Intrinsic Location Parameter of a Diffusion Process - arXiv

Intrinsic Location Parameter of a Diffusion Process R. W. R. Darling1 2

1.

P. O. Box 535, Annapolis Junction, Maryland 20701-0535, USA. E-mail: [email protected]

2.

Supported by the Air Force OfÞce of ScientiÞc Research. Dedicated to James Eells on his 72nd Birthday

ABSTRACT: For nonlinear functions f of a random vector Y, E[f(Y)] and f(E[Y]) usually differ. Consequently the mathematical expectation of Y is not intrinsic: when we change coordinate systems, it is not invariant.This article is about a fundamental and hitherto neglected property of random vectors of the form Y = f ( X ( t ) ) , where X(t) is the value at time t of a diffusion process X: namely that there exists a measure of location, called the Òintrinsic location parameterÓ (ILP), which coincides with mathematical expectation only in special cases, and which is invariant under change of coordinate systems. The construction uses martingales with respect to the intrinsic geometry of diffusion processes, and the heat ßow of harmonic mappings. We compute formulas which could be useful to statisticians, engineers, and others who use diffusion process models; these have immediate application, discussed in a separate article, to the construction of an intrinsic nonlinear analog to the Kalman Filter. We present here a numerical simulation of a nonlinear SDE, showing how well the ILP formula tracks the mean of the SDE for a Euclidean geometry. RESUME: Pour une fonction non linŽaire Ä dÕun vecteur alŽatoire, E [ f ( Y ) ] et f ( E [ Y ] ) sont usuellement diff•rents. Par consŽquent, lÕesp•rance mathŽmatique de Y nÕest pas intrins•que: quand nous changeons le syst•me des coordonnŽes, elle nÕest pas invariante. Cet article concerne une propriŽtŽ fondamentale, negligŽe jusquÕˆ maintenant, des vecteurs alŽatoires de la forme Y = f ( X ( t ) ) , o• X(t) est la valeur au temps t dÕun processus de diffusion X: cÕest ˆ dire quÕil existe une mesure de position, nommŽe le Òparam•tre intrins•que de centrageÓ (PIC), qui coincide avec lÕesp•rance mathŽmatique seulement dans des cas spŽciÞques, et qui est invariante par changement du syst•me des coordonnŽes. La construction utilise des martingales en rapport avec la gŽometrie intrins•que des processus de diffusion, et le ßot de chaleur des applications harmoniques. Nous calculons des formules qui peuvent •tre utiles aux statisticiens, aux ingŽnieurs, et ˆ toute autre personne qui utilise des mod•les fondŽs sur des processus de diffusion; ces formules se mettent en service ˆ la construction dÕune analogue non linŽaire intrins•que du Þltre de Kalman, discutŽe dans un autre article. Nous prŽsentons ici une simulation numŽrique dÕune EDS non linŽaire, qui montre la prŽcision avec laquelle la formule de PIC suit la moyenne de lÕEDS pour une gŽometrie EuclidŽenne. AMS (1991) SUBJECT CLASSIFICATION: Primary: 60H30, 58G32. KEY WORDS: intrinsic location parameter, gamma-martingale, stochastic differential equation, forward-backwards SDE, harmonic map, nonlinear heat equation

September 5, 1998 8:33 pm.

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2

Introduction

1

Introduction

1.1

Background

The relationship between martingales and parabolic partial differential equations was pointed out in the classic paper of Doob [11]: the solution u ( t, x ) to the one-dimensional heat equation, with a given function ψ as the boundary condition u ( 0, . ) , is given by u ( T, x ) = E x [ ψ ( W T ) ] ,

(1)

where { W t } is Brownian motion with W 0 ≡ x . This can also be expressed as the initial value V 0 of the martingale V t ≡ u ( T – t, W t ) which terminates at ψ ( W T ) at time T.

In the case of nonlinear parabolic PDE , the martingale { V t } must be replaced by the solution of an inverse problem for stochastic differential equations, also called a backwards SDE, as in the work of Pardoux and Peng: see [23], [10]. In the case of the system of elliptic PDE known as a harmonic mapping between Riemannian manifolds, this problem becomes one of constructing a martingale on a manifold with prescribed limit, which has been been solved in works by Kendall [20], [21], Picard [24], [25], Arnaudon [1], Darling [6], [7], and Thalmaier [29]. Thalmaier [30] studies the parabolic problem foir the nonlinear heat equation. The main point here is that the straightforward computation of an expectation as in (1) is no longer available in the nonlinear case. For discussion of the concept of using martingales on a manifold for determining barycentres, see Emery and Mokobodzki [18]. The aim of the present paper is to show the ideas mentioned in the previous paragraph have application to the question of determining the ÒmeanÓ of a diffusion process, or of its image under a smooth function, in an intrinsic way, and that furthermore it is possible to compute an approximation to such a mean without excessive effort.

1.2

Main Results p

Suppose X is a Markov diffusion process on R , or more generally on a manifold N. The diffusion variance of X induces a semi-deÞnite metric 〈.|.〉 on the cotangent bundle, a version of the LeviCivita connection Γ, and a Laplace-Beltrami operator ∆. We may treat X as a diffusion on N with generator ξ + ( 1 ⁄ 2 ) ∆ , where ξ is a vector Þeld. For sufÞciently small δ > 0 , X δ has an Òintrinsic location parameterÓ, deÞned to be the non-ran-

dom initial value V 0 of a Γ-martingale V terminating at X δ . It is obtained by solving a system of forward-backwards stochastic differential equations (FBSDE): a forward equation for X, and a backwards equation for V. This FBSDE is the stochastic equivalent of the heat equation (with drift ξ) for harmonic mappings, a well-known system of quasilinear PDE. Let { φ t :N → N , t ≥ 0 } be the ßow of the vector Þeld ξ, and let x t ≡ φ t ( x 0 ) ∈ N . Our main result is –1

that exp x V 0 can be intrinsically approximated to Þrst order in T x N by δ

δ

Geometry Induced by a Diffusion Process

3

δ

∇dφ δ ( x 0 ) ( Π δ ) – ∫ ( φ δ – t ) ( ∇dφ t ( x 0 ) ) dΠ t 0

where Π t =

*

t

More gener∫0 ( φ–s ) * 〈.|.〉xs ds ∈ T x0 N ⊗ T x0 N . This is computed in local coordinates. 2

ally, we Þnd an intrinsic location parameter for ψ ( X δ ) , if ψ:N → M is a C map into a Riemannian manifold M. We also treat the case where X 0 is random.

2

Geometry Induced by a Diffusion Process

2.1

Diffusion Process Model

Consider a Markov diffusion process { X t, t ≥ 0 } with values in a connected manifold N of dimension p, represented in coordinates by i dX t

where p

i

∂

∑ b ∂x

p

i

= b ( X t ) dt +

∑

j=1

i

j

σ j ( X t ) dW t , p

i

i

(2)

is a vector Þeld on N, σ ( x ) ≡ ( σ j ( x ) ) ∈ L ( R ;T x N ) , and W is a Wiener process in i

2

i

R .We assume for simplicity that the coefÞcients b , σ j are C with bounded Þrst derivative.

2.2

The diffusion variance semi-deÞnite metric

Given a stochastic differential equation of the form (2) in each chart, it is well known that one may 2

deÞne a C semi-deÞnite metric 〈.|.〉 on the cotangent bundle, which we call the diffusion variance semi-deÞnite metric, by the formula i

k

〈dx |dx 〉 x ≡ ( σ ⋅ σ )

ik

p

( x) ≡

∑

j=1

i

k

σ j ( x) σ j ( x) .

(3)

Note that 〈.|.〉 may be degenerate. This semi-deÞnite metric is actually intrinsic: changing coordii

nates for the diffusion will give a different matrix ( σ j ) , but the same semi-deÞnite metric. We postulate: Axiom A:

The appropriate metric for the study of X is the diffusion variance semi-deÞnite metric, not the Euclidean metric. ij

The p × p matrix ( ( σ ⋅ σ ) ) deÞned above induces a linear transformation α ( x ) :T x∗ N → T x N , namely i

ij

α ( x ) ( dx ) ≡ ∑ ( σ ⋅ σ ) ∂ ⁄ ∂x j . Let us make a constant-rank assumption, i.e. that there exists a rank r vector bundle E → N , a subbundle of the tangent bundle, such that E x = range ( σ ( x ) ) ⊆ T x N for all x ∈ N . In Section 7 below, we present a global geometric construction of what we call the canonical sub-Riemannian connection ∇° for 〈.|.〉 , with respect to a generalized inverse g, i.e. a vector bundle isomorphism g:TN → T∗ N such that

4

G-Martingales

α ( x) • g ( x) • α ( x) = α ( x) .

(4)

In local cošrdinates, g ( x ) is expressed by a Riemannian metric tensor ( g rs ) , such that if ij

ij

α ≡ ( σ ⋅ σ ) , then

∑α

r, s

ir

g rs α

sj

ij

= α .

(5)

s

The Christoffel symbols { Γ ij } for the canonical sub-Riemannian connection are speciÞed by (84) p

p

p

below. The corresponding local connector Γ ( x ) ∈ L ( T x R ⊗ T x R ;T x R ) can be written in the more compact notation: 2g ( Γ ( x ) ( u ⊗ v ) ) ⋅ w = D 〈g ( v ) |g ( w ) 〉 ( u ) + D 〈g ( w ) |g ( u ) 〉 ( v ) – D 〈g ( u ) |g ( v ) 〉 ( w ) ,

(6)

where g ( Γ ( x ) ( u ⊗ v ) ) is a 1-form, acting on the tangent vector w.

2.3

Intrinsic Description of the Process

The intrinsic version of (2) is to describe X as a diffusion process on the manifold N with generator 1 L ≡ ξ + --- ∆ 2

(7)

where ∆ is the (possibly degenerate) Laplace-Beltrami operator associated with the diffusion vari1

p

ance, and ξ is a vector Þeld, whose expressions in the local coordinate system { x , É, x } are as follows: ∆ = Note that

∑ ( σ ⋅ σ) i, j

∑ ( σ ⋅ σ)

ij k Γ ij

ij

k

{ D ij – ∑ Γ ij D k } , ξ = k

∑ {b k

k

ij k 1 + --- ∑ ( σ ⋅ σ ) Γ ij } D k . 2 i, j

(8)

has been speciÞed by (3) and (6).

Γ-Martingales

3

2

Let Γ be a connection on a manifold M. An H Γ-martingale is a kind of continuous semimartin2

q

gale on M which generalizes the notion of continuous L martingale on R : see Emery [17] and Darling [7]. We summarize the main ideas, using global coordinates for simplicity. q

Among continuous semimartingales in R , It™Õs formula shows that local martingales are characterized by 2

t

c

2

q

f ( X t ) – f ( X 0 ) – ( 1 ⁄ 2 ) ∫ D f ( X s ) ( dX ⊗ dX ) s ∈ M loc , ∀ f ∈ C ( R ) ,

(9)

0

where ( dX ⊗ dX )

ij

i

j

is the differential of the joint quadratic variation process of X and X , and

c

M loc refers to the space of real-valued, continuous local martingales (see Revuz and Yor [26]). For q

1

q

vector Þelds ξ, ζ on R , and ω ∈ Ω ( R ) , the smooth one-forms, a connection Γ gives an intrinsic way of differentiating ω along ξ to obtain

Taylor Approximation of a Gamma-Martingale Dirichlet Problem

1

5

q

∇ξ ω ∈ Ω ( R ) . ∇ ξ ω ⋅ ζ is also written ∇ω ( ξ ⊗ ζ ) . When ω = df , this gives the Hessian j

∇df ( D k ⊗ D i ) = D ki f – ∑ Γ ki D j f j

i

where the { Γ jk } are the Christoffel symbols. The intrinsic, geometric restatement of (9) is to characterize a Γ−martingale X by the requirement that t

2

c

q

f ( X t ) – f ( X 0 ) – ( 1 ⁄ 2 ) ∫ ∇df ( X s ) ( dX ⊗ dX ) s ∈ M loc , ∀ f ∈ C ( R ) . 0

k

(10)

c

This is equivalent to saying that M ∈ M loc for k = 1, É, p , where k

i

k

k

j

dM t = dX t + ( 1 ⁄ 2 ) ∑ Γ ij ( X t ) d 〈 X , X 〉 t .

(11)

i, j

2

If N has a metric g with metric tensor ( g ij ) , we say that X is an H Γ-martingale if (10) holds and also ∞

E 〈 X, X〉 ∞ ≡ E ∫

∑

g ( Xt) d 〈 X 0 i, j ij

i

j

, X 〉t < ∞ .

(12)

The Γ-martingale Dirichlet problem, which has been studied by, among others, Emery [16], Kendall [20], [21], Picard [24], [25], Arnaudon [1], Darling [6], [7], and Thalmaier [29], [30], is to construct a Γ-martingale, adapted to a given Þltration, and with a given terminal value; for the Euclidean connection this is achieved simply by taking conditional expectation with respect to every σ-Þeld in the Þltration, but for other connections this may be as difÞcult as solving a system of nonlinear partial differential equations, as we shall now see.

4

Taylor Approximation of a Gamma-Martingale Dirichlet Problem

4.1

Condition for the Intrinsic Location Parameter

Consider a diffusion process { X t, 0 ≤ t ≤ δ } on a p-dimensional manifold N with generator 1 ξ + --- ∆ where ∆ is the Laplace-Beltrami operator associated with the diffusion variance, and ξ is a 2 vector Þeld, as in (8). The coordinate-free construction of the diffusion X, given a Wiener process p

W on R , uses the linear or orthonormal frame bundle: see Elworthy [15] p. 252. We suppose X 0 = x0 ∈ N . Also suppose ( M, h ) is a Riemannian manifold, with Levi-Civita connection Γ , and ψ:N → M is 2

a C map. The case of particular interest is when M = N , ψ = identity , and the metric on N is a Ògeneralized inverseÓ to σ ⋅ σ in the sense of (5). The general case of ψ:N → M is needed in the context of nonlinear Þltering: see Darling [8].

6

Taylor Approximation of a Gamma-Martingale Dirichlet Problem

Following Emery and Mokobodzki [18], we assert the following: W

Any intrinsic location parameter for ψ ( X δ ) should be the initial value V 0 of an { ℑ t } -

Axiom B:

adapted H

2

Γ -martingale { V t, 0 ≤ t ≤ δ } on M, with terminal value V δ = ψ ( X δ ) .

This need not be unique, but we will specify a particular choice below. In the case where σ ⋅ σ ( x ) does not depend on x, then the local connector Γ, given by (6), is zero, and V 0 is simply

E [ ψ ( X δ ) ] . However our assertion is that, when Γ is not the Euclidean connection, the right mea-

sure of location is V 0 , and not E [ ψ ( X δ ) ] . We begin by indicating why an exact determination of V 0 is not computationally feasible in general.

4.2

Relationship with Harmonic Mappings p

For simplicity of exposition, let us assume that there are diffeomorphisms ϕ:N → R and 1

q

p

1

q

ϕ:M → R which induce global coordinate systems { x , É, x } for N and { y , É, y } for M, respectively. By abuse of notation, we will usually neglect the distinction between x ∈ N and p

p

ϕ ( x ) ∈ R , and write x for both. Γ ( x ) ( ( σ ⋅ σ ) ( x ) ) ∈ T x R is given by (3) and (6), and the local q

q

q

connector Γ ( y ) ∈ L ( T y R ⊗ T y R ;T y R ) comes from the Levi-Civita connection for ( M, h ) .

In order to Þnd { V t } , we need to construct an auxiliary adapted process { Z t } , with values in p q L  R ;T V R  , such that the processes { X t } and { ( V t, Z t ) } satisfy the following system of fort ward-backwards SDE: t

t

0

0

X t = x 0 + ∫ b ( X s ) ds + ∫ σ ( X s ) dW s , 0 ≤ t ≤ δ ; δ 1 V t = ψ ( X δ ) – ∫ Z s dW s + --2 t

δ

∫t Γ ( V s ) ( Zs ⋅ Zs ) ds , 0 ≤ t ≤ δ .

(13)

(14)

We also require that E

δ

∫0 ∑ hij ( V s ) ( Zs ⋅ Zs ) i, j

ij

ds < ∞ .

[Equation (14) and condition (15) together say that V is an H

2

(15)

Γ -martingale, in the sense of (11)

and (12).] Such systems are treated by Pardoux and Peng [23], but existence and uniqueness of solutions to (14) are outside the scope of their theory, because the coefÞcient Γ ( v ) ( z ⋅ z ) is not Lipschitz in z. 2

However consider the second fundamental form ∇dφ of a C mapping φ:N → M . Recall that ∇dφ ( x ) ∈ L ( T x N ⊗ T x N ;T φ ( x ) M ) may be expressed in local coordinates by: 2

∇dφ ( x ) ( v ⊗ w ) = D φ ( x ) ( v ⊗ w ) – Dφ ( x ) Γ ( x ) ( v ⊗ w ) + Γ ( y ) ( Dφ ( x ) v ⊗ Dφ ( x ) w ) p

(16)

p

for ( v, w ) ∈ T x R × T x R , y ≡ φ ( x ) . Let ξ be as in (8). Consider a system of quasilinear parabolic PDE (a Òheat equation with driftÓ for harmonic mappings - see Eells and Lemaire [13], [14]) consisting of a suitably differentiable family of mappings { u ( t, . ) :N → M } , for t ∈ [ 0, δ ] , such that

Taylor Approximation of a Gamma-Martingale Dirichlet Problem

7

1 ∂u = du ⋅ ξ + --- ∇du ( σ ⋅ σ ) , 0 ≤ t ≤ δ , 2 ∂t

(17)

u ( 0, . ) = ψ .

(18)

1 For x ∈ N , the right side of (17) is du ( t, . ) ⋅ ξ ( x ) + --- ∇du ( t, . ) ( σ ⋅ σ ( x ) ) ∈ T u ( t, x ) M . Following 2 the approach of Pardoux and Peng [23], It™Õs formula shows that V t = u ( δ – t, X t ) ∈ M ,

(19)

p Z ( t ) = du ( δ – t, X t ) • σ ( X t ) ∈ L  R ;T V M  t

(20)

solves (14). In particular u ( δ, x 0 ) = V 0 . (A similar idea was used by Thalmaier [29].) 4.2.a

Comments on the Local Solvability of (17) - (18)

Recall that the energy density of ψ:N → M is given by 1 e ( ψ ) ( x ) ≡ --- dψ ⊗ dψ ( σ ⋅ σ ) 2

2 ψ ( x)

β γ 1 = --- ∑ h βγ ( ψ ( x ) ) dψ ( x ) ⊗ dψ ( x ) ( σ ⋅ σ ( x ) ) . 2 β, γ

(21)

Note, incidentally, that this formula still makes sense when σ ⋅ σ is degenerate. In the case where σ ⋅ σ is non-degenerate and smooth, ξ = 0 , and ∫ e ( ψ ) d ( vol N ) < ∞ , the inverse function theorem method of Hamilton [19], page 122, sufÞces to show existence of a unique smooth solution to (17) - (18) when δ > 0 is sufÞciently small. For a more detailed account of the properties of the solutions when dim ( N ) = 2 , see Struwe [28], pages 221 - 235. Whereas Eells and Sampson [12] showed the existence of a unique global solution when ( M, h ) has non-positive curvature, Chen and Ding [4] showed that in certain other cases blow-up of solutions is inevitable. The case where σ ⋅ σ is degenerate appears not to have been studied in the literature of variational calculus, and indeed is not within the scope of the classical PDE theory of Ladyzenskaja, Solonnikov, and UralÕceva [22]. A probabilistic construction of a solution, which may or may not generalize to the case where σ ⋅ σ is degenerate, will appear in Thalmaier [30]. Work by other authors, using Hšr1 mander conditions on the generator ξ + --- ∆ , is in progress. For now we shall merely assume: 2 Hypothesis I

Assume conditions on ξ, σ ⋅ σ , ψ, and h sufÞcient to ensure existence and uniqueness of a solution { u ( t, . ) :N → M, 0 ≤ t ≤ δ 1 } , for some δ 1 > 0 .

4.3

DeÞnition: the Intrinsic Location Parameter

For 0 ≤ δ ≤ δ 1 , the intrinsic location parameter of ψ ( X δ ) is deÞned to be u ( δ, x 0 ) , where x 0 = X 0 .

1 This depends upon the generator ξ + --- ∆ , given in (8), where ∆ may be degenerate; on the map2 W ping ψ:N → M ; and on the metric h for M. It is precisely the initial value of an { ℑ t } -adapted H

2

Γ -martingale on M, with terminal value V δ = ψ ( X δ ) . However by using the solution of the

PDE, we force the intrinsic location parameter to be unique, and to have some regularity as a function of x 0 .

8

Taylor Approximation of a Gamma-Martingale Dirichlet Problem

The difÞculty with DeÞnition 4.3 is that, in Þltering applications, it is not feasible to compute solutions to (17) and (18) in real time. Instead we compute an approximation, as we now describe.

4.4

A Parametrized Family of Heat Flows γ

Consider a parametrized family { u , 0 ≤ γ ≤ 1 } of equations of the type (17), namely γ

γ γ γ ∂u = du ⋅ ξ + --- ∇du ( σ ⋅ σ ) , 0 ≤ t ≤ δ , 2 ∂t

(22)

γ

u ( 0, . ) = ψ .

(23) 0

Note that the case γ = 1 gives the system (17), while the case γ = 0 gives u ( t, x ) = ψ ( φ t ( x ) ) , where { φ t, t ≥ 0 } is the ßow of the vector Þeld ξ.

In a personal communication, Etienne Pardoux has indicated the possibility of a probabilistic conγ

struction, involving the system of FBSDE (47) and (56), of a unique family of solutions { u } for sufÞciently small γ ≥ 0 , and for small time δ > 0 , based on the results of Darling [6] and methods of Pardoux and Peng [23]. For now, it will sufÞce to replace Hypothesis I by the following: Hypothesis II

Assume conditions on ξ, σ ⋅ σ , ψ, and h sufÞcient to ensure existence of δ 1 > 0 and 2

γ

γ 1 > 0 such that there is a unique C mapping ( γ , t, x ) → u ( t, x ) from

[ 0, γ 1 ] × [ 0, δ 1 ] × N to M satisfying (22) and (23) for each γ ∈ [ 0, γ 1 ] .

4.4.a

Notation

For any vector Þeld ζ on N, and any differentiable map φ:N → P into a manifold P, the Òpush-forwardÓ φ * ζ takes the value dφ ⋅ ζ ( x ) ∈ T y P at y ≡ φ ( x ) ∈ P ; likewise φ * ( ζ ⊗ ζ′ ) ≡ φ * ζ ⊗ φ * ζ′ . We must also assume for the following theorem that we have chosen a generalized inverse g:T∗ N → TN to σ ⋅ σ , in the sense of (4), so that we may construct a canonical sub-Riemannian connection ∇° for 〈.|.〉 , with respect to g. We now state the Þrst result, which will later be subsumed by Theorem 4.7.

4.5

Theorem (PDE Version)

Assume Hypothesis II, and that 0 ≤ δ ≤ δ 1 . Then, in the tangent space T ψ ( x ) M , δ

∂ γ u ( δ, x 0 ) ∂γ

γ =0

δ 1 = --- { ∇dψ ( x δ ) ( φ δ Π ) + ψ * { ∇°dφ δ ( x 0 ) ( Π δ ) – ∫ ( φ δ – t ) ∇°dφ t ( x 0 ) dΠ t } } * δ * 2 0

(24)

where { φ t, t ≥ 0 } is the ßow of the vector Þeld ξ, x t ≡ φ t ( x 0 ) , and t

Π t ≡ ∫ ( φ – s ) 〈.|.〉 x ds ∈ T x N ⊗ T x N . 0

*

s

0

0

(25)

In the special case where M = N , ψ = identity, and h = g , the right side of (24) simpliÞes to the part in parentheses {É}.

Taylor Approximation of a Gamma-Martingale Dirichlet Problem

4.5.a

9

DeÞnition

The expression (24) is called the approximate intrinsic location parameter in the tangent space T ψ ( x ) M , δ

denoted I x [ ψ ( X δ ) ] . 0

4.5.b

Remark: How the Formula is Useful γ

First we solve the ODE for the ßow { φ t, 0 ≤ t ≤ δ } of the vector Þeld ξ, compute ∂u ( δ, x 0 ) ⁄ ∂γ at γ = 0 using (24) (or rather, using the local coordinate version (33)), then use the exponential map

to project the approximate location parameter on to N, giving exp ψ ( x ) { δ

∂ γ u ( δ, x 0 ) ∂γ

} ∈ M.

(26)

γ =0

Computation of the exponential map likewise involves solving an ODE, namely the geodesic ßow on M. In brief, we have replaced the task of solving a system of PDE by the much lighter task of solving two ODEÕs and performing an integration.

4.6

The Stochastic Version

We now prepare an alternative version of the Theorem, in terms of FBSDE, in which we give a local coordinate expression for the right side of (24). In this context it is natural to deÞne a new 2

parameter ε, so that γ = ε in (22). Instead of X in (13), we consider a family of diffusion processes ε

ε

2

{ X , ε ≥ 0 } on the time interval [ 0, δ ] , where X has generator ξ + ε ∆ ⁄ 2 . Likewise V in (14) ε

will be replaced by a family { V , ε ≥ 0 } of H V0

γ

γ

2

ε

ε

Γ -martingales, with V δ = ψ ( X δ ) , and

= u ( δ, x 0 ) . Note, incidentally, that such parametrized families of Γ -martingales are also

treated in recent work of Arnaudon and Thalmaier [2], [3]. 4.6.a

Generalization to the Case of Random Initial Value

Suppose that, instead of X 0 = x 0 ∈ N as in (13), we have X 0 = exp x ( U 0 ) , where U 0 is a zero0

mean random variable in T x N , independent of W, with covariance Σ 0 ∈ T x N ⊗ T x N ; the last 0

*

expression means that, for any pair of cotangent vectors β, λ ∈ T x N ,

0

0

0

E [ ( β ⋅ U 0) ( λ ⋅ U 0) ] = ( β ⊗ λ ) ⋅ Σ0 . ε

Now set up the family of diffusion processes { X , ε ≥ 0 } with initial values ε

X 0 = exp x ( εU 0 ) . 0

ε

Each Γ -martingale { V t , 0 ≤ t ≤ δ } is now adapted to the larger Þltration ˜ W } ≡ { ℑ W ∨ σ ( U ) } . In particular, {ℑ t

t

0

–1

ε

exp ψ ( x ) V 0 δ

is now a random variable in T ψ ( x ) M depending on U 0 . δ

(27)

10

Taylor Approximation of a Gamma-Martingale Dirichlet Problem

4.6.b

DeÞnition

In the case of a random initial value X 0 as above, the approximate intrinsic location parameter of ψ ( X δ ) in the tangent space T ψ ( x ) M , denoted I x δ

∂

0,

Σ0 [ ψ

( X δ ) ] , is deÞned to be –1

2

∂ (ε )

ε

E exp ψ ( x ) V 0 δ

ε=0

.

(28)

We will see in Section 6.3 below that this deÞnition makes sense. This is the same as γ –1 γ ∂ E exp ψ ( x ) u  δ, X 0  ∂γ δ

, γ =0

and coincides with I x [ ψ ( X δ ) ] , given by (24), in the case where Σ 0 = 0 . 0

4.6.c

Some Integral Formulas

Given the ßow { φ t, 0 ≤ t ≤ δ } of the vector Þeld ξ, the derivative ßow is given locally by –1

t

τ s ≡ d ( φ t • φ s ) ( x s ) ∈ L ( T x N ;T x N ) , s

(29)

t

t

where x s = φ s ( x ) , for 0 ≤ s ≤ δ . In local coordinates, we compute τ s as a p × p matrix, given by t

t

τ s = exp { ∫ Dξ ( x u ) du } . s

Introduce the deterministic functions t

χ t ≡ ∫ ( φ t – s ) 〈.|.〉 x ds = 0

*

s

t T

t t

∫0 τs ( σ ⋅ σ ) ( xs ) ( τs )

ds ∈ T x N ⊗ T x N , t

(30)

t

t T

t

Ξt ≡ χt + τ0 Σ0 ( τ0) ∈ T x N ⊗ T x N . t

(31)

t

Note, incidentally, that Ξ t could be called the intrinsic variance parameter of X t .

4.7

Theorem (Local Cošrdinates, Random Initial Value Version)

Under the conditions of Theorem 4.5, with random initial value X 0 as in Section 4.6.a, the approximate intrinsic location parameter I x

0,

Σ0 [ ψ

( X δ ) ] exists and is equal to the right side of (24), after redeÞning t

Π t ≡ Σ 0 + ∫ ( φ – s ) 〈.|.〉 x ds ∈ T x N ⊗ T x N . 0

In local coordinates, 2I x δ δ

2

0,

Σ0 [ ψ

*

s

0

(32)

0

( X δ ) ] is given by 2

δ

T

J ∫ τ t [ D ξ ( x t ) ( Ξ t ) – Γ ( x t ) ( σ ⋅ σ ( x t ) ) ] dt + D ψ ( x δ ) ( Ξ δ ) – Jτ 0 Γ ( x 0 ) ( Σ 0 ) + Γ ( y δ ) ( JΞ δ J ) 0

where y δ ≡ ψ ( x δ ) , J ≡ Dψ ( x δ ) , and Ξ t is given by (31).

(33)

Example of Computing an Intrinsic Location Parameter

11

4.7.a

Remarks

¥

Theorem 4.7 subsumes Theorem 4.5, which corresponds to the case Σ 0 = 0 .

¥

In the special case where M = N , ψ = identity, and h = g , formula (33) reduces to: Ix

[ Xδ] 0, Σ 0

δ 1 δ δ 2 = --- { ∫ τ t [ D ξ ( x t ) ( Ξ t ) – Γ ( x t ) ( σ ⋅ σ ( x t ) ) ] dt – τ 0 Γ ( x 0 ) ( Σ 0 ) + Γ ( x δ ) ( Ξ δ ) } . 2 0

¥

In the Þltering context [8], formulas (33) and (34) are of crucial importance.

5

Example of Computing an Intrinsic Location Parameter

(34)

The following example shows that Theorem 4.7 leads to feasible and accurate calculations.

5.1

Target Tracking

In target tracking applications, it is convenient to model target acceleration as an Ornstein-Uhlenbeck process, with the constraint that acceleration must be perpendicular to velocity. Thus 3

3

( v, a ) ∈ R × R must satisfy v ⋅ a = 0 , and the trajectory must lie within a set on which v 2

6

2

is

constant. Therefore we may identify the state space N with TS ⊂ R , since the v-component lies 2

on a sphere, and the a-component is perpendicular to v, and hence tangent to S . 6

Within a Cartesian frame, X is a process in R with components V (velocity) and A (acceleration), and the equations of motion take the nonlinear form: I3 03 × 3 dV = dA – ρ ( X ) I 3 – λP ( V )

V dt + 0 . A γP ( V ) dW ( t )

(35)

Here the square matrix consists of four 3 × 3 matrices, λ and γ are constants, W is a three-dimenT

T

T

sional Wiener process, and if x ≡ ( v , a ) , ρ ( x) ≡ a

2

⁄ v

2

,

(36)

T

3 3 vv P ( v ) ≡ I – ---------- ∈ L ( R ;R ) , 2 v

(37) 3

Note that P ( v ) is precisely the projection onto the orthogonal complement of v in R , and ρ ( x ) has been chosen so that d ( V ⋅ A ) = 0 .

5.2

Geometry of the State Space 2

The diffusion variance metric (3) is degenerate here; noting that P = P , we Þnd

α≡σ⋅σ≡

03 × 3 03 × 3 2

03 × 3 γ P ( v)

.

(38)

12

Example of Computing an Intrinsic Location Parameter

The rescaled Euclidean metric g = γ 2

–2

6

I 6 on R is a generalized inverse to α in the sense of (4), 6

since P = P . We break down a tangent vector ζ to R into two 3-dimensional components ζ v and ζ a . The constancy of v

2

implies that T T –1 DP ( v ) η v = ---------2- { η v v + vη v } . v

(39)

Referring to formula (6) for the local connector Γ ( x ) , T T –1 T D 〈g ( ζ ) |g ( ς ) 〉 ( η ) = ---------------2- ζ a ( η v v + vη v ) ς a , ( ζ, ς, η ) ∈ T x N × T x N × T x N . 2 γ v

Taking Þrst and second derivatives of the constraint v ⋅ a = 0 , we Þnd that T

T

T

T

ζa v + ζv a = 0 , ηa ζv + ηv ζa = 0 .

(40)

Using the last identity, we obtain from (6) the formula T

T

ζa ςa + ςa ζa S ( ζ ⊗ ς) v Γ ( x ) ( ζ ⊗ ς ) = --------------------------- , S ( ζ ⊗ ς ) ≡ . 2 T T 2 v – ζv ςa – ςv ζa

(41)

In order to compute (33), note that, in particular, 2

γ Γ ( x ) ( σ ⋅ σ ( x ) ) = ---------2- P ( v ) v = 0 . 0 0 v

5.3

(42)

Derivatives of the Dynamical System

It follows from (8), (35), and (42) that the formula for the intrinsic vector Þeld ξ is: ξ ( x) = Differentiate under the assumptions v Dξ ( x ) ( ζ ) =

2

a . – ρ ( x ) v – λP ( v ) a

(43)

is constant and v ⋅ a = 0 , to obtain

T ζv va 0 I , Q ≡ ---------2- . λQ – ρ – λP – 2Q ζ a v

(44)

Differentiating (44) and using the identities (40), 0 2 –2 D ξ ( x ) ( η ⊗ ζ ) = ---------2- T . T T v ηa ζa v + ( ζv ηa + ηv ζa ) a

(45)

Example of Computing an Intrinsic Location Parameter

5.3.a

13

Constraints in the Tangent Space of the State Manifold

Let us write a symmetric tensor χ ∈ T x N ⊗ T x N in 3 × 3 blocks as the matrix χ vv χ va χ av χ aa T

T

,

T

where χ av = χ va . Replacing η v ζ a by χ va , and η a ζ a by Tr ( χ aa ) , etc., in (45), we Þnd that 0 2 –2 D ξ ( x ) ( χ ) = ---------2. v Tr ( χ aa ) v + ( χ va + χ av ) a

5.4

(46)

Ingredients of the Intrinsic Location Parameter Formula 6

2

6

Let ψ:N → R be the inclusion of the state space N ≅ TS into Euclidean R . Thus in formula 2

(33), the local connector Γ ( . ) is zero on the target manifold, J is the identity, and D ψ is zero. When Σ 0 is taken to be zero, the formula for the approximate intrinsic location parameter m δ ≡ I x [ X δ ] for X δ becomes: 0

mt =

t t

–1

∫0 τu H u du , H t ≡ ------------2 v0

0 Tr ( χ aa ( t ) ) v t + ( χ va ( t ) + χ av ( t ) ) a t t

where v t and a t are the velocity and acceleration components of x t ≡ φ t ( x ) , for 0 ≤ t ≤ δ , and τ u

and χ t are given by (29) and (30). A straightforward integration scheme for calculating t

( τ u, χ t, m t ) at the same time, using a discretization of [ 0, δ ] , is:

t t–u τ u ≈ exp { ----------- [ Dξ ( x u ) + Dξ ( x t ) ] } , 2 t t T t–u t–u χ t ≈ ----------- ( σ ⋅ σ ) ( x t ) + τ u χ u + ----------- ( σ ⋅ σ ) ( x u ) ( τ u ) , 2 2 t t–u t–u m t ≈ τ u m u + ----------- H u + ----------- H t . 2 2

Since the local connector is zero on the target manifold, geodesics are simply straight lines, and x δ + I x [ X δ ] is a suitable estimate of the mean position of X δ . 0

14

Proof of Theorem 4.7

5.5

Simulation Results

FIGURE 1

SIMULATIONS OF THE MEAN OF AN SDE, VERSUS ITS APPROXIMATE ILP

x-acceleration: ODE "-", mean SDE +/- 1 s.e. "+", ILP "o" 2

0

-2

-4

-6

-8

-10

-12

0

5

10

15

4

20

25 3

We created 10 simulations of the process (35), with λ = 0.5 and γ = 5.2 × 10 , on the time interval [ 0, 1 ] , which was discretized into 25 subintervals for integration purposes. In each case V and A were initialized randomly, with magnitudes of 200 m/s and 50 m/s2, respectively. The plot shows the x-component of acceleration (the other two are similar): the Ò+Ó signs represent the 4

mean of the process (35) over 10 simulations , with bands showing plus and minus one standard error, the solid line is the solution { x t, 0 ≤ t ≤ 1 } of the discretized ODE dx t ⁄ dt = ξ ( x t ) , and the circles denote the approximate intrinsic location parameter (ILP). The reader will note that the ILP tracks the mean of the process better than the ODE does.

6

Proof of Theorem 4.7

The strategy of the proof will be to establish the formula (33) using It™ calculus, and then to show it is equivalent to (24) using differential geometric methods. While this may seem roundabout, the important formula for applications is really (33); converting it into (24) serves mainly as a check that formula (33) is indeed intrinsic. It will make no difference if we work in global coordinates, p

q

and identify N with R and M with R .

Proof of Theorem 4.7

6.1

15

Step I: Differentiation of the State Process with Respect to a Parameter ε

We consider a family of diffusion processes { X , ε ≥ 0 } on the time interval [ 0, δ ] , with initial ε

values X 0 = exp x ( εU 0 ) ; here U 0 is a zero-mean random variable in T x N , independent of W, 0

ε

0

2

with covariance Σ 0 ∈ T x N ⊗ T x N , and X has generator ξ + ε ∆ ⁄ 2 . 0

0

ε

ε

ε

ε

Note that, in local coordinates, the SDE for X is not Ò dX s = b ( X s ) ds + εσ ( X s ) dW s Ò, because the limiting case when ε = 0 would then be the ODE based on the vector Þeld not the same as ξ, which is given by (8). Instead the SDE is ε

ε

t

X t = exp x ( εU 0 ) + ∫ ξ ( X s ) ds – ε

2 t

ε

ε

t

∫0 ζ ( X s ) ds + ∫0 εσ ( X s ) dW s

0

0

i

∑ b Di , which is (47)

where we use the notation 1 ζ ( x ) ≡ --- Γ ( x ) ( σ ( x ) ⋅ σ ( x ) ) . 2

(48)

In the case ε = 0 , the solution is deterministic, namely { x t, 0 ≤ t ≤ δ } . Note that, in local cošrdinates, 2

2 ε exp x ( εU 0 ) = x 0 + εU 0 – ----- Γ ( x 0 ) ( U 0 ⊗ U 0 ) + o ( ε ) . 2 0

It is well known that, if the vector Þeld ξ and the semi-deÞnite metric 〈.|.〉 are sufÞciently differenε

2

ε

2

tiable, then the stochastic processes ∂X ⁄ ∂ε and ∂ X ⁄ ∂ε exist and satisfy the following SDEs: ε

∂X t ∂ε

 ∂X ε  t t ε ε s = U 0 + ∫ Dξ ( X s )  ds + ∫ σ ( X s ) dW s  ∂ε  0 0 ε ζ ( X s ) ds 0 t

– 2ε ∫ 2 ε

∂ Xt ∂ε

2

t

= – Γ ( x) ( U 0 ⊗ U 0) + ∫ D 0

2

 ε ε  ∂X s  Dσ ( X s ) dW s  ∂ε  0 t

+ ε∫

 ε ε  ∂X s ξ ( Xs )  ∂ε

(49)

2

+ O (ε ) ;

ε  2 ε ∂X s  ∂ Xs  t t ε ε  ds + Dξ ( X )  ds – 2 ∫ ζ ( X s ) ds ⊗  ∫ s 2 ∂ε  0 0  ∂ε 

 ε ε  ∂X s  Dσ ( X s ) dW s  ∂ε  0 t

+ 2∫

(50)

+ O ( ε) ,

where O ( ε ) denotes terms of order ε. DeÞne ε

Λt ≡ Now (49) and (50) give:

∂X t ∂ε

, ε=0

( 2) Λt

2 ε

≡

∂ Xt ∂ε

.

2

ε=0

(51)

16

Proof of Theorem 4.7

dΛ t = A t Λ t dt + σ ( x t ) dW t , Λ 0 = U 0 ; 2

( 2)

( 2)

= [ D ξ ( xt) ( Λt ⊗ Λt) + At Λt

dΛ t

( 2)

Λ0

(52)

– 2ζ ( x t ) ] dt + 2Dσ ( x t ) ( Λ t ) dW t ,

(53)

= –Γ ( x0) ( U 0 ⊗ U 0) ,

t

where A t ≡ Dξ ( x t ) . Let { τ s, 0 ≤ s, t ≤ δ } be the two-parameter semigroup of deterministic matrices given by (29), so that s

∂τ t ∂t 0

s

r

r t

= –τt At ; τs = τt τs .

0

Then (52) becomes d ( τ t Λ t ) = τ t σ ( x t ) dW t , which has a Gaussian solution t t 0

t

t t

t

Λ t = τ 0 U 0 + τ 0 ∫ τ s σ ( x s ) dW s = τ 0 U 0 + ∫ τ s σ ( x s ) dW s . 0

0

( 2)

Likewise (53) gives d ( τ t Λ t

(54)

0

0

0

2

) = τ t [ D ξ ( x t ) ( Λ t ⊗ Λ t ) – 2ζ ( x t ) ] dt + 2τ t Dσ ( x t ) ( Λ t ) dW t ,

whose solution is ( 2)

2

t t

t

Λt

= – τ 0 Γ ( x 0 ) ( U 0 ⊗ U 0 ) + ∫ τ s { [ D ξ ( x s ) ( Λ s ⊗ Λ s ) – 2ζ ( x s ) ] ds + 2Dσ ( x s ) ( Λ s ) dW s } . (55)

6.2

Step II: Differentiation of the Gamma-Martingale with Respect to a Parameter

0

ε

ε

ε

Consider the pair of processes ( V , Z ) obtained from (19) and (20), where u is replaced by u . As ε

ε

in the case where ε = 1 , ( V , Z ) gives an adapted solution to the backwards equation corresponding to (14), namely δ ε ε ε 1 V t = ψ ( X δ ) – ∫ Z s dW s + --2 t

δ

ε

ε

ε

∫t Γ ( V s ) ( Zs ⋅ Zs ) ds . ε

ε

However the version of (20) which applies here is Z t = Du ( δ – t, X t ) εσ ( X t ) , so we may replace ε

ε

Z s by εZ s , and the equation becomes

2

δ ε ε ε ε V t = ψ ( X δ ) – ε ∫ Z s dW s + ----2 t ε

ε

δ

ε

ε

ε

∫t Γ ( V s ) ( Zs ⋅ Zs ) ds .

ε

2

ε

2

(56) 2 ε

2

By the regularity of u , it follows that ∂V ⁄ ∂ε , ∂Z ⁄ ∂ε , ∂ V ⁄ ∂ε , and ∂ Z ⁄ ∂ε exist, and satisfy the following equations: ε

∂V t ∂ε

= Dψ

2 ε

∂ Vt ∂ε

2

= Dψ

 ε ε  ∂X δ  ( Xδ)  ∂ε 

 2 ε ε  ∂ Xδ ( Xδ)  2   ∂ε



2

δ

ε Z dW s t s

–∫

+D ψ

ε δ ∂Z s

– ε∫

 ε ε  ∂X δ ( Xδ)  ∂ε

t

∂ε

ε

dW s + ε

δ

ε

ε

ε

∫t Γ ( V s ) ( Zs ⋅ Zs ) ds+ O ( ε ε

∂X δ  δ ∂Z s  –2 ⊗ ∫t ∂ ε dW s + ∂ε 

δ

ε

ε

ε

2

);

∫t Γ ( V s ) ( Zs ⋅ Zs ) ds+ O ( ε )

(57)

(58)

Proof of Theorem 4.7

17

0 ˜ W ≡ ℑ W ∨ σ ( U ) . By combining (54) Note also that V t = y δ ≡ ψ ( x δ ) for all t ∈ [ 0, δ ] . Take ℑ t t 0

and (57), we see that, if J ≡ Dψ ( x δ ) , ε

Θt ≡

∂V t ∂ε

ε=0

˜ W = Jτ δ U + J t τ δ σ ( x ) dW ; = E JΛ δ ℑ t ∫ s s 0 0 s

(59)

0

0

δ

Z s = Jτ s σ ( x s ) .

(60)

DeÞne ( 2) Θt

( 2)

= E JΛ δ

2 ε

≡

∂ Vt ∂ε

(61)

2

ε=0 δ

2

0

0

∫t Γ ( yδ ) ( Zs ⋅ Zs ) ds

+ D ψ ( xδ) ( Λδ ⊗ Λδ) +

˜W ℑ t

(62)

From (55) and (62) we obtain: ( 2)

Θ0

δ δ

δ

2

= – Jτ 0 Γ ( x 0 ) ( U 0 ⊗ U 0 ) + JE { ∫ τ s [ D ξ ( x s ) ( Λ s ⊗ Λ s ) – 2ζ ( x s ) ] ds } U 0 0

2

+ E { D ψ ( xδ) ( Λδ ⊗ Λδ) +

δ

0

0

∫0 Γ ( yδ ) ( Zs ⋅ Zs ) ds }

U0 .

T

The expected value of a quadratic form η Aη in an N p ( µ, Σ ) random vector η is easily computed to be

∑ Aij Σ

ij

T

+ µ Aµ . In this case,

t

η = Λ s ∼ N p ( τ 0 U 0, χ s ) , where χ t is given by (30), so we obtain ( 2)

Θ0

δ

= – Jτ 0 Γ ( x 0 ) ( U 0 ⊗ U 0 ) + J 2

δ δ

∫0 τs [ D

2

2

s

s

ξ ( x s ) ( χ s ) + D ξ ( x s ) ( τ 0 U 0 ⊗ τ 0 U 0 ) – 2ζ ( x s ) ] ds

2

δ

δ

T

+ D ψ ( x δ ) ( χ δ ) + D ψ ( x δ ) ( τ 0 U 0 ⊗ τ 0 U 0 ) + Γ ( y δ ) ( Jχ δ J ) .

6.3

(63)

Step III: A Taylor Expansion Using the Exponential Map –1

ε

Let y δ ≡ ψ ( x δ ) , and deÞne β ( ε ) ≡ E exp y V 0 . Referring to (28), we are seeking ∂β ⁄ δ ε δ

2 ε=0

. It

follows immediately from the geodesic equation that D ( exp y )

–1

2

( y ) ( w ) = w , D ( exp y )

It follows from (59) and (63) that

–1

( y) ( v ⊗ w) = Γ ( y) ( v ⊗ w)

(64)

18

Proof of Theorem 4.7

2

2 ε δ ε ( 2) V 0 = y δ + εJτ 0 U 0 + ----- Θ 0 + o ( ε ) . 2

A Taylor expansion based on (64) gives 2  ε –1 ε ε ε ε 1 exp y V 0 = V 0 – y δ + --- Γ ( y δ ) ( ( V 0 – y δ ) ⊗ ( V 0 – y δ ) ) + o  V 0 – y δ  2 δ 2

2 δ ( 2) δ δ ε = εJτ 0 U 0 + ----- { Θ 0 + Γ ( y δ ) ( Jτ 0 U 0 ⊗ Jτ 0 U 0 ) } + o ( ε ) . 2

Taking expectations, and recalling that U 0 has mean zero and covariance Σ 0 ∈ T x N ⊗ T x N , we 0

0

obtain 2

 δ 2 ( 2) δ T ε β ( ε ) = ----- { E [ Θ 0 ] + Γ ( y δ )  Jτ 0 Σ 0 ( Jτ 0 )  } + o ( ε ) 2 It follows that

dβ dε

= 0 , and hence that ε=0

∂

2

2

∂ (ε )

β ( ε)

ε=0

1d β = --- 2 2dε

ε=0

 δ ( 2) δ T 1 = --- { E [ Θ 0 ] + Γ ( y δ )  Jτ 0 Σ 0 ( Jτ 0 )  } 2 δ δ 2 2 T 1 = --- { J ∫ τ s [ D ξ ( x s ) ( χ s ) – Γ ( x s ) ( σ ⋅ σ ( x s ) ) ] ds + D ψ ( x δ ) ( χ δ ) + Γ ( y δ ) ( Jχ δ J ) 2 0 δ δ

2

2

δ

T

+ J ∫ τ s D ξ ( x s ) ( Σ s ) ds – Jτ 0 Γ ( x 0 ) ( Σ 0 ) + D ψ ( x δ ) ( Σ δ ) + Γ ( y δ ) ( JΣ δ J ) } 0

s

s T

where Σ s ≡ τ 0 Σ 0 ( τ 0 ) . If we write Ξs ≡ χs + Σs ,

(65)

then the formula becomes δ δ 2 2 T 1 --- { J ∫ τ s [ D ξ ( x s ) ( Ξ s ) ] ds + D ψ ( x δ ) ( Ξ δ ) + Γ ( y δ ) ( JΞ δ J ) } 2 0 δ δ δ 1 – --- J { ∫ τ s Γ ( x s ) ( σ ⋅ σ ( x s ) ) ds + τ 0 Γ ( x 0 ) ( Σ 0 ) } . 2 0

This establishes the formula (33).

6.4

◊

Step IV: Intrinsic Version of the Formula

It remains to prove that (24), with Π t as in (32), is the intrinsic version of (33). We abbreviate here by writing ψ • φ t as ψ t . By deÞnition of the ßow of ξ,

Proof of Theorem 4.7

19

∂ ψ ( x ) = dψ • ξ ( φ t ( x 0 ) ) , ∂t t 0 and so, differentiating with respect to x, and exchanging the order of differentiation, ∂ ∂ Dψ t ( x 0 ) = D  ψ t ( x 0 )  ∂ t ∂t

(66) –1

t

t

or, by analogy with (29), taking θ s ≡ Dψ t – s ( x s ) = Dψ ( x t ) • D ( φ t • φ s ) ( x s ) = Dψ ( x t ) τ s , t

∂τ s ∂t δ t

δ

t

t

= Dξ ( x t ) τ s ;

δ t

∂θ s

t

= Dψ ( x t ) • Dξ ( x t ) τ s .

∂t

δ t

(67)

δ

Since τ t τ 0 = τ 0 , we have θ t τ 0 = Dψ ( x δ ) τ t τ 0 = θ 0 , which upon differentiation yields  ∂θ δ  t  ∂τ t  δ t   τ = – θ  0  = – θ δ Dξ ( x ) τ t , t  ∂t  t t 0  ∂t  0 which gives δ

∂τ t

=

∂t

δ – τ t Dξ ( x t )

δ

;

∂θ t

δ

= – θ t Dξ ( x t ) .

∂t

(68)

A further differentiation of (66) when ψ is the identity yields  ∂ D 2 φ ( x )  ( v ⊗ w ) = D 2 ξ ( x ) ( τ t v ⊗ τ t w ) + Dξ ( x ) D 2 φ ( x ) ( v ⊗ w ). t 0  t 0 0 t t 0  ∂t

(69)

Combining (68) and (69), we have 2 2 2 δ δ t t ∂ δ 2 ( θ D φ t ( x 0 ) ) = – θ t Dξ ( x t ) D φ t ( x 0 ) + θ t { D ξ ( x t ) ( τ 0 ( . ) ⊗ τ 0 ( . ) ) + Dξ ( x t ) D φ t ( x 0 ) } ∂t t δ 2 t t ∂ δ 2 ( θ D φt ( x0) ) ( v ⊗ w) = θt D ξ ( xt) ( τ0 v ⊗ τ0 w) . ∂t t

(70)

The formula (16) for ∇°dφ t can be written as 2

t

t

t

∇°dφ t ( x 0 ) ( v ⊗ w ) = D φ t ( x 0 ) ( v ⊗ w ) – τ 0 Γ ( x 0 ) ( v ⊗ w ) + Γ ( x t ) ( τ 0 v ⊗ τ 0 w ) It is clear that ∂ δ t ∂ δ ( θ τ Γ ( x0) ) = ( θ Γ ( x0) ) = 0 . ∂t t 0 ∂t 0 Hence from (70) and (71) it follows that 2 δ t t ∂ ∂ { ( ψ δ – t ) ∇°dφ t ( x 0 ) } ( v ⊗ w ) = { θ [ D φt ( x0) ( v ⊗ w) + Γ ( xt) ( τ0 v ⊗ τ0 w) ] } * ∂t ∂t t

(71)

20

Proof of Theorem 4.7

δ

2

t

t

= θt D ξ ( xt) ( τ0 v ⊗ τ0 w) +

δ t t ∂ { θt Γ ( xt) ( τ0 v ⊗ τ0 w) } . ∂t

(72)

The last term in (72) can be written, using (67), as δ t t δ t t t t ∂ { θ Γ ( x t ) } ( τ 0 v ⊗ τ 0 w ) + θ t Γ ( x t )  Dξ ( x t ) τ 0 v ⊗ τ 0 w + τ 0 v ⊗ Dξ ( x t ) τ w  , 0 ∂t t

(73)

for v ⊗ w ∈ T x N ⊗ T x N . We will replace v ⊗ w by 0

0

0 T

0

0 T

t 0

= Σ 0 + ∫ τ s ( σ ⋅ σ ) ( x s ) ( τ s ) ds ∈ T x N ⊗ T x N

Πt ≡ τt Ξt ( τt )

0

0

0

(74)

t T

t

where Ξ t ≡ χ t + τ 0 Σ 0 ( τ 0 ) . Observe that t

t T

τ0 Πt ( τ0)

= Ξt ∈ T x N ⊗ T x N . t

(75)

t

Moreover from (75) and (67), it is easily checked that dΞ t dt

T

= ( σ ⋅ σ ) ( x t ) + Dξ ( x t ) Ξ t + Ξ t { Dξ ( x t ) } .

(76)

It follows from (72) - (76) that  dΠ  ∂ ∂ t { ( ψ δ – t ) ∇°dφ t ( x 0 ) ( Π t ) } – ( ψ δ – t ) ∇°dφ t ( x 0 )  = { ( ψ δ – t ) ∇°dφ t ( x 0 ) } ( Π t ) * * *  dt  ∂t ∂t

=

δ

2

θt D ξ ( xt) + δ

2

δ

2

T δ δ ∂ { θ Γ ( x t ) } ( Ξ t ) + θ t Γ ( x t ) [ Dξ ( x t ) Ξ t + Ξ t ( Dξ ( x t ) ) ] ∂t t

θt D ξ ( xt) +

=

 dΞ  δ δ ∂ t { θt Γ ( xt) } ( Ξt) + θt Γ ( xt)  – ( σ ⋅ σ) ( xt)   dt  ∂t δ

= θt D ξ ( xt) ( Ξt) – θt Γ ( xt) ( ( σ ⋅ σ) ( xt) ) +

δ ∂ { θ Γ ( xt) ( Ξt) } . ∂t t

Since ∇°dφ 0 = 0 , it follows upon integration from 0 to δ that in T ψ ( x ) M , δ

δ

ψ * ∇°dφ δ ( x 0 ) ( Π δ ) – ∫ ( ψ δ – t ) ( ∇°dφ t ( x 0 ) ) dΠ t = 0

δ δ

*

2

δ

Dψ ( x δ ) { ∫ τ t [ D ξ ( x t ) ( Ξ t ) – Γ ( x t ) ( ( σ ⋅ σ ) ( x t ) ) ] dt + Γ ( x δ ) ( Ξ δ ) – τ 0 Γ ( x 0 ) ( Ξ 0 ) } . 0

(77)

However the formula (16) for ∇dψ ( x δ ) ( v ⊗ w ) can be written as 2

D ψ ( x δ ) ( v ⊗ w ) – JΓ ( x δ ) ( v ⊗ w ) + Γ ( y δ ) ( Jv ⊗ Jw ) .

(78)

The Canonical Sub-Riemannian Connection

21

where y δ ≡ ψ ( x δ ) , and J ≡ Dψ ( x δ ) . We take v ⊗ w = Ξ δ ∈ T x N ⊗ T x N , and add (77) and (78): δ

δ δ

2

δ

δ

2

T

J ∫ τ t [ D ξ ( x t ) ( Ξ t ) – Γ ( x t ) ( ( σ ⋅ σ ) ( x t ) ) ] dt + D ψ ( x δ ) ( Ξ δ ) – Jτ 0 Γ ( x 0 ) ( Ξ 0 ) + Γ ( y δ ) ( JΞ δ J ) 0

δ

= ∇dψ ( x δ ) ( ( φ δ ) Π ) + ψ * { ∇°dφ δ ( x 0 ) ( Π δ ) – ∫ ( φ δ – t ) ( ∇°dφ t ( x 0 ) ) dΠ t } . *

δ

0

*

◊

The equivalence of (24) and (33) is established, completing the proofs of Theorems 4.5 and 4.7.

7

The Canonical Sub-Riemannian Connection

The purpose of this section is to present a global geometric construction of a torsion-free connec2

tion ∇° on the tangent bundle TN which preserves, in some sense, a C semi-deÞnite metric 〈.|.〉 p on the cotangent bundle T∗ N induced by a section σ of Hom ( R ;TN ) of constant rank. In other words, we assume that there exists a rank r vector bundle E → N , a sub-bundle of the tangent bundle, such that E x = range ( σ ( x ) ) ⊆ T x N for all x ∈ N . Given such a section σ, we obtain a vector bundle morphism α:T∗ N → TN by the formula α ( x) ≡ σ ( x) • ι • σ ( x) ∗ , x ∈ N ,

(79)

p ∗ p where ι: ( R ) → R is the canonical isomorphism induced by the Euclidean inner product. The

relation between α and 〈.|.〉 is that, omitting x, µ ⋅ α ( λ ) ≡ 〈µ|λ〉 , ∀µ ∈ T∗ N .

7.1

(80)

Lemma

Under the constant-rank assumption, any Riemannian metric on N induces an orthogonal splitting of the cotangent bundle of the form T x∗ N = Ker ( α ( x ) ) ⊕ F x ,

(81)

where F → N is a rank r sub-bundle of the cotangent bundle on which 〈.|.〉 is non-degenerate. There exists –1 a vector bundle isomorphism α°:T∗ N → TN such that F = Ker ( α° ( x ) – α ( x ) ) , and α° ( x ) is a x

generalized inverse to α ( x ) , in the sense that α ( x ) • α° ( x )

–1

• α ( x) = α ( x) . T

Proof: For any matrix A, range ( A ) = range ( AA ) , and so range ( α ( x ) ) = range ( σ ( x ) ) . It follows that dim Ker ( α ( x ) ) = p – dimE x = p – r . Given a Riemannian metric g on N (which always exists), let 〈.|.〉° be the dual metric on the cotangent bundle. DeÞne

22

The Canonical Sub-Riemannian Connection

F x ≡ { θ ∈ T x∗ N : 〈θ|λ〉 x ° = 0 ∀λ ∈ Ker ( α ( x ) ) } . We omit the proof that F → N is a vector bundle. Since dim Ker ( α ( x ) ) = p – r , it follows that dimF x = r ; since the rank of 〈.|.〉 is r, we see that 〈θ|θ〉 > 0 for all non-zero θ ∈ F x . This shows that

〈.|.〉 is non-degenerate on the sub-bundle F → N .

Now (81) results from the orthogonal decomposition of T x∗ N with respect to 〈.|.〉° . Hence an arbitrary λ ∈ T ∗ N can be decomposed as λ = λ ⊕ λ , with λ ∈ Ker ( α ( x ) ) and λ ∈ F . The met0

x

1

0

ric 〈.|.〉° induces a vector bundle isomorphism β:T∗ N → TN , namely

1

x

µ ⋅ β ( λ ) ≡ 〈µ|λ〉° , ∀µ ∈ T∗ N . Now deÞne α°:T∗ N → TN by α° ( λ ) ≡ β ( λ 0 ) + α ( λ 1 ) . It is clearly linear, and a vector bundle morphism. Since β is injective, { λ ∈ T x∗ N : α° ( λ ) = α ( λ ) } = { λ 0 ⊕ λ 1 ∈ T x∗ N : β ( λ 0 ) = 0 } = F x , which shows that F x = Ker ( α° ( x ) – α ( x ) ) . To show α° is an isomorphism, it sufÞces to show that α° ( λ ) ≠ 0 whenever λ ≠ 0 . When λ 0 ≠ 0 , non-degeneracy of 〈.|.〉° implies that λ 0 ⋅ α° ( λ ) = 〈λ 0|λ 0〉° + 〈λ 0|λ 1〉 = 〈λ 0|λ 0〉° ≠ 0 . On the other hand when λ 0 = 0 , and λ 1 ≠ 0 , non-degeneracy of 〈.|.〉 on F → N implies that λ 1 ⋅ α° ( λ ) = λ 1 ⋅ α ( λ 1 ) = 〈λ 1|λ 1〉 ≠ 0 . Hence α°:T∗ N → TN is a vector bundle isomorphism as claimed. The generalized inverse property follows from the fact that α ( x ) • α° ( x )

7.2

–1

• α ( x) ( λ) = λ1 .

◊

Proposition

p Suppose σ is a constant-rank section of Hom ( R ;TN ) , inducing a semi-deÞnite metric 〈.|.〉 on T∗ N and a vector bundle morphism α:T∗ N → TN as in (79) and (80). Suppose furthermore that α°:T∗ N → TN is

a vector bundle isomorphism such that α ( x ) • α° ( x )

–1

• α ( x ) = α ( x ) , as in Lemma 7.1. Then TN

admits a canonical sub-Riemannian connection ∇° for 〈.|.〉 , with respect to α° , which is torsion-free, and ˆ preserves 〈.|.〉 in the following sense: for vector Þelds V in the range of α, such that the dual connection ∇ and for 1-forms θ, λ which lie in the sub-bundle F ≡ Ker ( α° – α ) , ˆ θ|λ〉 + 〈θ|∇ ˆ λ〉 . V 〈θ|λ〉 = 〈∇ V V ˆ θ ⋅ W = Z ( θ ⋅ W ) – θ ⋅ ∇° W .] For any 1-forms θ, µ, λ , and corresponding vector Þelds [Here ∇ Z Z Y ≡ α° ( θ ) , Z ≡ α° ( µ ) , W ≡ α° ( λ ) ,

(82)

The Canonical Sub-Riemannian Connection

23

the formula for ∇° is: 1 µ ⋅ ∇° Y W ≡ --- { Y 〈λ|µ〉 + W 〈µ|θ〉 – Z 〈θ|λ〉 + λ ⋅ [ Z, Y ] + µ ⋅ [ Y, W ] – θ ⋅ [ W , Z ] } . 2 7.2.a

(83)

Expression in Local Cošrdinates

Take local cošrdinates for N, so that α° ( x )

–1

jk

matrix ( α ) , where by Lemma 7.2,

∑α

jk

k, r

is represented by a matrix ( g lm ) , and α ( x ) by a

g kr α

rm

= α

jm

.

Take Y ≡ ∂ ⁄ ∂x i , W ≡ ∂ ⁄ ∂x j , and Z ≡ ∂ ⁄ ∂x k in (83), so that µ = s

1

∂

∑ Γ ij gsk = --2- ∑ { ∂ x s

r, s

rs

i

When 〈.|.〉 is non-degenerate, then

( g jr α g sk ) +

∑ g jr α

rs

r

∑ gks dx

s

, etc.; (83) becomes

rs rs ∂ ∂ ( g ir α g sk ) – ( g ir α g sj ) } . ∂x j ∂ xk

(84)

s

= δ j , and (84) reduces to the standard formula for the

Levi-Civita connection for g, namely s

∑ Γ ij gsk = s

7.2.b

1  ∂g jk ∂g ik ∂g ij  --+ – . 2  ∂ xi ∂ x j ∂ xk 

Remark

A similar construction appears in formula (2.2) of Strichartz [27], where he cites unpublished work of N. C. GŸnther. 7.2.c

Proof of Proposition 7.2

First we check that the formula (83) deÞnes a connection. The R-bilinearity of ( Y, W ) → ∇° Y W is ∞

immediate. To prove that ∇° Y fW = f ∇° Y W + ( Yf ) W for all f ∈ C ( N ) , we replace W by fW

and λ by fλ on the right side of (83), and the required identity holds. VeriÞcation that (83) is torsion-free is likewise a straightforward calculation. Next we shall verify (82). By (80), and the deÞnition of duality, taking V ≡ α ( θ ) , ˆ µ ⋅ α ( λ ) = V 〈λ|µ〉 – 〈λ|∇ ˆ µ〉 . µ ⋅ ∇° V α ( λ ) = V ( µ ⋅ α ( λ ) ) – ∇ V V Switch µ and λ in the last expression to obtain: ˆ λ〉 . λ ⋅ ∇° V α ( µ ) = V 〈λ|µ〉 – 〈µ|∇ V Taking U ≡ α ( µ ) and S ≡ α ( λ ) , we see that ˆ µ〉 – 〈µ|∇ ˆ λ〉 = µ ⋅ ∇° S + λ ⋅ ∇° U – V 〈λ|µ〉 . V 〈λ|µ〉 – 〈λ|∇ V V V V

(85)

In terms of the splitting T x∗ N = Ker ( α ( x ) ) ⊕ F x of Lemma 7.1, we may write θ ≡ θ 0 + θ 1 , etc., and we Þnd that, if V ≡ α ( θ ) , then V = α ( θ 1 ) = α° ( θ 1 ) , etc. It follows from (83) that

24

Future Directions

1 µ 1 ⋅ ∇° V S ≡ --- { V 〈λ 1|µ 1〉 + S 〈µ 1|θ 1〉 – U 〈θ 1|λ 1〉 + λ 1 ⋅ [ U , V ] + µ 1 ⋅ [ V, S ] – θ 1 ⋅ [ S, U ] } ; 2 1 λ 1 ⋅ ∇° V U ≡ --- { V 〈λ 1|µ 1〉 + U 〈λ 1|θ 1〉 – S 〈θ 1|µ 1〉 + µ 1 ⋅ [ S, V ] + λ 1 ⋅ [ V, U ] – θ 1 ⋅ [ U , S ] } . 2 To prove (82), we can assume λ 0 = µ 0 = 0 , and now the right side of (85) becomes µ 1 ⋅ ∇° V S + λ 1 ⋅ ∇° V U – V 〈λ 1|µ 1〉 = 0 , as desired.

8

◊

Future Directions

We would like to Þnd out under what conditions on ξ, σ ⋅ σ , ψ, and h the system of PDE (17) - (18) has a unique solution for small time, other than the well-known case where σ ⋅ σ is non-degenerate, and ξ = 0 ; likewise for the parametrized family (22) - (23). It is likely that the conditions will involve the energy of the composite maps { ψ • φ t, 0 ≤ t ≤ δ } . Both stochastic and geometric methods should be considered. Another valuable project would be to derive bounds on the error of approximation involved in the linearization used in Theorem 4.7. This is likely to involve the curvature under the diffusion variance semi-deÞnite metric Ð see Darling [9]. Acknowledgments: The author thanks the Statistics Department at the University of California at Berkeley for its hospitality during the writing of this article, Anton Thalmaier for allowing access to unpublished work, and James Cloutier, James Eells, Etienne Pardoux, and Richard Schoen for their advice.

9

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25

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