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SÉMINAIRE DE PROBABILITÉS (STRASBOURG)
JAMESR.NORRIS
Acompletedifferentialformalismforstochastic
calculusinmanifolds
Séminaire de probabilités (Strasbourg), tome 26 (1992), p. 189-209
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differential formalism
A
complete
manifolds
in
calculus
stochastic
for
Norris
J.R.
Statistical Cambridge
of
Laboratory, University
Mill UK
16 CB2 1SB, England,
Lane, Cambridge
no for the of stochastic
This will contain practitioner
essay surprises experienced
certain
useful others to do
in but be found
calculus manifolds, may by wishing calculations,
take the view
on a manifold. We that
motion Riemannian
relating say to Brownian
old
roles - to construct new from and
the stochastic calculus has two main processes
Ito and differentials for real-
of Stratonovich
to determine The formalism
martingales.
these
valued a and effective to handle
processes provides flexible, complete highly way
for
formalism
a effective
account of
of no
roles. We know
two similarly processes taking
as
manifold. But we do based
values in a differentiable
their certainly below,
everything
on
the notion of a horizontal is at least in the literature
it is on implicit quite
lift, large
is in with
calculus in manifolds. of our main contentions
stochastic One that, dealing
of in
to differentials
one and should not avoid
in cannot try processes
processes manifolds,
to form a line a
vector bundles. For if one wants along semimartingale
example, integral
in a the natural is a in the and one knows
space
manifold, integrand process cotangent
between and Ito involves
the real case that the difference the
from Stratonovich integrals
the differentials be handled
differential of Such horizontal
integrand. may effectively by
to from our
but we it a make the horizontal lift
lift, have made principle disappear formulae
stochastic second
whenever This is done covariant differentials. Our
possible. by
using
contention once these covariant
main is differentials are the formalism
that, introduced,
becomes flexible and as
as as the real-valued case.
complete
We the basic elements
of stochastic calculus to
by in fix no-
begin reviewing R,
tation and
make it clear we
what are to manifolds. Then we show how
extending the
notions of and differential to
semimartingale Stratonovich extend manifolds. Next we
a
consider vector bundle with
connection and the
associated notions
of
horizontal lift of
a and translation. 4
Section to the case a
semimartingale parallel specializes where con-
nection is on
the bundle of our
manifold: this the
definition of
given tangent permits Ito
we a
introduce and show it transforms a
differentials, Doob-Meyer under
decomposition
of
measure the
familiar
Girsanov formula. So is well
change by far, everything known and
is covered in in book in we
greater depth Emery’s [Em] ; particular refer to for the
[Em]
existence of the
stochastic
development.
190
in we
Section introduce notions of covariant stochastic differential
Beginning 5, for
in a vector bundle notions also
semimartingales with connection. These may be found
in book The calculus of these which does not
Elworthy’s [El]. differentials, Elworthy
to
is but use and illuminat-
pursue, remarkably simple, trivial,
perhaps consequently easy
in The fundamental to Ito conversion formulae are
ing applications. Stratonovich given
in In 7 we discuss stochastic differential a
Section 6. Section covariant equations over
in a
manifold. These arise when one consid-
semimartingale equations naturally
given
ers smooth variations of the base in a as for in the
semimartingale parameter, example
of or
stochastic flows the Malliavin calculus. of connection are
study Also, changes seen
to to a class of linear covariant and this leads to sim-
correspond particular equations,
formulae translation and covariant differentials two
ple parallel to
relating corresponding
different connections.
The three
final sections of the differential formalism. In
present Section
applications
we a
8 derive formula on
generalized Feynman-Kac for heat semigroups sections of a
vector bundle. The
fact that we can work
with connections allows a
general unification
of results: in we
include the
previous particular Cameron-Martin formula. The to
ability
switch connections in
the
differential formalism is in a
technical
readily exploited lemma.
This is in In and we
Sections 9 10 ‘stochastic
application pursued simple calculus’
[N]. give
of some known
results on the
effect of on and on
proofs mappings martingales Brownian
a
factorization
result of and
motion, Elworthy Kendall.
including
1. Review of stochastic calculus in R
work on a
We a
probability space 0, with filtration sat-
(S~, equipped
P)
the usual conditions. We shall consider continuous which
isfying only semimartingales,
now called
from on are understood. The
simply continuity same
semimartingales, being
for of finite variation.
goes martingales processes
and
be T
a a and let
Let zi continuous 7, be random
adapted
semimartingale, yt process
T. as ~V 2014~
times with 7 Then, oo,
is in a almost The
The in subsequence surely.
convergence probability, particular converges
is fact
of for the Ito in
natural class
limit is the Ito The integral larger,
integral. integrands
sum fails
all bounded but the Riemann approximation
including locally previsible process,
in Let
general.
zt = +
z0 t0ysdxs,
191
write
a We
is itself
then ~ semimartingale.
dzt =
zt has a decomposition
Doob-Meyer
Every unique
semimartingale
= + +
Zt Zo ,
~~ ~?
where is a local from is a of finite varia-
starting 0, process
xmt martingale and xft (locally)
write in
tion from It is sometimes convenient to the differential
starting 0. decomposition
notation
. dzt
=~+~.
Ito
The the has
respects Doob-Meyer decomposition: ~ decomposition
integral
dzi = +
This is the merit of the
principal Ito
integral.
now is a then
Suppose that ~ semimartingale;
S~(’~)~(~))(’(~)-.(~))~f~.
This limit
is the
Stratonovich Also
integral.
(y(k 2N)-y (k+1 2N)) (x (k+1) 2N)-x (k 2N)) ~ 03C3 ~xs~ys.
We call
this
limit
the quadratic
integral. Clearly
ys~xs = ~xs~ys.
03C3 03C3 ysdxs + 03C3
If either are of
xt or yt finite variation, then, the the
by Cauchy-Schwarz
inequality,
vanishes
and
the
Ito and
quadratic Stratonovich In
integral integrals the
agree. general
process
qt ==
. /
is of
finite
variation. We write
=
~ .
Equivalently
=
dqt dxtdyt.
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