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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 malliavincalculusfordiusionprocesses 18 ...

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                            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.
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...Introduction to malliavin calculus march m hairer imperial college london contents whitenoiseandwienerchaos themalliavinderivative and its adjoint smoothdensities malliavincalculusfordiusionprocesses hormander stheorem hypercontractivity graphicalnotationsandthefourthmomenttheorem constructionofthe eld one of the main tools modern stochastic analysis is in a nutshell this theory providing way dierentiating random variables dened on gaussian probability space typically wiener with respect underlying noise allows develop an innite dimensional generalisation usual analytical concepts we are familiar r fourier sobolev spaces etc themaingoalofthiscourseistodevelopthistheorywiththeproofofhorman der stheoreminmind thiswasactuallytheoriginalmotivationforthedevelopment states following consider dierential equation given by c o oo where smooth functions i d processes denotes stratonovich integration ito also write shorthand notation vector elds all derivatives bounded might then ask under what c...

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