Introduction to stochastic calculus
                               Rough lecture notes
                              Masterofmathematics
                           UniversitéParis-Dauphine–PSL
                                  2020–2021
                                 Correctedversion
                              CompiledJanuary13,2023
                         Coursehomepageonhttps://djalil.chafai.net/
ii
        Suggestedscheduleofthelectures.
        Werecommendthattheinclassororallecturesdifferfromthelecturenotes,ideallytheyshouldcontain
      less details and should be focused ontheessentialaspects,thestructure,theculture,andtheintuition.
        • Lecture1(2x1.5h)
         Chapter1(Preliminaries)
        • Lecture2(2x1.5h)
         Chapter2(Processes,filtrations,stoppingtimes,martingales)
        • Lecture3(2x1.5h)
         Chapter3(Brownianmotion)
        • Lecture4(2x1.5h)
         Chapter3(Brownianmotion)
        • Lecture5(2x1.5h)
         Chapter4(Moreonmartingales)
        • Lecture6(2x1.5h)
         Chapter5(ItôstochasticintegralwithrespecttoBM)
        • Lecture7(2x1.5h)
         Chapter5(Itôstochasticintegralandsemi-martingales)
        • Lecture8(2x1.5h)
         Chapter6(Itôformulaandapplications)
        • Lecture9(2x1.5h)
         Chapter6(Itôformulaandapplications)
        • Lecture10(2x1.5h)
         Chapter7(Stochasticdifferentialequations)
        • Lecture11(2x1.5h)
         Chapter7(Stochasticdifferentialequations)
        • Lecture12(2x1.5h)
         Chapter8(Morelinkswithpartialdifferentialequations)
        • Exam
        Therearealsoseparateexcercisessessions(séancesdetravauxdirigés).
                             ii/144
                                                                                                                             iii
               These are the lecture notes of an introduction course on stochastic calculus, given at Université Paris-Dauphine – PSL, for
                                           1
           secondyearmasterstudentsinmathematics .TheprerequisiteisaprobabilitytheorycoursebasedonLebesgueintegral,including
           conditionalexpectation,gaussianrandomvectors,andstandardnotionsofconvergence. Theinitialversionoftheselecturenotes
           was based on a course given by Halim Doss, inspired from the book by Nobuyuki Ikeda and Sinzo Watanabe [20]. The current
           versionisalsoinspiredinpartfromthebooksbyFabriceBaudoin[4]andJean-FrançoisLeGall[31],andbyplentyofothersources.
           Somebitsaretrulyoriginal. Bewarethattheselecturenotesaredesignedtoconstitutearichwrittenreferenceforthelivecourse.
           Thelivecourseconcernsonlyastrictsubpartfocusingonintuition,selectedforbeingessentialforunderstandingtheconceptsand
           techniques. Atthetimeofwriting,herearethemaindifferenceswiththewrittenlecturenotesbyHalimDossbefore2018:
               • Moreonprobabilitybasics,uniformintegrability,Lebesgue–Stieltjesintegral
               • Moreonmartingalesandlocalmartingales
               • Moreonexamplesandapplicationseverywhere
               • Moreonhistory,intuition,linkwithphysics,programming
               • PropertiesofBrownianmotion,Dubins–Schwarztheorem,Feynman–Kacformula,Langevinprocesses
               • Moreonsemi-martingales,stochasticintegral,andItôformula
           Theselecturenotesdonotcoverseveralimportanttopicsrelatedtostochasticcalculus, suchasfineanalysisofBrownianmotion
           : regularity, excursions, zeros, recurrence and transcience, etc, random time change, Euler–Maruyama schemes for numerical
           analysis of stochastic differential equations, applications of stochastic calculus to finance, physics, biology, statistics, stochastic
           control, and Monte Carlo methods, Malliavin calculus, Stroock–Varadhan support theorems, local times and Tanaka formula,
           Schilder large deviation principle, additive functionals : law of large numbers, ergodic theorems, central limit theorems, large
           deviation principles, link with entropy and Poisson equation, Doob H-transforms, Friedlin–Wentzell large deviations principle
           for perturbation of dynamical systems, Feller branching diffusions, branching Brownian motion, Fisher–Wright diffusion, diffu-
           sionswithjumps,space/timewhitenoise,Bakry–Émerynon-explosioncriterionandlinkwithPoincaré,logarithmicSobolev,and
           isoperimetric functional inequalities, diffusions on manifolds, Eyrings-Kramers formula, etc. On the other hand, some topics are
           consideredintheexams,suchasCox–Ingersoll–RossandBesselprocesses,LévyareaofplanarBrownianmotion,etc.
               There are many other references on the subject. An accessible introduction are the books by Laurence Craig Evans [16] and
           byBerntØksendal[49]. ThebooksbyRichardDurrett[13],PhilipProtter[42], andHui-HsiungKuo[28]arealsoaccessible. More
           advanced references include the books by Michel Métivier [36], Chris Rogers and David Williams [44, 45], Daniel Stroock and
           Srinivasa Varadhan [47], Ioannis Karatzas and Steven Shreve [24], Daniel Revuz and Marc Yor [43], Jean Jacod [21], Iosif Gikhman
           andAnatoliSkorokhod[18], andbyClaudeDelacherieandPaul-AndréMeyer[9,10]. Finally, accessible references with exercises
           includethebookbyFrancisCometsandThierryMeyre[8](inFrench)andPaoloBaldi[3]forinstance.
           Contributors.
               • 2018–2022: DjalilChafaï
               • –2018: HalimDoss
           Glitcheshunters.
               • 2021–2022: PaulineAmrouche,FanirianaRakotoEndor,JustinSalez
               • 2020–2021: OskarBataillon,YiHan,QiaoyuLuo,GabrielMoreira-Nogueira,DiegoAlejandroMurillo
                 Taborda,LyesTifoun,WalidElWahabi
               • 2019–2020: OscarCosserat,ŁukaszMad˛ ry,AlejandroRosalesOrtiz,ZiyuZhou
               • 2018–2019: ClémentBerenfeld
             1MASEF(Mathématiquespourl’économieetlafinance)andMATH(Mathématiquesappliquéesetthéoriques).
                                                       iii/144
iv
                   Notation.
                                                    R+      [0,+∞)
                                                   BM Brownianmotion
                                                  O,o       Landaunotation
                                                     iff    if and only if
                                                   a.s.     almostsurely
                                                   u.i.     uniformlyintegrable
                                                 w.r.t.     withrespectto
                                                    1A      indicatorof A
                                      x·y or〈x,y〉           x1y1+···+xdyd ifx,y ∈Rd
                                                    |x|     qx2+···+x2 ifx∈Rd
                                                                 1            d
                                                   BE Borelσ-algebraofE
                                                      e     exponential
                                                      d differentialelement
                                                       i    thecomplexnumber(0,1)
                                  d,i,j,k,m,n,ℓ             integernumbers
                          p,q,r,s,t,u,v,α,β,ε               real numbers
                                    s ∧t and s∨t            min(s,t)andmax(s,t)
                                   f is increasing          f (y)≥ f (x) if y ≥ x
                                            p                             d                             k kp
                                           L d(Ω,P)         X :Ω→R measurablewithE( X                           ) <∞
                                            R
                                              〈x,y〉H        scalar productintheHilbertspaceH
                                              〈M,N〉 anglebracketoflocalmartingalesM,N
                                                  〈M〉 〈M,M〉
                                              [M,N]         squarebracketoflocalmartignalesM,N
                                                  [M]       [M,M]
                                                X ∼µ X haslawµ
                                                                                            iv/144