Stochastic Calculus
                                                              HWS2021
                                                                            ∗
                                                            David Prömel
                                                       Universität Mannheim
                                                          October 22, 2021
                                                               Abstract
                            This is a preliminary version of the lecture notes for the course “Stochastic Cal-
                         culus”, which will be continuously updated during the semester and may contain many
                         mistakes. Please be careful when using it and let me know if you find mistakes or have
                         suggestions for improvements.
                     ∗The present lecture notes are written jointly with Mathias Trabs, Karlsruher Institut für Technologie.
                                                                   1
                    2                                                                                         CONTENTS
                    Contents
                    1 Introduction                                                                                          4
                    2 Brownian motion and local martingales                                                                 6
                        2.1   Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         7
                        2.2   Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     10
                        2.3   Local martingales      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  17
                    3 Stochastic Itô integration                                                                          20
                        3.1   Construction of the Itô integral . . . . . . . . . . . . . . . . . . . . . . . . . . .      21
                        3.2   Extension of Itô integration via localization . . . . . . . . . . . . . . . . . . . .       27
                        3.3   Itô formula for Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . .         31
                    4 Itô processes                                                                                       35
                        4.1   Itô’s integration for Itô processes . . . . . . . . . . . . . . . . . . . . . . . . . .     35
                        4.2   Itô’s formula for Itô processes . . . . . . . . . . . . . . . . . . . . . . . . . . . .     38
                    5 Stochastic differential equations                                                                    43
                        5.1   Linear SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     43
                        5.2   SDEs with Lipschitz continuous coefficients . . . . . . . . . . . . . . . . . . . .           45
                    6 Martingale representation                                                                           51
                        6.1   Martingale representation theorem . . . . . . . . . . . . . . . . . . . . . . . . .         51
                        6.2   Time-changes and Lévy’s characterization of Brownian motion . . . . . . . . . . 54
                    7 Girsanov’s theorem                                                                                  55
                    8 Application: Bachelier model                                                                        58
                    A Mathematical Foundation                                                                             62
                        A.1 Conditional expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         62
                        A.2 Filtration, stochastic processes and stopping times . . . . . . . . . . . . . . . .           63
                        A.3 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       63
                        A.4 Backward martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          65
                        A.5 Grönwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        65
                        A.6 Radon-Nikodym theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
                    B Miscellaneous                                                                                       67
                        B.1 Dictionary English-German . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           67
                        B.2 English abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       69
        CONTENTS                                3
        Recommended literature
          • Kuo, H.-H., Introduction to Stochastic Integration, Springer, 2006. [Main recommenda-
           tion - very accessible textbook on a similar level as the lecture.]
          • Steele, J.M., Stochastic Calculus and Financial Applications, Springer, 2001. [Main
           recommendation - nice textbook on a similar level as the lecture.]
          • Karatzas, I., and Shreve S.E. Brownian Motion and Stochastic Calculus, Springer, 1998.
           [Classical textbook also covering the material of this lecture course.]
          • Protter, P.E., Stochastic Integration and Differential Equations, Springer, 2005. [More
           general classical textbook about stochastic calculus.]
          • Revuz, D., and Yor, M., Continuous Martingales and Brownian Motion, Springer, 1999.
           [More general classical textbook about stochastic calculus.]
          • Øksendal, B., Stochastic Differential Equations, Springer, 2003. [Good textbook treating
           further applications of stochastic calculus.]
          • Klenke, A., Probability Theory. Springer-Verlag, 2006. [Very good textbook for the
           necessary foundation in probability theory. There is a German version of the book
           called “Wahrscheinlichkeitstheorie”.]
                   4                                                                          1 INTRODUCTION
                   1    Introduction
    Lecture 1
                   Stochastic calculus provides the mathematical theory required for probabilistic modeling of
                   real-world phenomena in continuous time, as they appear in various areas like mathematical
                   finance, engineering and physics. Let us start by briefly discussing, on a heuristic level, two
                   areas of applications where stochastic calculus is naturally required and which may serve us
                   as motivation to develop it. Of course, there is a long list of further applications of stochastic
                   calculus like stochastic control, stochastic filtering and optimal stopping, just to name a few.
                   Application 1: mathematical finance in continuous time. Let us consider a very
                   simple financial market consisting of a risky asset and a risk-free asset (“bank account”), which
                   both can be traded in continuous time. To capture the unpredictability and randomness of
                   future prices, the price process is modeled by a one-dimensional stochastic process (S )
                                                                                                             t t∈[0,T]
                   and the price evolution of the risk-free asset (B )        is given B := 1 for t ∈ [0,T], i.e. we
                                                                     t t∈[0,T]          t
                   assume that the interest rate is r = 0.
                      Forsimplicitywerestricttradingonthisfinancialmarkettoself-financingtradingstrategies
                          0   1
                   ϕ=(ϕ ,ϕ )          of the form:
                          t   t t∈[0,T]
                          1
                      • ϕ =f(S ), for some f ∈ C(R;R), stands for the numbers of shares of risky assets hold
                          t        t
                         at time t,
                          0
                      • ϕ stands for the numbers of risk-free assets held at time t, chosen such that ϕ =
                          t
                           0   1
                         (ϕ ,ϕ )        is self-financing.
                           t   t t∈[0,T]
                                                                                                         0   1
                   Hence, the capital process (Vt(ϕ))        generated by trading according to ϕ = (ϕ ,ϕ )
                   satisfies                            t∈[0,T]                                           t   t t∈[0,T]
                                Z t           Z t                N−1                     N−1                  
                                      0             1              X 0                     X 1
                       V (ϕ) =      ϕ dB +        ϕ dS          ≈      ϕ (B      −B )+         ϕ (S      −S )
                         t            s   s         s   s               t    ti+1    ti          t   ti+1    ti
                                                                         i                       i
                                  0             0                  i=0                     i=0
                              =Z tf(S )dS ,
                                        s    s
                                  0
                   for 0 ≤ t0 ≤ ··· ≤ t  ≤T and N ∈N.
                                       N
                      To keep our life simple, let us choose the stochastic process (St)            with sufficiently
                                                                                             t∈[0,T]
                   smooth sample paths so that classical analysis can be applied to the sample paths of the
                   stochastic process (S )      . In particular, this would allow us to use the fundamental theorem
                                        t t∈[0,T]
                   of calculus, i.e.                         Z                  Z
                                                               T                  T
                                          F(S )−F(S )=           f(S )S′ ds =       f(S )dS
                                              T         0            s   s              s    s
                                                              0                  0
                   for any function F : R → R such that F′(x) = f(x) ∈ C(R;R), and the integral RT f(S )dS
                                                                                                         0     s    s
                   is well-defined.
                                                                                      0   1
                      Now, we can take, for instance, the trading strategy ϕ = (ϕ ,ϕ )            with
                                                                                      t   t t∈[0,T]
                                 1                                                                     2
                               ϕ :=2(S −S ),        i.e.  f(x) = 2(x−S ) and F(x):=(x−S ) ,
                                 t        t    0                          0                          0
                   and obtain that the corresponding capital process (Vt(ϕ))           satisfies
                                                                                t∈[0,T]
                                  V (ϕ) = Z T 2(S −S )dS = (S −S )2 ≥ 0 and V (ϕ) = 0,
                                   T               s     0    s      T     0                0
                                             0