Intrinsic Location Parameter of a Diffusion Process

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"intrinsic location parameter"(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 flow 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.


Background
The relationship between martingales and parabolic partial differential equations was pointed out in the classic paper of Doob [11]: the solution to the one-dimensional heat equation, with a given function ψ as the boundary condition , is given by , (1) where is Brownian motion with . This can also be expressed as the initial value of the martingale which terminates at at time T .
In the case of nonlinear parabolic PDE , the martingale 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.

Main Results
Suppose X is a Markov diffusion process on , 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 Levi-Civita connection Γ , and a Laplace-Beltrami operator ∆ . We may treat X as a diffusion on N with generator , where ξ is a vector Þeld.
For sufÞciently small , has an Òintrinsic location parameterÓ, deÞned to be the non-random initial value of a Γ -martingale V terminating at . 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 be the ßow of the vector Þeld ξ , and let . Our main result is that can be intrinsically approximated to Þrst order in by where . This is computed in local coordinates. More generally, we Þnd an intrinsic location parameter for , if is a map into a Riemannian manifold M . We also treat the case where is random.

Diffusion Process Model
Consider a Markov diffusion process with values in a connected manifold N of dimension p , represented in coordinates by , (2) where is a vector Þeld on N , , and W is a Wiener process in .We assume for simplicity that the coefÞcients , are with bounded Þrst derivative.

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 deÞne a semi-deÞnite metric on the cotangent bundle , which we call the diffusion variance semi-deÞnite metric, by the formula . (3) Note that may be degenerate. This semi-deÞnite metric is actually intrinsic: changing coordinates for the diffusion will give a different matrix , but the same semi-deÞnite metric. We postulate:  (4) In local cošrdinates, is expressed by a Riemannian metric tensor , such that if , then . (5) The Christoffel symbols for the canonical sub-Riemannian connection are speciÞed by (84) below. The corresponding local connector can be written in the more compact notation: , (6) where is a 1-form, acting on the tangent vector w.

Intrinsic Description of the Process
The intrinsic version of (2) is to describe X as a diffusion process on the manifold N with generator (7) where ∆ is the (possibly degenerate) Laplace-Beltrami operator associated with the diffusion variance, and ξ is a vector Þeld, whose expressions in the local coordinate system are as follows: , . (8) Note that has been speciÞed by (3) and (6).

Γ-Martingales
Let be a connection on a manifold M. An Γ-martingale is a kind of continuous semimartingale on M which generalizes the notion of continuous martingale on : see Emery [17] and Darling [7]. We summarize the main ideas, using global coordinates for simplicity.
Among continuous semimartingales in , It™Õs formula shows that local martingales are characterized by , , (9) where is the differential of the joint quadratic variation process of and , and refers to the space of real-valued, continuous local martingales (see Revuz and Yor [26]). For vector Þelds ξ, ζ on , and , the smooth one-forms, a connection Γ gives an intrinsic way of differentiating ω along ξ to obtain is also written . When , this gives the Hessian where the are the Christoffel symbols. The intrinsic, geometric restatement of (9) is to characterize a Γ−martingale X by the requirement that , . (10) This is equivalent to saying that for , where . (11) If N has a metric g with metric tensor , we say that X is an Γ-martingale if (10) holds and also . (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.

Condition for the Intrinsic Location Parameter
Consider a diffusion process on a p-dimensional manifold N with generator where ∆ is the Laplace-Beltrami operator associated with the diffusion variance, and ξ is a vector Þeld, as in (8). The coordinate-free construction of the diffusion X, given a Wiener process W on , uses the linear or orthonormal frame bundle: see Elworthy [15] p. 252. We suppose .
Also suppose is a Riemannian manifold, with Levi-Civita connection , and is a map. The case of particular interest is when , , and the metric on N is a Ògeneralized inverseÓ to in the sense of (5). The general case of is needed in the context of nonlinear Þltering: see Darling [8].
Following Emery and Mokobodzki [18], we assert the following: This need not be unique, but we will specify a particular choice below. In the case where does not depend on x, then the local connector Γ, given by (6), is zero, and is simply . However our assertion is that, when Γ is not the Euclidean connection, the right measure of location is , and not . We begin by indicating why an exact determination of is not computationally feasible in general.

Relationship with Harmonic Mappings
For simplicity of exposition, let us assume that there are diffeomorphisms and which induce global coordinate systems for N and for M, respectively. By abuse of notation, we will usually neglect the distinction between and , and write x for both. is given by (3) and (6), and the local connector comes from the Levi-Civita connection for .
In order to Þnd , we need to construct an auxiliary adapted process , with values in , such that the processes and satisfy the following system of forward-backwards SDE: , ; (13) , . (14) We also require that . (15) [Equation (14) and condition (15) together say that V is an -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 is not However consider the second fundamental form of a mapping . Recall that may be expressed in local coordinates by: (16) for , . 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 , for , such that . (18) For , the right side of (17) is . Following the approach of Pardoux and Peng [23], It™Õs formula shows that , (20) solves (14). In particular . (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 is given by . (21) Note, incidentally, that this formula still makes sense when is degenerate. In the case where is non-degenerate and smooth, , and , the inverse function theorem method of Hamilton [19], page 122, sufÞces to show existence of a unique smooth solution to (17) - (18) when is sufÞciently small. For a more detailed account of the properties of the solutions when , see Struwe [28], pages 221 -235. Whereas Eells and Sampson [12] showed the existence of a unique global solution when has non-positive curvature, Chen and Ding [4] showed that in certain other cases blow-up of solutions is inevitable. The case where 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 is degenerate, will appear in Thalmaier [30]. Work by other authors, using Hšrmander conditions on the generator , is in progress. For now we shall merely assume:

Hypothesis I
Assume conditions on ξ, , ψ, and h sufÞcient to ensure existence and uniqueness of a solution , for some .

DeÞnition: the Intrinsic Location Parameter
For , the intrinsic location parameter of is deÞned to be , where .
This depends upon the generator , given in (8), where ∆ may be degenerate; on the mapping ; and on the metric h for M. It is precisely the initial value of an -adapted -martingale on M, with terminal value . However by using the solution of the PDE, we force the intrinsic location parameter to be unique, and to have some regularity as a func- 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.

A Parametrized Family of Heat Flows
Consider a parametrized family of equations of the type (17), namely , , (22) . (23) Note that the case gives the system (17), while the case gives , where is the ßow of the vector Þeld ξ.
In a personal communication, Etienne Pardoux has indicated the possibility of a probabilistic construction, involving the system of FBSDE (47) and (56), of a unique family of solutions for sufÞciently small , and for small time , 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, , ψ, and h sufÞcient to ensure existence of and such that there is a unique mapping from to M satisfying (22) and (23) for each .

4.4.a Notation
For any vector Þeld ζ on N, and any differentiable map into a manifold P, the Òpush-for-wardÓ takes the value at ; likewise .
We must also assume for the following theorem that we have chosen a generalized inverse to , 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.

Theorem (PDE Version)
Assume Hypothesis II, and that . Then, in the tangent space , (24) where is the ßow of the vector Þeld ξ, , and . (25) In the special case where , , and , the right side of (24) simpliÞes to the part in parentheses {É}.

4.5.a DeÞnition
The expression (24) is called the approximate intrinsic location parameter in the tangent space , denoted .

4.5.b Remark: How the Formula is Useful
First we solve the ODE for the ßow of the vector Þeld ξ, compute at 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 . (26) 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.

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 parameter ε, so that in (22). Instead of X in (13), we consider a family of diffusion processes on the time interval , where has generator . Likewise V in (14) will be replaced by a family of -martingales, with , and . Note, incidentally, that such parametrized families of -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 as in (13) (27) Each -martingale is now adapted to the larger Þltration . In particular, is now a random variable in depending on .

4.6.b DeÞnition
In the case of a random initial value as above, the approximate intrinsic location parameter of in the tangent space , denoted , is deÞned to be . (28) We will see in Section 6.3 below that this deÞnition makes sense. This is the same as , and coincides with , given by (24), in the case where .

4.6.c Some Integral Formulas
Given the ßow of the vector Þeld ξ, the derivative ßow is given locally by , (29) where , for . In local coordinates, we compute as a matrix, given by .
Note, incidentally, that could be called the intrinsic variance parameter of .

Theorem (Local Cošrdinates, Random Initial Value Version)
Under the conditions of Theorem 4.5, with random initial value as in Section 4.6.a, the approximate intrinsic location parameter exists and is equal to the right side of (24), after redeÞning .
In local coordinates, is given by (33) where , , and is given by (31).

Example of Computing an Intrinsic Location Parameter
The following example shows that Theorem 4.7 leads to feasible and accurate calculations.

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 must satisfy , and the trajectory must lie within a set on which is constant. Therefore we may identify the state space N with , since the v-component lies on a sphere, and the a-component is perpendicular to v, and hence tangent to .
Within a Cartesian frame, X is a process in with components V (velocity) and A (acceleration), and the equations of motion take the nonlinear form: .
Here the square matrix consists of four matrices, λ and γ are constants, W is a three-dimensional Wiener process, and if , , , Note that is precisely the projection onto the orthogonal complement of v in , and has been chosen so that .

Geometry of the State Space
The diffusion variance metric (3) is degenerate here; noting that , we Þnd .
The rescaled Euclidean metric on is a generalized inverse to α in the sense of (4), since . We break down a tangent vector ζ to into two 3-dimensional components and . The constancy of implies that .
Referring to formula (6) for the local connector , , .
Taking Þrst and second derivatives of the constraint , we Þnd that , .
Using the last identity, we obtain from (6) the formula , .
In order to compute (33), note that, in particular, .

Derivatives of the Dynamical System
It follows from (8), (35), and (42) that the formula for the intrinsic vector Þeld ξ is: .
Differentiate under the assumptions is constant and , to obtain , .

Ingredients of the Intrinsic Location Parameter Formula
Let be the inclusion of the state space into Euclidean . Thus in formula (33), the local connector is zero on the target manifold, J is the identity, and is zero.
When is taken to be zero, the formula for the approximate intrinsic location parameter for becomes: , where and are the velocity and acceleration components of , for , and and are given by (29) and (30). A straightforward integration scheme for calculating at the same time, using a discretization of , is: , , .
Since the local connector is zero on the target manifold, geodesics are simply straight lines, and is a suitable estimate of the mean position of .

FIGURE 1 SIMULATIONS OF THE MEAN OF AN SDE, VERSUS ITS APPROXIMATE ILP
We created simulations of the process (35), with and , on the time interval , which was discretized into 25 subintervals for integration purposes. In each case V and

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, and identify N with and M with . Step

I: Differentiation of the State Process with Respect to a Parameter
We consider a family of diffusion processes on the time interval , with initial values ; here is a zero-mean random variable in , independent of W, with covariance , and has generator .
Note that, in local coordinates, the SDE for is not Ò Ò, because the limiting case when would then be the ODE based on the vector Þeld , which is not the same as ξ, which is given by (8). Instead the SDE is (47) where we use the notation . (48) In the case , the solution is deterministic, namely . Note that, in local cošrdinates, .
It is well known that, if the vector Þeld ξ and the semi-deÞnite metric are sufÞciently differentiable, then the stochastic processes and exist and satisfy the following SDEs: Now (49) and (50) give: , (53) , where . Let be the two-parameter semigroup of deterministic matrices given by (29), so that ; .

Step II: Differentiation of the Gamma-Martingale with Respect to a Parameter
Consider the pair of processes obtained from (19) and (20), where u is replaced by . As in the case where , gives an adapted solution to the backwards equation corresponding to (14), namely .
However the version of (20) which applies here is , so we may replace by , and the equation becomes .
By the regularity of , it follows that , , , and exist, and satisfy the following equations: Note also that for all . Take . By combining (54) and (57), we see that, if , ; . (60) From (55) and (62) we obtain: .
The expected value of a quadratic form in an random vector η is easily computed to be . In this case, , where is given by (30), so we obtain . (63)

Step III: A Taylor Expansion Using the Exponential Map
Let , and deÞne . Referring to (28), we are seeking . It follows immediately from the geodesic equation that , It follows from (59) and (63) that A Taylor expansion based on (64) then the formula becomes . This establishes the formula (33). ◊

Step IV: Intrinsic Version of the Formula
It remains to prove that (24), with as in (32), is the intrinsic version of (33). We abbreviate here by writing as . By deÞnition of the ßow of ξ, and so, differentiating with respect to x, and exchanging the order of differentiation, (66) or, by analogy with (29) The last term in (72) can be written, using (67), as , (73) for . We will replace by (74) where . Observe that .
Moreover from (75) and (67), it is easily checked that .

Since
, it follows upon integration from 0 to δ that in , .
However the formula (16) for can be written as , and . We take , and add (77) and (78): .
The equivalence of (24) and (33) where is the canonical isomorphism induced by the Euclidean inner product. The relation between α and is that, omitting x, , . (80)

Lemma
Under the constant-rank assumption, any Riemannian metric on N induces an orthogonal splitting of the .
We omit the proof that is a vector bundle.

Proposition
Suppose σ is a constant-rank section of , inducing a semi-deÞnite metric on and a vector bundle morphism as in (79)    When is non-degenerate, then , and (84) reduces to the standard formula for the Levi-Civita connection for g, namely .

7.2.c Proof of Proposition 7.2
First we check that the formula (83) deÞnes a connection. The R-bilinearity of is immediate. To prove that for all , we replace W by and λ by on the right side of (83), and the required identity holds. VeriÞcation that (83) is torsion-free is likewise a straightforward calculation.
Switch µ and λ in the last expression to obtain: .