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STOCHASTIC CALCULUS
JASONMILLERANDVITTORIASILVESTRI
Contents
Preface 1
1. Introduction 1
2. Preliminaries 4
3. Local martingales 10
4. The stochastic integral 16
5. Stochastic calculus 36
6. Applications 44
7. Stochastic differential equations 49
8. Diffusion processes 59
Preface
These lecture notes are for the University of Cambridge Part III course Stochastic Calculus,
given Lent 2017. The contents are very closely based on a set of lecture notes for this course
due to Christina Goldschmidt. Please notify vs358@cam.ac.uk for comments and corrections.
1. Introduction
In ordinary calculus, one learns how to integrate, differentiate, and solve ordinary differential
equations. In this course, we will develop the theory for the stochastic analogs of these
constructions: the Itˆo integral, the Itˆo derivative, and stochastic differential equations (SDEs).
Let us start with an example. Suppose a gambler is repeatedly tossing a fair coin, while betting
£1 on head at each coin toss. Let ξ ,ξ ,ξ ... be independent random variables denoting the
1 2 3
consecutive outcomes, i.e.
ξ =+1 if head at kth coin toss, (ξ ) i.i.d.
k −1 otherwise, k k≥1
1
2 JASONMILLERANDVITTORIASILVESTRI
Then the gambler’s net winnings after n coin tosses are given by X := ξ + ξ + ... + ξ .
n 1 2 n
Note that (X ) is a simple random walk on Z starting at X = 0, and it is therefore a
n n≥0 0
discrete time martingale with respect to the filtration (F ) generated by the outcomes, i.e.
th n n≥0
F =σ(ξ ,...,ξ ). Now suppose that, at the m coin toss, the gambler bets an amount H
n 1 n m
on head (we can also allow Hm to be negative by interpreting it as a bet on tail). Take, for
now, (H ) to be deterministic (for example, the gambler could have decided in advance
m m≥1
the sequence of bets). Then it is easy to see that
n
(1.1) (H·X) :=XH (X −X )
n k k k−1
k=1
gives the net winnings after n coin tosses. We claim that the stochastic process H · X
is a martingale with respect to (F ) . Indeed, integrability and adaptedness are clear.
n n≥0
Furthermore,
(1.2) E (H ·X) −(H·X) F =H E(X −X |F )=0
n+1 n n n+1 n+1 n n
for all n ≥ 1, which shows that H · X is a martingale. In fact, this property is much more
general. Indeed, instead of deciding his bets in advance, the gambler might want to allow its
(n+1)th bet to depend on the first n outcomes. That is, we can take Hn+1 to be random, as long
as it is Fn-measurable. Such processes are called previsible, and they will play a fundamental
role in this course. In this more general setting, (H · X)n still represents the net amount of
winnings after n bets, and (1.2) still holds, so that H · X is again a discrete time martingale.
For this reason, the process H · X is usually referred to as discrete martingale transform.
Remark 1.1. This teaches us that we cannot make money by gambling on a fair game!
Our goal for the first part of the course is to generalise the above reasoning to the continuous
time setting, i.e. to define the process
Z tH dX
s s
0
1
for (H ) suitable previsible process and (X ) continuous martingale (think, for example,
t t≥0 t t≥0
of X as being standard one-dimensional Brownian Motion, the analogue to a simple random
walk in the continuum). To achieve this, as a first attempt one could try to generalise the
Lebesgue-Stieltjes theory of integration to more general integrators: this will lead us to talk
about finite variation processes. Unfortunately, as we will see, the only continuous martingales
which have finite variation are the constant ones, so a new theory is needed in order to integrate,
for example, with respect to Brownian Motion. This is what is commonly called Itˆo’s theory
of integration. To see how one could go to build this new theory, note that definition (1.1) is
1
At this point we have not yet defined the notion of previsible process in the continuous time setting: this
will be clarified later in the course.
STOCHASTIC CALCULUS 3
perfectly well-posed in continuous time whenever the integrand H is piecewise constant taking
only finitely many values. In order to accommodate more general integrands, one could then
think to take limits, setting, for example,
⌊t/ε⌋
(1.3) (H·X) ”:=” lim XH (X −X ).
t ε→0 kε (k+1)ε kε
k=0
Onthe other hand, one quickly realises that the r.h.s. above in general does not converge in the
usual sense, due to the roughness of X (think, for example, to Brownian Motion). What is the
right notion of convergence that makes the above limit finite? As we will see in the construction
of Itˆo integral, to get convergence one has to exploit the cancellations in the above sum. As an
example, take X to be Brownian Motion, and observe that
⌊t/ǫ⌋
X 2
E H (X −X ) =
kε (k+1)ε kε
k=0
⌊t/ε⌋ ⌊t/ε⌋
=E XH2 X −X 2+XH H (X −X )(X −X )
kε (k+1)ε kε jε kε (k+1)ε kε (j+1)ε jε
k=0 j6=k
⌊t/ε⌋ ⌊t/ε⌋ Z t
=E XH2 X −X 2 =ε XH2 → H2ds as ε→0,
kε (k+1)ε kε kε s
k=0 k=0 0
where the cancellations are due to orthogonality of the martingale increments. These type
of considerations will allow us to provide a rigorous construction of the Itˆo integral for X
continuous martingale (such as Brownian Motion), and in fact even for slightly more general
integrands, namely local martingales and semimartingales.
Oncethestochastic integral has been defined, we will discuss its properties and applications. We
will look, for example, at iterated integrals, the stochastic chain rule and stochastic integration
by part. Writing, to shorten the notation,
dY =HdX
t t t
to mean Y = RtH dX , we will learn how to express df(Y ) in terms of dY by mean of the
t 0 s s t t
celebrated Itˆo’s formula. This is an extremely useful tool, of which we will present several
applications. For example, we will be able to show that any continuous martingale is a
time-changed Brownian Motion (this is the so-called Dubins-Schwarz theorem).
In the second part of the course we will then focus on the study of Stochastic Differential
Equations (SDEs in short), that is equations of the form
(1.4) dX =b(t,X )dt+σ(t,X )dB , X =x ,
t t t t 0 0
4 JASONMILLERANDVITTORIASILVESTRI
for suitably nice functions a,σ. These can be thought of as arising from randomly perturbed
ODEs. To clarify this point, suppose we are given the ODE
dx(t)
(1.5) dt =b(t,x(t)),
x(0) = x ,
0
which writes equivalently as x(t) = x +Rtb(s,x(s))ds. In many applications we may wish to
0 0
perturb (1.5) by adding some noise, say Brownian noise. This gives a new (stochastic) equation
X =x +Z tb(s,X )ds+σB ,
t 0 s t
0
where the real parameter σ controls the intensity of the noise. In fact, we may also want to
allow such intensity to depend on the current time and state of the system, to get
X =x +Z tb(s,X )ds+Z tσ(s,X )dB ,
t 0 s s s
0 0
where the last term is an Itˆo integral. This, written in differential form, gives (1.4). Of such
SDEs we will present several notions of solutions, and discuss under which conditions on the
functions b and σ such solutions exist and are unique.
Finally, in the last part of the course we will talk about diffusion processes, which are Markov
processes characterised by martingales properties. We will explain how such processes can be
constructed from SDEs, and how they can be used to build solutions to certain (deterministic)
PDEs involving second order elliptic operators.
2. Preliminaries
2.1. Finite variation integrals. Recall that a function a: [0,∞) → R is said to be c´adl´ag if
it is right-continuous and has left limits. In other words, for all x ∈ (0,∞) we have both
lim a(y) exists and lim a(y) = a(x).
− +
y→x y→x
− −
We write a(x ) for the left limit at x, and let ∆a(x) := a(x) − a(x ). Assume that a is
non-decreasing and c´adl´ag with a(0) = 0. Then there exists a unique Borel measure da on
[0,∞) such that for all s < t, we have that
da((s,t]) = a(t) − a(s).
For any f : R → R measurable and integrable function we can define the Lebesgue-Stieltjes
integral f · a by setting Z
(f · a)(t) = (0,t] f(s)da(s)
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