Stochastic analysis on manifolds explained
In mathematics, stochastic analysis on manifolds or stochastic differential geometry is the study of stochastic analysis over smooth manifolds. It is therefore a synthesis of stochastic analysis (the extension of calculus to stochastic processes) and of differential geometry.
The connection between analysis and stochastic processes stems from the fundamental relation that the infinitesimal generator of a continuous strong Markov process is a second-order elliptic operator. The infinitesimal generator of Brownian motion is the Laplace operator and the transition probability density
of Brownian motion is the minimal heat kernel of the
heat equation. Interpreting the paths of Brownian motion as
characteristic curves of the operator, Brownian motion can be seen as a stochastic counterpart of a flow to a second-order partial differential operator.
Stochastic analysis on manifolds investigates stochastic processes on non-linear state spaces or manifolds. Classical theory can be reformulated in a coordinate-free representation. In that, it is often complicated (or not possible) to formulate objects with coordinates of
. Thus, we require an additional structure in form of a linear
connection or
Riemannian metric to define martingales and Brownian motion on manifolds. Therefore, controlled by the Riemannian metric, Brownian motion will be a local object by definition. However, its stochastic behaviour determines global aspects of the topology and geometry of the manifold.
with respect to a manifold
and can be constructed as the solution to a non-canonical
stochastic differential equation on a Riemannian manifold. As there is no Hörmander representation of the operator
if the manifold is not
parallelizable, i.e. if the
tangent bundle is not trivial, there is no canonical procedure to construct Brownian motion. However, this obstacle can be overcome if the manifold is equipped with a connection: We can then introduce the
stochastic horizontal lift of a
semimartingale and the
stochastic development by the so-called
Eells-Elworthy-Malliavin construction.
[1] [2] The latter is a generalisation of a horizontal lift of smooth curves to horizontal curves in the frame bundle, such that the anti-development and the horizontal lift are connected by a stochastic differential equation. Using this, we can consider an SDE on the orthonormal frame bundle of a Riemannian manifold, whose solution is Brownian motion, and projects down to the (base) manifold via stochastic development. A visual representation of this construction corresponds to the construction of a spherical Brownian motion by rolling without slipping the manifold along the paths (or footprints) of Brownian motion left in Euclidean space.[3]
Stochastic differential geometry provides insight into classical analytic problems, and offers new approaches to prove results by means of probability. For example, one can apply Brownian motion to the Dirichlet problem at infinity for Cartan-Hadamard manifolds[4] or give a probabilistic proof of the Atiyah-Singer index theorem.[5] Stochastic differential geometry also applies in other areas of mathematics (e.g. mathematical finance). For example, we can convert classical arbitrage theory into differential-geometric language (also called geometric arbitrage theory).[6]
Preface
For the reader's convenience and if not stated otherwise, let
(\Omega,l{A},(l{F}t)t\geq,P)
be a
filtered probability space and
be a smooth manifold. The filtration satisfies the usual conditions, i.e. it is right-continuous and complete. We use the
Stratonovich integral which obeys the classical
chain rule (compared to
Itô calculus). The main advantage for us lies in the fact that stochastic differential equations are then stable under
diffeomorphisms
between manifolds, i.e. if
is a solution, then also
is a solution under transformations of the stochastic differential equation.
Notation:
is. the
tangent bundle of
.
is the
cotangent bundle of
.
is the
-module of
vector fields on
.
is the Stratonovich integral.
is the space of test functions on
, i.e.
is smooth and has compact support.
\widehat{M}:=M\cup\{infty\}
is the
one-point compactification (or Alexandroff compactification).
Flow processes
Flow processes (also called
-diffusions) are the probabilistic counterpart of
integral curves (
flow lines) of vector fields. In contrast, a flow process is defined with respect to a second-order differential operator, and thus, generalises the notion of deterministic flows being defined with respect to a
first-order operator.
Partial differential operator in Hörmander form
Let
be a vector field, understood as a derivation by the
-
isomorphism\Gamma(TM)\to\operatorname{Der}RCinfty(M), A\mapsto(f\mapstoAf)
for some
. The map
is defined by
. For the composition, we set
for some
.
A partial differential operator (PDO)
is given in
Hörmander form if and only there are vector fields
A0,A1,...,Ar\in\Gamma(TM)
and
can be written in the form
.
Flow process
Let
be a PDO in Hörmander form on
and
a starting point. An adapted and continuous
-valued process
with
is called a
flow process to
starting in
, if for every test function
and
the process
N(f)t:=f(Xt)-f(X0)-\int
Lf(Xr)dr
is a martingale, i.e.
E\left(N(f)t\midl{F}s\right)=N(f)s, \foralls\leqt
.
Remark
For a test function
, a PDO
in Hörmander form and a flow process
(starting in
) also holds the flow equation, but in comparison to the deterministic case
only in mean
.
and we can recover the PDO by taking the time derivative at time 0, i.e.
.
Lifetime and explosion time
Let
be open und
a predictable
stopping time. We call
the
lifetime of a continuous semimartingale
on
if
- there is a sequence of stopping times
with
, such that
-almost surely on
.
is a semimartingale.
Moreover, if
for almost all
, we call
explosion time.
A flow process
can have a finite lifetime
. By this we mean that
is defined such that if
, then
-almost surely on
we have
in the one-point compactification
\widehat{M}:=M\cup\{infty\}
. In that case we extend the process path-wise by
for
.
Semimartingales on a manifold
A process
is a s
emimartingale on
, if for every
the random variable
is an
-semimartingale, i.e. the composition of any smooth function
with the process
is a real-valued semimartingale. It can be shown that any
-semimartingale is a solution of a stochastic differential equation on
. If the semimartingale is only defined up to a finite lifetime
, we can always construct a semimartingale with infinite lifetime by a transformation of time. A semimartingale has a quadratic variation with respect to a section in the bundle of
bilinear forms on
.
Introducing the Stratonovich Integral of a differential form
along the semimartingale
we can study the so called winding behaviour
of
, i.e. a generalisation of the winding number.Stratonovich integral of a 1-form
Let
be an
-valued semimartingale and
be a
1-form. We call the integral
\intX\alpha:=\int\alpha(\circdX)
the
Stratonovich integral of
along
. For
we define
f(X)\circ\alpha(\circdX):=f(X)\circd(\intX\alpha)
.
SDEs on a manifold
A stochastic differential equation on a manifold
, denoted
SDE on
, is defined by the pair
including a bundle homomorphism (i.e. a
homomorphism of vector bundles) or the (
)-tuple
with vector fields
given. Using the
Whitney embedding, we can show that there is a unique maximal solution to every SDE on
with initial condition
. If we have identified the maximal solution, we recover directly a flow process
to the operator
.
Definition
An SDE on
is a pair
, where
is a continuous semimartingale on a finite-dimensional
-vector space
; and
is a (smooth)
homomorphism of
vector bundles over
where
is a linear map.
The stochastic differential equation
is denoted by
or
The latter follows from setting
with respect to a basis
and
-valued semimartingales
with
.
As for given vector fields
there is exactly one bundle homomorphism
such that
, our definition of an SDE on
as
is plausible.
If
has only finite life time, we can transform the time horizon into the infinite case.
[7] Solutions
Let
be an SDE on
and
an
-measurable random variable. Let
be a continuous adapted
-valued process with life time
on the same probability space such as
. Then
is called a
solution to the SDE
with initial condition
up to the life time
, if for every test function
the process
is an
-valued semimartingale and for every stopping time
with
, it holds
-almost surely
f(X\tau)=f(x0)+
A(Xs)\circdZs
,
where
is the
push-forward (or differential) at the point
. Following the idea from above, by definition
is a semimartingale for every test function
, so that
is a
semimartingale on
.
If the lifetime is maximal, i.e.
\{\zeta<infty\}\subset\left\{\lim\limitst\nearrowXt=inftyin\widehat{M}\right\}
-almost surely, we call this solution the
maximal solution. The lifetime of a maximal solution
can be extended to all of
, and after extending
to the whole of
, the equation
f(Xt)=f(X0)+\int
(df)XA(X)\circdZ, t\geq0
,
holdsup to indistinguishability.
Remark
Let
with a
-dimensional Brownian motion
, then we can show that every maximal solution starting in
is a flow process to the operator
.
Martingales and Brownian motion
. However, to follow a canonical
ansatz, we need some additional structure. Let
be the
orthogonal group; we consider the canonical SDE on the
orthonormal frame bundle
over
, whose solution is Brownian motion. The orthonormal frame bundle is the collection of all sets
of
orthonormal frames of the tangent space
O(M):=cup\limitsx\inOx(M)
or in other words, the
-
principal bundle associated to
.Let
be an
-valued semimartingale. The solution
of the SDE
defined by the projection
of a Brownian motion
on the Riemannian manifold, is the
stochastic development from
on
. Conversely we call
the
anti-development of
or, respectively,
. In short, we get the following relations:
W\leftrightarrowU\leftrightarrowX
, where
is an
-valued semimartingale; and
is an
-valued semimartingale.
For a Riemannian manifold we always use the Levi-Civita connection and the corresponding Laplace-Beltrami operator
. The key observation is that there exists a lifted version of the Laplace-Beltrami operator on the orthonormal frame bundle. The fundamental relation reads, for
,
\DeltaMf(x)=\DeltaO(M)(f\circ\pi)(u)
for all
with
, and the operator
on
is well-defined for so-called
horizontal vector fields. The operator
is called
Bochner's horizontal Laplace operator.
Martingales with linear connection
To define martingales, we need a linear connection
. Using the connection, we can characterise
-martingales, if their anti-development is a local martingale. It is also possible to define
-martingales without using the anti-development.
We write
to indicate that equality holds
modulo differentials of
local martingales.
Let
be an
-valued semimartingale. Then
is a
martingale or
-martingale, if and only if for every
, it holds that
d(f\circX)\stackrel{m}{=}\tfrac{1}{2}(\nabladf)(dX,dX).
Brownian motion on a Riemannian manifold
Let
be a
Riemannian manifold with Laplace-Beltrami operator
. An adapted
-valued process
with maximal lifetime
is called a
Brownian motion
, if for every
is a local
-martingale with life time
. Hence, Brownian motion Bewegung is the diffusion process to
. Note that this characterisation does not provide a canonical procedure to define Brownian motion.
References and notes
- Book: Stochastic differential equations on manifolds . 1982 . 70.
- Book: Géométrie différentielle stochastique . 1978.
- Book: Stochastische Analysis: Eine Einführung in die Theorie der stetigen Semimartingale . 978-3-519-02229-9 . 349-544.
- Book: Brownian Motion and the Dirichlet Problem at Infinity on Two-dimensional Cartan-Hadamard Manifolds . 2014 . 41 . 443–462 . 10.1007/s11118-013-9376-3.
- Book: Stochastic Analysis on Manifolds . 38.
- Book: Geometric Arbitrage Theory and Market Dynamics . 2015 . 7 . 10.3934/jgm.2015.7.431 . 4.
- Book: Stochastische Analysis: Eine Einführung in die Theorie der stetigen Semimartingale . 978-3-519-02229-9 . 364.