoted dBt(X) where dBt(X(p» are the formal vectors of a standard n dimensional Brownian motion at time t. This denotation contains some ambiguity because it has the usual interpretation of dB only at p. However, it is clear from the taplacian how to insert the appropriate drift term for this local frame field description by the differentiation of the frame. In the applications here one is basically interested in the description at p. It should be recalled that even in Euclidean space the stochastic differential description of a process is a formal device. The precise object is the integral of the stochastic differential equation. A similar prescription can be given for a stochastic differential equation that includes a drift term. Let Z be a smooth vector field over M. The formal stochastic differential equation with drift Z is dY t

0:

dYt(X(p»

Z dt + dB t - Z(X(p»dt+dBt(X(p»

(1) •

Given the stochastic differential equation (1) on a compact manifold M it has been shown by Duncan (1977) that there is a unique Mvalued process (Y t ) that satisfies the stochastic differential equation. This stochastic process is an M-valued diffusion process.

n Dxi!>x i flp +~ Z !>xi flp i=1 ie l ~

A STOCHASTIC EQUATION FOR THE CONDITIONAL PROBABILITY DENSITY Some years ago a number of derivations were given for a stochastic differential equation for the conditional probability density for a filtering problem in Euclidean space described by stochastic differential equations. Mortensen (1966) was apparently the first to use explicitly absolute continuity techniques to obtain the equation for the conditional probability density. Subsequently, Duncan (1967) solved a more general problem making more explicit use of martingale methods. Independently, Zakai (1969) subsequently solved a related problem. A few results have been obtained where some of the processes describing the filtering problem are in manifolds, e.g. Duncan (1977). Let M be a compact Riemannian manifold of dimension m and let N 0: G/H be a Riemannian symmetric space of dimension n where G is a real semisimple Lie group and H is a maximal compact subgroup. From Duncan (1977, 1979, 1981) there are an M and an N valued Brownian motion. Denote these as (B(t» and (~(t» respectively. The state process will evolve in M and the observations will evolve in N. M has been assumed to be compact to avoid technicalities associated with smoothness of the fundamental solution of a parabolic operator, that is the forward Kolmogorov operator for the diffusion. Let K be a smooth vector field over M and let L be a smooth map L : M .. VF(N) that is uniformly bounded where VF(N) is the collection of smooth vector fields over N with the topology of uniform convergence of the vector fields and their derivatives. Let pt M and qt N and let U'" (UI' ••• 'Um) and V - (VI' ••• ,Vn ) be local frame fields adapted to normal coordinate systems at p and q respectively. The state and the observation stochastic processes satisfy the stochastic differential equations

St ochas ti c Filt e rin g in Ma nifo l ds

where Xo '" a, Yo - b and t f [O,lJ. It follows from Duncan (1977) that such equations have solutions that exist and are unique and that parallelism is defined along these processes. The probability space will be denoted (rl,1,p) where rl - rl) x rl 2 , n) .. Ca ([O,l] ;M) and rl2 Cb([O,lJ;N), 1-~)

6Q12

where:li is the com-

pletion with respect to Wiener measure of the Borel a-algebra of rl i , i'" 1,2. P is the product Wiener measure. The a-algebra d"(t) is the completion of the Borel a-algebra of Ca([O,tJ;M) x Cb([O,tJ;N) and 2 (t) is the

:r

completion of the Borel a-algebra of Cb([O,tJ;N). The collection of subsets rl) x:12 is a sub-a-algebra of ~) 6Q1;. and by abuse of notation it is identified (via the natural isomorphism) with~. A conditional probability is a map

p! : rl) x~

+

[O,lJ

that satisfies i) for each w) is a probability on (n 2

,3i)

f

n),

p! (w),·)

and ii) for each

Af~, P!(.,A) is measurable on (rl),~).

Let

be the measure for the process

~XY

«Xt,Y t » described by (2-3) and let ~XB be the measure for the process «Xt,B t ». By an elementary extension of Proposition 2.2 of Duncan (1979) it follows that ~xy«~xB and that the Radon-Nikodym derivative W= d~xy / d~xjJ is Wt-exp[J

t

o

-~

t 0

J.

555

To obtain a stochastic equation for the conditional probability density it is important to know some smoothness properties of the fundamental solution of the forward Kolmogorov equation for the diffusion process (X t ). Since M is compact these smoothness properties can be obtained from the smoothness of fundamental solutions of second order linear parabolic equations in Euclidean space. The usual estimates from second order linear parabolic equations in Euclidean space with bounded coefficients can be used in M. Definition 2.

Let M be a manifold.

An

(~t)­

predictable vector field f along an M-valued continuous stochastic process (X t ) is a vector field such that f t (w)

f

TX t (w)M

and the assignment of f to the fibres of the tangent bundle is an ~t)-predictable process. The usual definition of predictability (Meyer, 1968) can be applied to a countable cover of the Banach manifold of M-valued continuous functions. Definition 3.

Let f be an crt)-predictable

vector field along the M-valued continuous process (Y t ) for which parallelism is defined and let 11 be a sub-a-algebra of~. The conditional expectation of f with respect to B,E[fl~J, is the (equivalence class) vector field over (Y t ) such that for any ~)-predictable vector field g that is A'-measurable, the real-valued function Yt is h-measurable and

where Af/f.

Real-valued stochastic integrals for a manifold-valued Brownian motion have also been defined by Duncan (1979).

By parallelism it is easy to show the existence of E[fI.!'J (Duncan, 1977).

With these preliminaries an expression can be given for the conditional probability density.

Define the function q, which is the unnormalized conditional probability density for the filtering problem (2-3) as

Let 1\

Lemma 1.

f

~ (X t ) and let P(·, t I~ (t»

be the conditional probability. The conditional probability satisfies the equation (4)

q(x,tla,O,~(t» =E[Wt I ~(Xt.x) 6Q~(t)Jpx(O,a;t,x)

where Px is the transition density for the Mvalued diffusion (X t ). The notation ~(Xt - x) means the sub-a-algebra

where

Eux

measure

is integration with respect to the

~X.

The proof for processes in Euclidean space (Duncan, 1967) is a proof from abatract measure theory so that it applies in this case verifying the lemma.

(5)

~(Xt)

where the :B(X t )

measurable function in the n1 variable is written as a function of x. The following result gives a stochastic integral equation for q. Lemma 4. Let t t (O,lJ. A version of q satisfies the stochastic equation q(x,tla,O,~(t» - px(O,a;t,x) +

556

T. E. Dun ca n

I

t

I
o

(6)

Proof. Initially it will be shown that the following equality is satisfied

t

.. 10
I ;8(Xt

.ox) ~ ~(t)].dYs·> a.s.

from the stochastic integral equation (6) for q. The only thing that is required is to justify the interchanges of differentiation and integration. The estimates on the derivatives of the fundamental solution of a second order linear parabolic equation in Euclidean space can be used here. These estimates on the derivatives ensure that the sequences that are formed as the approximations to the derivatives can be interchanged with the integrations. These operations give the desired stochastic differential equation (8).

(7)

Real-valued stochastic integrals have been defined by parallelism to TbM by Duncan (1979). Finite sum approximations to the stochastic integral in TbM can be shown to satisfy the equality (7) from which (7) follows by passage to the limit. The following computations can be performed in local trivializations of TM or in TTbM.

I ~(Xt ., x) 6() ~ (t)] E[LuE[1/Iu I "(Xu = x'. Xt

E[1/IuLu -

.. x)

6Q3i'(u)] 11J(X t -x) 6Q~(t)] .. E[LuE[1/Iu

I ~(Xu" x')

~~(u)] ,.(Xt-x) 6Q~(t)]

.. I

M

px(u.x';t.x)pX(O.a;u.x') PX(O.a;t.x)

L(x'.Y u )E[1/Iu 115(~-x') £~(u)]dP(x') 1 PX(O .a;t .x)

I Px(u.x' ;t.x) M

L(x' 'Yu)q(x' .ula.O.~(u»dP(x') Using the definition of q and applying the stochastic differential rule to 1/1. the equation (6) follows immediately. Finally. the stochastic differential equation for q can be given. Theorem 5. Let xe M be fixed and let (U1 ••••• Um) be a local orthonormal frame field that is adapted to a normal coordinate system at x. The unnormalized conditional probability denaity q satisfies the stochastic differential equation m dq(x.t I a.O.~(t» -~ l: DuiDu i qlx i-I

Proof.

It is clear that formally this ato-

~ic differential equation can be obtained

REFERENCES

Brockett. R.W. (1981). Remarks on finite dimensional non-linear estimation. To appear in Asterisque. Debiard. A•• B. Gaveau. and E. Hazet. (1974). Temps d'arret des diffusions riemanniennes C.R. Acad. Sc. Paris 278. Ser. A. 723725.

Duncan. T.E. (1967). Probability densities for diffusion processes with applications to nonlinear filtering theory and detection theory. Thesis. Stanford Univ. Duncan. T.E. (1979). Stochastic systems in Riemannian msnifolds. J. Opt. Theory ~ . .E. 399-426. Duncan. T.E. (1977). Some filtering results in Riemann manifolds. Info. and Control • 35. 182-195. Duncan7 T.E. (1981). Brownian motion in some noncompact manifolds with applications. Proc . Conf. Info. Sci. and Systems. Baltimore • Meyer. P.A. (1968). Guide detaille de la theorie "generale". Sem. de Probabilites, 11. Lecture Notes in Math. 51. 140-165. Springer-Verlag. Mitter. S.K. (1981). Filtering theory and quantum fields. To appear in Asterisque. Mortensen. R.E. (1966). Optimal control of continuous-time stochastic systems. Thesis. Univ. of California. Berkeley. Zakai. M. (1969). On the optimal filtering of diffusion processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 11. 230-243.

Discus sion t o Paper 23 . 2 J . S . Baras (USA) : In your observati on equa t i on

-

dYt = h (xt , Yt ) dt + L(xt , Yt)dB t how c ri tical is the d ependence on Yt in t he case o f p r ocesses and observat i ons on mani f o lds? T.E. Duncan (USA ) : I n general the dr i ft t e rm in t he obse rvat i on equation will depend on the obser va ti o n s whi ch r e fl ect t h e nontriv i al i ty o f the t angent bundl e o f t he man if o ld t hat the observati ons evo l ve in. M. Hazew i nke l (Ne t he r lands): How do t hese constructi ons work f or compact ma ni f o l ds and how do t hey rel ate t o th e c o n str ucti o n v i a the heat ke rne l (o f Ee ll s , Elwo r thy ) ? T.E . Duncan (US A) : Brownian moti on with infinite life t ime ca n be cons tructed i n an arb itra r y compact Riemann i an man ifo l d . I gav e s uch a c onstruction some yea r s ago .