Stochastic Calculus via Regularizations

656 Pages·2022·7.853 MB·English

by  Francesco Russo, Pierre Vallois

Bocconi & Springer Series 11
Mathematics, Statistics, Finance and Economics
Francesco Russo
Pierre Vallois
Stochastic
Calculus via
Regularizations
Bocconi & Springer Series
Mathematics, Statistics, Finance and Economics
Volume 11
Editors-in-Chief
Lorenzo Peccati, Dipartimento di Scienze delle Decisioni, Università Bocconi,
Milan,Italy
SandroSalsa,DepartmentofMathematics,PolitecnicodiMilano,Milan,Italy
SeriesEditors
Beatrice Acciaio, Department of Mathematics, Eidgenössische Technische
HochschuleZürich,Zürich,Switzerland
CarloA.Favero,DipartimentodiFinanza,UniversitàBocconi,Milan,Italy
Eckhard Platen, Department of Mathematical Sciences, University of Technology
Sydney,Sydney,NSW,Australia
WolfgangJ.Runggaldier,DipartimentodiMatematicaPuraeApplicata,Università
degliStudidiPadova,Padova,Italy
Erik Schlögl, School of Mathematical and Physical Sciences, University of
TechnologySydney,CityCampus,NSW,Australia
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THESERIESISINDEXEDINSCOPUS
Francesco Russo (cid:129) Pierre Vallois
Stochastic Calculus via
Regularizations
FrancescoRusso PierreVallois
ENSTAParis FacultédesSciencesetTechnologies
InstitutPolytechniquedeParis InsitutÉlieCartan,UniversitédeLorraine
UnitédeMathématiquesAppliquées Vandœuvre-lès-Nancy,France
Palaiseau,France
ISSN2039-1471 ISSN2039-148X (electronic)
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Preface
In physics, classical analysis plays a central role. For instance in Newtonian
mechanics, thermodynamics,and electricity, many phenomenaare well explained
by deterministic models involving either ordinary differential equations or partial
differentialequations.
Inthedeterministicworld,theoretically,thefutureevolutionofasystemdepends
on initial conditions and on some parameters driving its own dynamics; never-
theless, in practice, their values are often measured with uncertainty. Moreover,
the theory of chaos explains that small variations in the initial data can generate
big fluctuations later on. In addition, deterministic models used for representing
complex phenomena currently require a huge number of equations which are
either difficultor impossible to solve within a reasonabletime. In factcomplexity
and uncertainty can be integrated directly via a probabilistic model. Mention can
be made of statistical mechanics, networks (Internet, gene interaction), financial
markets,andpercolation.Inprobabilitytheory,arandomquantitywhichfluctuates
as a function depending on time is modeled by a stochastic process. On a given
probability space ((cid:2),F,P), a process X = (X (ω),t ∈ [0,T],ω ∈ (cid:2)) is
t
a “measurable” collection of random variables which are defined on the same
underlying space and take their values in some space E. A process depends on
time t and on the random realization ω. Fixing ω, t (cid:3)→ X (ω) is a real function,
t
oftencalled pathor trajectory of the process.In the sequel,the processX will be
denotedeither(X )or(X(t)),omittingthevariableω.Forthesakeofsimplicity,in
t
thisintroduction,weessentiallyonlydealwithE =R.
In deterministicanalysis,differentiationandintegrationplayan importantrole.
Consider two functions f,x : (cid:2)R →(cid:3) R which are a(cid:2)ssum(cid:3)ed to be differentiable.
Thus, the derivative of t (cid:3)→ f x(t) at time t is f(cid:5) x(t) x(cid:5)(t). This property is
equivalentto
(cid:4) (cid:4)
(cid:2) (cid:3) (cid:2) (cid:3) t (cid:2) (cid:3) t (cid:2) (cid:3)
f x(t) =f x(0) + f(cid:5) x(s) x(cid:5)(s)ds = f(cid:5) x(s) dx(s),t ∈[0,T], (1)
0 0
wherethelatterintegralisintendedinthesenseofLebesgue-Stieltjes.
vii
viii Preface
(cid:2) (cid:3)
Now, let (X(t),t ∈ [0,T] be a real valued stochastic process. Fixing ω,
the path t (cid:3)→ X(t,ω) is generally continuous but is neither differentiable nor
with boundedvariation as it happensin particular when X is a Brow(cid:2)nian(cid:3)motion.
Therefore, even if f is C1, (1) cannot be applied since t (cid:3)→ f X(t) is not
differentiable. An important field in probability theory is the so called stochastic
calculuswhichcombinesprobabilityandcalculus,andallowsinparticular
(cid:4)
t (cid:2) (cid:3)
(cid:5)
1. givingasensetointegrals(intime)ofthetype f X(s) dX(s),
0
2. developing a us(cid:2)eful (cid:3)and efficient (stochastic) calculus, i.e., a formula for
differentiatingf X(t) as(1),
foralargeclassofnon-differentiableproc(cid:2)essesXandfu(cid:3)nctionsf.
Infact,foragivenintegratorprocess X ,t ∈[0,T] ,moregenerally,onegoal
t (cid:4)
T
of stochastic integration is to define the stochastic integral Y(s)dX(s) for a
(cid:2) (cid:3) 0
large class of integrands Y ,t ∈ [0,T] also defined on the same probability
t
space. Obviously, if for almost all ω, the path t (cid:3)→ X (ω) were differentiable
(cid:4) t
T
and |Y (ω)X˙ (ω)|dt < ∞, the above-mentioned stochastic integral could be
t t
0
defined as the Lebesgue integral in time for almost all ω. As mentioned earlier,
mostofthesignificantprocessesinterveninginclassicalstochasticmodelsarenot
(cid:4)
T
differentiable,sothatthepreviousschemedoesnotpermitdefining Y(s)dX(s).
0
ClassicalstochasticintegrationisanicecombinationofLebesgueintegrationand
martingaletheory.Themostfamous(stochastic)integralistheItôintegralwhichis
definedforinstancewhentheintegratoristheBrownianmotionX =W.However,
modern (Itô’s) stochastic integration can be developed in a more general setting,
whichreferstothecasewhenX isasemimartingalewithrespecttoanunderlying
filtration(Ft)t≥0,i.e.,thesumofa(local)(Ft)-martingaleMandan(Ft)-“adapted”
(i.e.,non-anticipative)boundedvariationprocessV.Here,weessentiallydealwith
continuous semimartingales, i.e., when M and V are continuous processes. If Y
and X are both semimartingales, an(cid:5)other celebrated integral is the so called Fisk-
Stratonovich integral, denoted by tY ◦ dX which coincides with the sum of
(cid:5) 0
Itô’sintegral tYdXandhalfthe(oblique)bracket(cid:9)Y,X(cid:10)betweenthemartingale
0
components of Y and X. When X = Y, one also denotes (cid:9)X(cid:10) := (cid:9)Y,X(cid:10). Itô’s
(and Fisk-Stratonovich’s) integral only allows non-anticipating integrands Y with
respecttothesemimartingaleintegratorX.Aprocess(Y )issaidtobeadaptedif,
t
for any t, the random variable Y is F -measurable. In many situations (Y ) must
t t t
be progressivelymeasurable,which is a more stringentnotion,see Definition 2.9.
Timeevolutionplaysacrucialroleindynamicalphysicalsystemsandmostofthe
interesting (random)quantities Y are adapted to the informationcarried by X. At
timet,thisinformationisconcentratedintheσ-fieldF ,whichisgeneratedbythe
t
collectionofrandomvariables(X ,0≤u≤t).Instochasticmodelsforfinance,at
u
eachtimet, theinvestorknowstheassets(stocks,bonds,interestrates)price(X )
t
Preface ix
inagivenmarket.Thus,theportfoliocompositionY ofth(cid:2)ein(cid:3)vestorattimet,must
be non-a(cid:2)ntic(cid:3)ipatingfor the increasingfamily of σ-fields Ft . If X is a Brownian
motion, F iscalledaBrownianfiltration.
t
Givena progressivelymeasu(cid:4)rableintegrandprocess(Yt), andas(cid:4)emimartingale
t t
X = M +V, the Itô integral Y dX is defined as the sum of Y dM and
s s s s
(cid:4) 0 0
t
Y dV . The formerone makesmajor use of the martingalepropertyof M and
s s
0
thesecondintegralisdefinedforanyωastheclassicalLebesgueintegralunderthe
(cid:4)
T
assumption |Y |d(cid:12)V(cid:12) < ∞,where(cid:12)V(cid:12)standsforthetotalvariationprocess
s s
0
ofV.Inordertogiveasensetothestochasticintegralwithrespecttothemartingale
M, let us start with a bounded and piecewise constant process of the type Y =
t
N(cid:6)−1
Y−110(t)+ Yi1]ti,ti+1](t),t ∈ [0,T], where Yi is Fti-measurable, Y−1 is F0-
i=0
measurableand0 = t < ··· < t < ··· < t = T is a subdivisionof[0,T]. It
0 i N
seemsreasonabletoset
(cid:4)
t
YsdMs :=Y−1M0+Y0(Mt1 −Mt0)+···+Yi(Mt −Mti), (2)
0
foranyt in[ti,ti+1].We denotebyE theclassofsuchelementaryprocessesY.It
can be proved, see(cid:7)S(cid:5)ect. 5.1, that the ma(cid:8)p I, which to any Y ∈ E associates the
stochastic integral tY dM ,t ∈[0,T] is linear, takes its values in the set of
0 s s
continuouslocalmartingales(seeDefinition2.21),wherethatspaceofcontinuous
stochastic processes is equipped with the topologyof the uniformconvergencein
probability. Moreover, the map I can be prolonged by continuity to the space of
(cid:4)
T
progressively measurable processes Y such that Y2d < M > < ∞ almost
s s
0
surely.IntheparticularcaseofBrownianmotionM =W,wehave<W > =t for
t
anyt ≥0.
The class of progressively measurable processes (resp. semimartingale(cid:4)s) is the
·
rightsettingforintegrands(resp.integrators).IndeedthemapY ∈ E (cid:3)→ YdX
0
is continuous if and only if X is a semimartingale, according to the celebrated
Bichteler–Dellacherie theorem, see Section III.7 in [273], or Section IV-2.16 in
[276]orSectionVIII.4in[78].
Asalreadypointedout,classicalobjectsofrealanalysisareordinarydifferential
equations, which can be expressed in the differential form or equivalently in the
integral form. Their stochastic counterpartare the so called stochastic differential
equations (SDEs). They appear in an integral formulation, since there is no
naturalpath-wise differentiationforstochastic integratorslike semimartingales,in
particularBrownianmotion.
x Preface
ThemostcelebratedSDEsarethosedrivenbyaclassicalBrownianmotionW =
(W ),whosecoefficientsa,b : [0,T]×R → RareBorelfunctions,andaninitial
t
valueξ whichisarandomvariableindependentof(W ),i.e.,
t
(cid:4) (cid:4)
t t
X =ξ + a(s,X )dW + b(s,X )ds, t ∈[0,T], (3)
t s s s
0 0
(cid:4)
t
where a(s,X )dW isanItôintegral,whichisa(local)martingale.Inparticular
s s
0
a(cid:4)solution is a semimartingale whose value at 0 is ξ, its local(cid:4)martingale part is
t t
a(s,X )dW and its bounded variation componentequals b(s,X )ds. For
s s s
0 0
them,itisoftenconvenienttoadopttheimproperdifferentialformulation
(cid:9)
dX = a(t,X )dW +b(t,X )dt
t t t t (4)
X = ξ.
0
ThesolutionsofSDEsof thetype(4), arecalled diffusionprocessesordiffusions.
Thefamilyofdiffusionsprocessesisanimportantsubclassoftheoneofsemimartin-
gales. In physics, the motion of a microscopic particle in a medium with velocity
b(t,x)attimet andpositionx,subjectedtonoiseperturbation,withintensitya,is
oftendescribedbyadiffusion.InChaps.12and13wedeveloptheclassicaltheory
of SDEs and mainly study existence and uniqueness. The SDE (4) admits multi-
dimensionalandinfinitedimensionalformulations,see[73].
Notethatifthediffusioncoefficientavanishesthen(4)isanordinarydifferential
equation, for which the differential calculus is the main device. In order to study
stochastic differential equations, one needs the aforementioned Itô’s stochastic
calculus,supportingItôstochasticintegration.Thecentralfeatureofthatpowerful
calculusisthesocalledItô’sformula,whichisinfactachangeofvariableformula.
Thatinstrumentconstitutesthestochasticextensionofthefundamentaltheoremof
integralcalculusorchainrulepropertystatedin(1).Letusassumethatf :R→R
isofclassC2 andXisasemimartingale.Then(1)canbegeneralizedinto
(cid:4) (cid:4)
t 1 t
f(X )=f(X )+ f(cid:5)(X )dX + f(cid:5)(cid:5)(X )d <X> , 0≤t ≤T. (5)
t 0 s s s s
2
0 0
IfXisadifferentiableprocess,then<X >=0sinceitsmartingalepartisconstant.
IntheStratonovichformulation(5)canbealsowrittenas
(cid:4)
t 1
f(X )=f(X )+ f(cid:5)(X )dX + (cid:9)f(cid:5)(X),X(cid:10) , 0≤t ≤T. (6)
t 0 s s t
2
0