2.2 El Derecho Penal de la Tercera Velocidad:

We say that an _Rm-valued process X = (X1, . . . , Xm) is a semimartingale if each of its components Xi _{is a semimartingale in the sense of the previous section. As}

mentioned above, we are concerned only with continuous semimartingales. If X =

X1, . . . , Xm

is such a process then Itô’s formula states that for f ∈ C2(Rm) we

have f(Xt) =f(X0) + Z t 0 Dif(Xs)dXsi+ 1 2 Z t 0 Djkf(Xs)d[Xj, Xk]s (2.8)

for all t≥0, almost surely, whereDi and Djk stand for the first and second partial

derivatives off and where we employ the usual summation convention over repeated up-down indices.

2.2.2 Semimartingales on M

Suppose that M is a smooth metrizable manifold of dimension m. We say that an M-valued process X is a semimartingale if for each f ∈ C2(M) the real-valued process f(X) is a semimartingale in the sense of the previous section. Note that if M = Rm then this definition agrees with the one in the previous subsection.

The collection of continuous semimartingales exhibits certain stability properties. For example, if F : M → M˜ is a smooth map between manifolds and if X is a continuous semimartingale on M thenF(X) is a continuous semimartingale on M˜. Furthermore, if X is a continuous semimartingale with respect to _P and if _Q is a probability measure which is absolutely continuous with respect to P then X is a

then the collection of processes which are continuous semimartingales with respect to _Pcoincides with the collection of those which are with respect to _Q.

2.2.3 The Orthonormal Frame Bundle

Now suppose that M is a Riemannian manifold equipped with its Levi-Civita con- nection. In this setting we wish to write down a version of Itô’s formula for M- valued continuous semimartingales. In order to do this we must first introduce some auxilliary objects. An orthonormal frame at p ∈ M is an R-linear isometry u:Rm →TpM and the collection of all orthonormal frames at a pointp is denoted

by Op(M). The orthonormal frame bundle is then defined to be the disjoint union

O(M) :=F

p∈MOp(M). Since each fibre Op(M) is diffeomorphic to the orthogonal

group O(m,R), it follows thatO(M) can be made into a differentiable manifold of

dimension m₂(m+ 1) and the canonical projection Π :O(M)→M is a smooth map between manifolds.

2.2.4 Horizontal Lifts and Antidevelopment

A smooth curve U taking values in O(M) is called horizontal if for each e ∈ _Rm

the vector field U e is parallel along the curve ΠU. If u ∈ O(M) then a vector in

TuO(M) is calledhorizontal if it is the tangent vector to a horizontal curve starting at u. We denote by HuO(M) the space of all horizontal tangent vectors at u. It

follows thatTuO(M) = ker(DuΠ)⊕HuO(M)and foru∈ O(M)there is a canonical lift map H(u) :Rm →HuO(M) given by

H(u)e= (DuΠ|HuO(M))

−1_(ue).

An O(M)-valued continuous semimartingale U is called horizontal if there exists an F₀-measurable O(M)-valued random variable U0 and an Rm-valued continuous

semimartingaleZ such thatU solves the Stratonovich equation

with initial conditionU0. For the sense in which this equation should be interpreted,

see Elworthy [1982]. The process Z, which if it exists can be shown to be unique, is called the antidevelopment of U. IfX is anM-valued continuous semimartingale then anO(M)-valued horizontal continuous semimartingale U is called a horizontal liftofXifΠU =X. If one specifies anF₀-measurableO(M)-valued random variable

U0 such thatΠU0=X0 then there exists aunique horizontal liftU ofX with initial

conditionU0and the dependence of the processU (and of its antidevelopmentZ) on

the choice of U0 commutes with the action of the orthogonal group. Alternatively,

if we begin with an F₀-measurable O(M)-valued random variable U0 and an Rm-

valued continuous semimartingale Z then we say that the projection onto M of the maximal solution to equation (2.9) with inital condition U0 is the development of

the semimartingaleZ onto M with respect toU0.

2.2.5 Itô’s Formula

We are now in a position to write down an intrinsic version of Itô’s formula. In particular, iff ∈C2(M)withXa continuous semimartingale onM with a horizontal lift U and anti-development Z and if (e1, . . . , em) is an orthonormal basis for Rm

then f(Xt) =f(X0) + Z t 0 Useif(Xs)dZ_si+1 2 Z t 0 UsejUsekf(Xs)d[Zj, Zk]s

for all t≥0, almost surely, which can can be written more succinctly as

f(Xt) =f(X0) + Z t 0 h∇f(Xs), UsdZsi+ 1 2 Z t 0 tr HessXsf(Us, Us)d[Z]s. (2.10)

Note that if M = Rm then, after the usual identifications, we can choose Us =

id_Rm and Z = X and this formula reduces to formula (2.8). Just as the basic Itô

formula (2.1) can be extended from C2 functions to those which can be written as the difference of two convex functions, so formula (2.10) can be extended to a wider class of possibly non-differentiable functions.

2.3

Local Time for Semimartingales on Manifolds