Filetype PDF | Posted on 26 Jan 2023 | 2 years ago
Authentication
digital resources
123x Filetype PDF File size 0.45 MB Source: www.hairer.org
Introduction to Malliavin Calculus
March25,2021
M.Hairer
Imperial College London
Contents
1 Introduction 1
2 WhitenoiseandWienerchaos 3
3 TheMalliavinderivative and its adjoint 9
4 Smoothdensities 16
5 MalliavinCalculusforDiffusionProcesses 18
6 Hörmander’sTheorem 22
7 Hypercontractivity 27
8 Graphicalnotationsandthefourthmomenttheorem 33
9 ConstructionoftheΦ4 field 40
2
1 Introduction
One of the main tools of modern stochastic analysis is Malliavin calculus. In
a nutshell, this is a theory providing a way of differentiating random variables
defined on a Gaussian probability space (typically Wiener space) with respect to
the underlying noise. This allows to develop an “analysis on Wiener space”, an
infinite-dimensional generalisation of the usual analytical concepts we are familiar
with on R=. (Fourier analysis, Sobolev spaces, etc.)
ThemaingoalofthiscourseistodevelopthistheorywiththeproofofHörman-
der’stheoreminmind. Thiswasactuallytheoriginalmotivationforthedevelopment
of the theory and states the following. Consider a stochastic differential equation
Introduction 2
on R= given by
<
3- (C) = + (-(C)) 3C + Õ+ (-(C)) ◦ 3, (C)
9 9,0 9,8 8
8=1
< 3 <
=+ (-)3C+ 1ÕÕ+ (-)m + (-)3C+Õ+ (-)3,(C),
9,0 2 :,8 : 9,8 9,8 8
8=1 :=1 8=1
where the + are smooth functions, the , are i.i.d. Wiener processes, ◦3,
9,: 8 8
denotes Stratonovich integration, and 3, denotes Itô integration. We also write
8
this in the shorthand notation
<
3- =+ (-)3C+Õ+(-)◦3,(C), (1.1)
C 0 C 8 C 8
8=1
where the + are smooth vector fields on R= with all derivatives bounded. One
8
might then ask under what conditions it is the case that the law of - has a density
C
with respect to Lebesgue measure for C > 0. One clear obstruction would be the
existence of a (possibly time-dependent) submanifold of R= of strictly smaller
dimension (say : < =) which is invariant for the solution, at least locally. Indeed,
=-dimensional Lebesgue measure does not charge any such submanifold, thus
ruling out that transition probabilities are absolutely continuous with respect to it.
If such a submanifold exists, call it say M ⊂ R × R=, then it must be the case
that the vector fields m − + and {+ }< are all tangent to M. This implies in
C 0 8 8=1
particular that all Lie brackets between the +9’s (including 9 = 0) are tangent to
M, so that the vector space spanned by them is of dimension strictly less than
=+1. Since the vector field m −+ is the only one spanning the “time” direction,
C 0
we conclude that if such a submanifold exists, then the dimension of the vector
space V(G) spanned by {+ (G)}< as well as all the Lie brackets between the + ’s
8 8=1 9
evaluated at G, is strictly less than = for some values of G.
This suggests the following definition. Define V = {+ }< and then set
0 8 8=1
recursively Ø
V =V∪{[+,+] : + ∈ V, 8 ≥ 0} , V= V,
=+1 = 8 = =≥0 =
as well as V(G) = span{+(G) : + ∈ V}.
Definition 1.1 Given a collection of vector fields as above, we say that it satisfies
the parabolic Hörmander condition if dim V(G) = = for every G ∈ R=.
Conversely,Frobenius’stheorem(seeforexample[Law77])isadeeptheoremin
Riemanniangeometrywhichcanbeinterpretedasstatingthatifdim V(G) = : < =
WhitenoiseandWienerchaos 3
for all G in some open set Oof R=, then R×Ocan be foliated into : +1-dimensional
submanifolds with the property that m − + and {+ }< are all tangent to this
C 0 8 8=1
foliation. This discussion points towards the following theorem.
Theorem1.2(Hörmander) Consider(1.1), as well as the vector spaces V(G) ⊂
R= constructed as above. If the parabolic Hörmander condition is satisfied, then
the transition probabilities for (1.1) have smooth densities with respect to Lebesgue
measure.
The original proof of this result goes back to [Hör67] and relied on purely
analytical techniques. However, since it has a clear probabilistic interpretation,
a more “pathwise” proof of Theorem 1.2 was sought for quite some time. The
breakthrough came with Malliavin’s seminal work [Mal78], where he laid the
foundations of what is now known as the “Malliavin calculus”, a differential
calculus in Wiener space, and used it to give a probabilistic proof of Hörmander’s
theorem. This new approach proved to be extremely successful and soon a
number of authors studied variants and simplifications of the original proof
[Bis81b, Bis81a, KS84, KS85, KS87, Nor86]. Even now, more than three decades
after Malliavin’s original work, his techniques prove to be sufficiently flexible to
obtain related results for a number of extensions of the original problem, including
for example SDEs with jumps [Tak02, IK06, Cas09, Tak10], infinite-dimensional
systems [Oco88, BT05, MP06, HM06, HM11], and SDEs driven by Gaussian
processes other than Brownian motion [BH07, CF10, HP11, CHLT15].
1.1 Original references
The material for these lecture notes was taken mostly from the monographs
[Nua06, Mal97], as well as from the note [Hai11]. Additional references to some
of the original literature can be found at the end.
2 WhitenoiseandWienerchaos
Let = !2(R+,R<)(butforthepurposeofmuchofthissection, couldbeanyreal
separable Hilbert space), then white noise is a linear isometry , : → !2(Ω,P)
for some probability space (Ω,P), such that each ,(ℎ) is a real-valued centred
Gaussian random variable. In other words, for all 5,6 ∈ , one has
E,(ℎ) = 0 , E,(ℎ),(6) = hℎ,6i ,
and each ,(ℎ) is Gaussian. Such a construct can easily be shown to exist.
Indeed, it suffices to take a sequence {b=}=≥0 of i.i.d. normal random variables
and an orthonormal basis {4=}=≥0 of . For ℎ = Í=≥0ℎ=4= ∈ , it then
sufficestoset,(ℎ) = Í=≥0ℎ=b=,withtheconvergence taking place in !2(Ω,P).
WhitenoiseandWienerchaos 4
Conversely, given a white noise, it can always be recast in this form (modulo
possible modifications on sets of measure 0) by setting b= = ,(4=).
Awhite noise determines an <-dimensional Wiener process, which we call
again ,, in the following way. Write 1(8) for the element of given by
[0,C)
(1(8) )9(B) = 1 if B ∈ [0,C) and 9 = 8, (2.1)
[0,C) 0 otherwise,
and set , (C) = ,(1(8) ). It is then immediate to check that one has indeed
8 [0,C)
E,(B), (C) = X (B∧C) ,
8 9 8 9
so that this is a standard Wiener process. For arbitrary ℎ ∈ , one then has
< ¹ ∞
,(ℎ) = Õ ℎ (B) 3, (B) , (2.2)
0 8 8
8=1
with the right hand side being given by the usual Wiener–Itô integral.
Let now = denote the =th Hermite polynomial. One way of defining these is
to set 0 = 1 and then recursively by imposing that
′(G) = = =−1(G) (2.3)
=
and that, for = ≥ 1, E =(-) = 0 for a normal Gaussian random variable - with
variance1. Thisdeterminesthe = uniquely,sincethefirstconditiondetermines =
uptoaconstant, with the second condition determining the value of this constant
uniquely. The first few Hermite polynomials are given by
1(G) = G , 2(G) = G2 −1 , 3(G) = G3 −3G .
Remark2.1 Beware that the definition given here differs from the one given in
[Nua06] by a factor =!, but coincides with the one given in most other parts of
the mathematical literature, for example in [Mal97]. In the physical literature,
they tend to be defined in the same way, but with - of variance 1/2, so that they
are orthogonal with respect to the measure with density exp(−G2) rather than
exp(−G2/2).
There is an analogy between expansions in Hermite polynomials and expansion
in Fourier series. In this analogy, the factor =! plays the same role as the factor 2c
that appears in Fourier analysis. Just like there, one can shift it around to simplify
certain expressions, but one can never quite get rid of it.
no reviews yet
Please Login to review.
