Notes on Brownian motion and stochastic processes

Ana E. de Orellana

Contents
1. Stochastic processes 1
1.1. Filtrations 2
1.2. Martingales 2
1.3. Stopping times 3
1.4. Sationary processes with independent increments 3
2. Brownian motion 3
2.1. Markov processes 4
2.2. Modulus of continuity 5
3. Diffusion processes 5
4. Stochastic integration 6
4.1. Itô calculus 7
4.2. Properties of the stochastic integral 8
4.3. Itô isometry and a generalisation 8
4.4. Itô’s formula 9
4.5. A note on quadratic variation 10
4.6. Examples 11
5. d-dimensional Brownian motion and higher dimensional Itô calculus 13
References 14

Most of this material is based on [KS91], [Gar88], and [Eva12].

1. Stochastic processes
A stochastic process is a model of a random phenomenon that depends on time. The random-
ness is captured by the introduction of a measurable space (Ω, F ) called the sample space, on
which probability measures can be defined.
Formally, a stochastic process is a collection of random variables X = {Xt : 0 ⩽ t < ∞} that
take values on a second measurable space called the state space (S, S ) (which usually is S = Rd ,
S = B(Rd )). For each t ∈ [0, ∞) (or t ∈ [0, T ] for some time T ), Xt : Ω → S. Fixing ω ∈ Ω, the
function t 7→ Xt (ω) is the sample path of the process X associated with omega.

AEdO was financially supported by the University of St Andrews.

1
2 ANA E. DE ORELLANA

Example 1.1. Take the height of people as our random variable. In this case Ω is the population
and S is R+ . We can measure heights at different times, so fixing one person ω ∈ Ω, Xt (ω) is
that person’s height for all the times t ⩾ 0. That is a stochastic process.

It is a model of a random experiment whose outcome can be observed continuously in time.

Intuition. There’s nothing ‘random’ in the definition of a random variable. A random variable
is a measurable function that goes from Ω to S, that we don’t know how it works, all we know
are the outputs. So we can ‘see’ how this function works by taking the measure of preimages
P ({ω : X(ω) = a}), we don’t know which ω are being mapped to a, but we can learn their
proportion. So a stochastic process a collection of random variables that depend on both ω and t.
We usually fix ω and study Xt (ω) as a function of t.

1.1. Filtrations. We equip our sample space (Ω, F ) with a filtration. That is, a non-decreasing
family {Ft : t ⩾
0} of sub σ-Algebras of F . Fs ⊆ Ft ⊆ F for 0 ⩽ s < t < ∞. We define
F∞ = σ ∪t⩾0 Ft 1.
The simplest choice of a filtration is the one generated by the stochastic process itself. Defined
as
FtX = σ(Xs : 0 ⩽ s ⩽ t),
is the smallest σ-algebra with respect to Xs is measurable for every s ∈ [0, t].
We say that the stochastic process X is adapted to the filtration {Ft } if, for each t ⩾ 0, Xt is
an Ft -measurable random variable.
Intuition. Filtrations are a way to encode the information contained in the history of a stochastic
process. If a process is adapted, then all information about the process up to a certain time is
contained in the corresponding filtration.

1.2. Martingales. The conditional probability of some event A given another event B, P (A|B),
can be thought of changing the probability space to Ωe = B, F f = {C ∩ B : C ∈ F } and the
e e e
probability to P = P/P (B), so that P (Ω) = 1. Thus, if we want to calculate the expected value
of a variable X given the event B, we should set
Z
1
E(X|B) = X dP.
P (B) B
And what about the expected value of X given another random variable Y ? In other words, if
‘chance’ selects a sample point ω ∈ Ω and all we know about ω is the value Y (ω), what is our
best guess as to the value of X(ω)?
The conditional expectation of X given Y is a F Y -measurable random variable Z such that
Z Z
X dP = Z dP, ∀A ∈ F Y .
A A

We denote Z by E(X|Y ).
Intuition. We can understand E(X|Y ) as the information available in the σ-algebra generated
by Y , F Y (which is the smallest σ-algebra that contains all the information of Y ) that we want
to use to estimate the values of X.

1When X is a set, we denote by σ(X) to the σ-algebra generated by X. When X is a random variable, σ(X)
( −1 )
is the sigma-algebra generated by the random variable, that is σ(X) = σ {X (A) : A ∈ F }
 NOTES ON BROWNIAN MOTION AND STOCHASTIC PROCESSES 3

A discrete-time martingale is a discrete-time stochastic process such that for any time t,
E|Xt | < ∞ and E(Xt+1 |X1 , X2 , . . . , Xt ) = Xt . That is, the conditional expected value of the
next observation given all the past observations is equal to the most recent observation. Where
the conditional expectation of X given A is
X X P ({X = x} ∩ A)
E(X|A) = xP (X = x|A) = x .
x x
P (A)

Consider a real-valued stochastic process X on a probability space (Ω, F , P ) adapted to a

filtration {Ft } and such that E|Xt | < ∞ for any t ⩾ 0. The process {Xt , Ft : 0 ⩽ t < ∞} is a
sub(super)martingale if for every 0 ⩽ s < t < ∞, that P -a.e. E(Xt |Fs ) ⩾ Xs (⩽). We say that
{Xt , Ft : 0 ⩽ t < ∞} is a martingale if it’s both a sub and super martingale.
Example 1.2. (From Wikipedia) Consider a gambler who wins £1 when a coin copes up heads
and loses £1 when the coin comes up tails. Suppose that heads comes up with probability p.
· If p = 1/2, the gambler on average neither wins nor loses money, and the gambler’s
fortune over time is a martingale.
· If p < 1/2, the gambler loses on average, so his fortune over time is a supermartingale.
· If p > 1/2, the gambler wins on average, so his fortune over time is a submartingale.
1.3. Stopping times. Consider a measurable space (Ω, F ) with a filtration {Ft }. A random
variable T is a stopping time of the filtration, if the event {ω : T (ω) < t} ∈ Ft for every t ⩾ 0.
Intuition. The occurrence of the event T = t only depends on the values of X1 , . . . , Xt . The
time a gambler leaves the table depends only on the past winnings and loses, not on the future
ones.
1.4. Sationary processes with independent increments. A stochastic process Xt is said to
be strictly stationary if its finite-dimensional distributions are invariant under time displacements.
That is, for any 0 ⩽ t < ∞ such that tj , tj + t are in [0, ∞) for all j,
Ft1 +t,...,tn +t (x1 , . . . , xn ) = Ft1 ,...,tn (x1 , . . . , xn ).
If also Xt ∈ then for each s, t ∈ [0, ∞), E(Xt ) = µ and Cov(Xt , Xs ) = C(t − s).
L2 ,
Another type of process are those with independent increments, also called additive process,
which mean that, for any finite sequence {tj } ⊂ [0, ∞) with tj < tj+1 , the differences Xtj+1 − Xtj
are independent. In these cases the probability is determined by the distribution of Xt and
Xt − Xs for t > s.

2. Brownian motion
A standard one-dimensional Brownian motion is a continuous, adapted process W = {Wt , Ft :
0 ⩽ t < ∞}, defined on some probability space (Ω, F , P ) with the properties that W0 = 0
almost surely and for any two pairs of times 0 ⩽ s < t, the increment Wt − Ws is independent
of Fs 2, and is normally distributed with mean zero and variance t − s. The filtration {Ft } is
not necessarily the one induced by the stochastic process W . In fact, for some applications (such
as stochastic differential equations), a larger filtration is needed. It’s an example of a strictly
stationary process with independent increments (see Section 1.4)
For each n ∈ N the Brownian motion satisfies
E|Wt − Ws |2n ≈n Cn |t − s|n . (2.1)
This is useful because of the following continuity theorem by Kolmogorov.
2That means that if 0 ⩽ s < t ⩽ s < t then W − W is independent of W − W
1 1 2 2 t1 s1 t2 s2
4 ANA E. DE ORELLANA

Theorem 2.1. Suppose that a process X = {Xt : 0 ⩽ t ⩽ T } on a probability space (Ω, F , P )

satisfies the condition
E|Xt − Xs |α ≲ |t − s|1+β , 0 ⩽ s, t ⩽ T,
for some α, β > 0. Then there exists a continuous modification X e = {X
et : 0 ⩽ t ⩽ T } of X,
which is locally Hölder continuous with exponent γ for every γ ∈ (0, β/α), i.e.
 |Xet (w) − Xes (w)| 
P w: sup ⩽ δ = 1,
0<t−s<h(w) |t − s|γ
where h(w) is an a.s. positive random variable and δ > 0 is some constant.

Given different times t1 , . . . , tn we are interested in knowing the probability that a sample path
of Brownian motion takes values between ai and bi for each time ti . That is, what is
P (a1 ⩽ Wt1 ⩽ b1 , . . . , an ⩽ Wtn ⩽ bn )?
The following theorem will give an answer to a more general problem.
Theorem 2.2. Let Wt be the standard Brownian motion. Then for all n ∈ N, all choices of
times 0 = t0 < t1 < · · · < tn and each function f : Rn → R,


E f (Wt1 , . . . , Wtn ) =
Z ∞ Z ∞ x2 (xn −x )2
1 − 2t1 1 − 2(t −tn−1 )
··· f (x1 , . . . , xn ) √ e 1 ··· p e n n−1 dx · · · dx .
n 1
−∞ −∞ 2πt1 2π(tn − tn−1 )
This comes from the fact that
Z b
1 x2
P (a ⩽ Wt ⩽ b) = √ e− 2t dx.
2πt a

Setting f (x1 , . . . , xn ) = 1[a1 ,b1 ] (x1 ) · · · 1[an ,bn ] we get

P (a1 ⩽ Wt1 ⩽ b1 , . . . ,an ⩽ Wtn ⩽ bn ) =
Z b1 Z bn x2 (xn −x )2
1 − 2t1 1 − 2(t −tn−1 )
··· √ e 1 ··· p e n n−1 dx · · · dx .
n 1
a1 an 2πt1 2π(tn − tn−1 )

2.1. Markov processes. Let Xt be a stochastic process with state space Rd and times 0 ⩽ t <
∞. For t1 ⩽ t2 , define
F ([t1 , t2 ]) = σ(Xt : t1 ⩽ t ⩽ t2 ).
That is, F ([t1 , t2 ]) is the σ-algebra that encodes all the information between the times t1 and t2 .
The process Xt is called a Markov process if for t0 ⩽ s ⩽ t ⩽ T and all B ∈ B,
P (Xt ∈ B|F ([t0 , s])) = P (Xt ∈ B|Xs ),
with probability 1.
Intuition. The Markov property says that the probable future state of the system at any time
t > s is independent of the past behaviour of the system at times t < s, given the present state at
time s.
Intuition. The process only ‘knows’ its value at time s and does not ‘remember’ how it got there.

The following definition will establish some notation that we will use from now on.
Definition 2.3 (Probability kernel). A measure kernel from a measurable space (X, FX ) to
another measurable space (Y, FY ) is a function P : X × Y → R+ such that
 NOTES ON BROWNIAN MOTION AND STOCHASTIC PROCESSES 5

· For any Y ∈ Y, P (x, Y ) is FX measurable.

· For any x ∈ X, P (x, Y ) is a measure on (Y, FY ).
We will write the integral of a function f : Y → R with respect to this measure as
Z Z
f (Y )P (x, dY ) := f (Y )P (x, Y ) dY.

A well-known equation by Chapman–Kolmogorov says that for fixed s, t, u, with s ⩽ u ⩽ t,

Z
P (s, x, t, B) = P (u, y, t, B)P (s, x, u, dy).
Rd
where P (s, Xs , t, B) = P (Xt ∈ B|Xs ) is the transition probability between the times s < t. The
Chapman–Kolmogorov equation indicates that the one-step transition probability can be written
in terms of all possible combinations of two-step transition probabilities with respect to any
arbitrary intermediate time.
Brownian motion is a Markov process with stationary transition probability
Z
|y−x|2
(eπt)− 2 e− 2t dy.
d
P (t, x, B) =
B
And
E|Wt − Ws |4 = 3(t − s)2 ,
this shows by 2.1 that W has a version with continuous sample paths. We’ll see a bit more of
this in the next subsection.

2.2. Modulus of continuity. A function g is called a modulus of continuity for the function f
if g(δ) → 0 as δ → 0 and |t − s| ⩽ δ imply |f (t) − f (s)| ⩽ g(δ) for all sufficiently small δ. Because
of the Hölder condition that we know Brownian motion satisfies (see (2.1) and Theorem 2.1), its
modulus of continuity cannot be any larger than a constant multiple of δ γ for any γ ∈ (0, 1/2).
Levy proved that with p
g(δ) = 2δ log(1/δ), δ > 0,
cg(δ) is a modulus of continuity for almost every Brownian path on [0, 1] if c > 1, but is a modulus
for almost no Brownian path on [0, 1] if 0 < c < 1.

3. Diffusion processes
Brownian motion has the property that the distribution of Xt − Xs given the σ-algebra Fs of
information up to time s, is Gaussian with mean 0 and variance t − s. One can visualise a Markov
process Xs for which the corresponding conditional distribution of the increment is approximately
Gaussian with mean ha(s, Xs ) and variance hB(s, Xs ). In such case, a(s, x) would be a d-
dimensional vector and B(s, x) would be a symmetric positive semi-definite matrix. hB(s, Xs )
would then be approximately the conditional variance matrix of the vector Xt − Xs .

Z
1
lim P (s, x, t, dy) = 0,
t→s t − s |y−x|>ε
Z
1
lim (y − x)P (s, x, t, dy) = a(s, x),
t→s t − s |y−x|⩽ε
Z
1
lim (y − x)(y − x)T P (s, x, t, dy) = B(s, x).
t→s t − s |y−x|⩽ε
6 ANA E. DE ORELLANA

The functions a(s, Xs ) and B(s, Xs ) are called the coefficients of the diffusion process. In partic-
ular, a is referred to as the drift vector and B the diffusion matrix.
The transition probability P (s, x, t, B) of a diffusion process is uniquely determined by the
drift and the diffusion coefficients of the process (under some regularity conditions).

4. Stochastic integration
We will replace ordinary differential equations of the form dx
dt = f (t, x) with a random differ-
ential equation
dX
= F (t, X, Y ), (4.1)
dt
where Y = Yt represents some stochastic input process explicitly. A solution to (4.1) is an
indexed family of functions depending on time. If the sample path structure of Y is sufficiently
pathological (e.g. not integrable), then (4.1) must be reinterpreted, that is, we cannot interpret
(4.1) as an ordinary differential equation along each path.
For example, a solution of the equation
dX
= f (t, X) + g(t, X)N ,
dt
where N is a Gaussian white noise process, should be the solution of
Z t Z t
Xt = Xt0 + f (s, Xs ) ds + g(s, Xs )Ns ds.
t0 t0

However, the last integral cannot be defined given the irregularity of N .

To deal with this problem, the last integral will be replaced with
Z t
g(s, Xs ) dWs , (4.2)
t0

dt = Nt (white
where Ws is the Brownian motion. This is motivated by the fact that (formally) dW t

noise is the time derivative of Brownian motion). So now we need only to worry about how to
interpret (4.2).
Intuition (Why white noise?). If Xt is a stochastic process with E(Xt2 ) < ∞ for all t ⩾ 0, we
define the autocorrelation function of Xt by
r(t, s) := E(Xt Xs ).
If r(t, s) = c(t − s) for some function c : R → R, then Xt is stationary. A white noise process is
a Gaussian, stationary process, with c(t) = δ0 (t). In general, we define the Fourier transform of
the autocorrelation function,
Z ∞
1
f (ξ) := e−iξt c(t) dt,
2π −∞
to be the spectral density of the process.
For white noise,
Z ∞
1 1
f (ξ) = e−iξt δ0 (t) dt = ∀ξ.
2π −∞ 2π
Thus, the spectral density of the white noise N is flat, that is, all frequencies contribute equally
in the correlation function. Just as all colours contribute equally to make white light.
 NOTES ON BROWNIAN MOTION AND STOCHASTIC PROCESSES 7

Recall that to define the Riemann-Stieltjes integration we take increments with respect to a
function F , F (tj ) − F (tj−1 ) and define
Z b X
n
g(t) dF (t) ≈ g(tj−1 )F (tj ) − F (tj−1 ).
a j=1

The condition that we need on F in the Riemann-Stieltjes construction is for it to have bounded
variation. Any differentiable function with continuous derivative f (t) = F ′ (t) has finite variation,
Rb Rb
and a g(t) dF (t) = a g(t)f (t) dt, so we write dF (t) = f (t) dt.
Rb
PnWe would hope to do the same for integrals of the form a g(t) dWt by using approximations
i=1 g(ti )(Wti − Wti−1 ), where each g(ti ) is a random variable. There are two difficulties with
this. The first one is that the random variable g(ti ) may not be measurable with respect to
the σ-algebra generated by Wti−1 , and the convergence of the sum is dependent on the choice of
points ti . The second problem is clear from the following proposition.
Proposition 4.1. The paths of Brownian motion are a.s. not of bounded variation.

So defining the integral as we know using classic calculus is quite difficult. We will study the
solution of Itô, which is choosing ti to be the left extremes of the intervals.

4.1. Itô calculus. Let {Wt : 0 ⩽ t ⩽ T < ∞} be the standard Brownian motion process. We
know that Wt − Ws ∼ N (0, t − s), then E(Wt − Ws )2 = V(Wt − Ws ) = t − s. Therefore, if
0 = t0 < t 1 < · · · < t n = T ,
X
n 2 X∞
E Wtk − Wtk−1 = (tk − tk−1 ) = T. (4.3)
k=1 k=1

And so, since

Z T X 1X
n n
1
Ws dWs ≈ Wtk−1 (Wtk − Wtk−1 ) = Wt2 − (Wtk − Wtk−1 )2 . (4.4)
0 2 2
k=1 k=1

Letting n → ∞, the last term of (4.4) converges to T from (4.3). Therefore,

Z T
1 1
Ws dWs (s) = Wt2 − T
0 2 2
Intuition. We are saying that (dWt )2 = dt, because the definition of Itô’s integral involves
convergence in the quadratic sense.
Rb
Theorem 4.2. The integral a Wt dWt interpreted in the Itô sense satisfies
Z b 
E Wt dWt = 0
a
Z b 2
E Wt dWt = 12 (b2 − a2 ).
a

What differences Itô’s calculus from others is the choice of h(tk ) = Wtk . If we had chosen
instead the midpoint h(tk ) = B(tk−1 + 21 (tk − tk−1 )) we would’ve got the Stratonovich integral.
The advantage in the choice of ti to be the left extreme of the interval is that we don’t require
to know the future of the process, only the present time.
8 ANA E. DE ORELLANA

Rb
Formally, the definition of the Itô integral is done via random step functions f with a E(f )2 <
∞ (when this happens we say that f ∈ L2 ), and using a density argument to extend it to random
functions in L2 . Then the Itô integral
Z b
I(f ) = f (t) dWt ,
a
can be defined as a linear mapping I : L2 → 2
L . The next theorem summarises this construction.
Theorem 4.3. The integral I defined for random step functions f in L2 as
X
n
I(f ) = f (tk−1 )(Wtk − Wtk−1 ),
k=1

extends to a continuous linear random functional from L2 into L2 which satisfies

E(I(f )) = 0,
kI(f )k2 = kf k.

Note that the properties given in this last theorem are the extension of those in Theorem 4.2
to the entire space L2 (the same can be done for more general spaces, but this will be sufficient
for us).

4.2. Properties of the stochastic integral. Throughout this section we will assume that f is
some random function in L2 .
For any disjoint Borel sets W1 and W2 ,
Z Z Z
f (t) dWt = f (t) dWt + f (t) dWt .
W1 ∪W2 W1 W2
In particular, if a ⩽ t1 ⩽ t2 ⩽ b,
Z t2 Z t1 Z t2
f (t) dWt = f (t) dWt + f (t) dWt ,
a a t1
Rt
and setting Xt = a f (s) dWs it’s clear that Xt is a Markov process. It is also a martingale since
by independence of the increments with respect to Ft , E(Xt+s − Xt |Ft ) = E(Xt+s − Xt ) = 0.
Rt
Proposition 4.4. Let f ∈ L2 and Xt = a f (s) dWs . Then for any r > 0,
  Z b 
−2
P sup |Xt | > r ⩽ r E 2
f (t) dt ,
[a,b] a

and   Z 
b
E sup |Xt | 2
⩽ 4E 2
f (t) dt .
[a,b] a

4.3. Itô isometry and a generalisation. Itô isometry is a crucial fact in Itô calculus. Let Wt
be the standard Brownian motion and Xt be a stochastic process adapted to the natural filtration
of Brownian motion. Then
 Z t 2  Z t 
E Xs dWs =E 2
Xs ds .
0 0

This formula is generalised in the following theorem by Burkholder–Davis–Gundy. The con-

stant (10)p follows from [Peš96, (2.5) and (2.23)].
 NOTES ON BROWNIAN MOTION AND STOCHASTIC PROCESSES 9

Theorem 4.5 ([BDG72]). Let Xt be a real-valued Wt integrable adapted process. Then for all
1 ⩽ p < ∞,
 Z t 2p   Z 1 p 
E sup Xs dWs ⩽ (10p) E
p 2
Xs ds .
0⩽t⩽1 0 0
R
4.4. Itô’s formula. Let f, g ∈ L2 (this can be done more generally, e.g. we don’t need (Ef )2 <
∞) with f ∈ L1 [a, b]. Then the equation
Z t Z t
Xt = Xa + f (s) ds + g(s) dWs
a a

defines a stochastic process with continuous sample paths a.s.. Written in differential form
dXt = f (t) dt + g(t) dWt . (4.5)
Rt Rt
By Theorem 4.3, since Xt − Xs = s f (u) du + s g(u) dWu ,
Z t
E(Xt − Xs ) = f (u) du,
s

and
 Z 2   Z 2 
  t t
E (Xt − Xs ) 2
=E f (u) du +E g(u) dWu
s s
 Z t  Z t 
− 2E f (u) du g(u) dWu
s s
Z t 2  Z t 2  Z t  Z t 
= f (u) du +E g(u) dWu −2 f (u) du E g(u) dWu ,
s s s s
h R
2 i Rb 
b
by Theorem 4.3, and using the fact that E a Yu dWs =E a Yu2 du , we get
Z t
V(Xt − Xs ) = g(u)2 du
s

If F (t, x) is a sufficiently smooth deterministic function defined for all t ∈ [a, b] and Xt is a
process with stochastic differential (4.5). Then F (t, Xt ) determines a process with stochastic
differential
dF (t, Xt ) = fe(t, Xt ) dt + ge(t, Xt ) dWt . (4.6)

Itô’s formula gives analytic expressions for fe and ge in terms of the partial derivatives of F and
the functions f and g.
Theorem 4.6. Let X = {Xt : 0 ⩽ t < ∞} be a continuous martingale. Let F : [0, T ] × R → R
be a continuous function with continuous partial derivatives ∂F/∂t, ∂F/∂x and ∂ 2 F/∂x2 . Then
the process F (t, Xt ) has a stochastic differential
∂F ∂F 1 ∂2F
dF (t, Xt ) = (t, Xt ) dt + (t, Xt ) dXt + g(t) 2 (t, Xt ) dt.
∂t ∂x 2 ∂x
In particular, if Xt has stochastic differential (4.5), then
∂F ∂F 1 ∂2F ∂F
dF (t, Xt ) = (t, Xt ) dt + f (t) (t, Xt ) dt + g(t)2 2 (t, Xt ) dt + g(t) (t, Xt ) dWt ,
∂t ∂x 2 ∂x ∂x
10 ANA E. DE ORELLANA

and
F (t, Xt ) − F (0, X0 ) =
Z t 2 Z t
∂F ∂F 1 2∂ F ∂F
(x, Xs ) + f (s) (s, Xs ) + g(s) 2
(s, Xs ) ds + g(s) (s, Xs ) dWs .
0 ∂t ∂x 2 ∂x 0 ∂x
Itô’s formula gives
∂F ∂F 1 ∂2F
dF = dt + dX + g 2 2 dt,
∂t ∂x 2 ∂x
where the first two terms are what we expect from classical calculus, the third term is the new
addition. The reason for this is that if we have dx = f dt + g dB, it is not independent of dt, and
so
(dx)2 = f 2 (dt)2 + 2f g dt dB + g 2 (dB)2 .
The key point is that (dB)2 behaves like dt in mean square calculus.

4.5. A note on quadratic variation. Here I’ve tried to maintain a simple argument, but these
things can be done in far more generality. In what I wrote before we depend a lot on ‘the fact’
that (dWt )2 = dt, but what does that mean? And how can we define all these things when we
are not working with something so nice as classical Brownian motion?
We know that one of the problems is that the paths of Brownian motion are of unbounded
variation, and its because of that, that we choose to work with the quadratic variation (see (4.3)).
I’m writing this section only to be aware of the notation that I’ll introduce and to also know
that all these things can be done in more general settings. What is more, this is the way that
stochastic integration should be defined, and is used, for example, in [KS91].
Let Xt be a stochastic process. Fix t > 0 and let Π = {t0 , t1 , . . . , tn } be a partition of [0, t].
Define the p-th variation of X over the partition Π to be

(p)
X
m
Vt (Π) = |Xtk − Xtk−1 |p .
k=1

Then, in probability,
(2)
lim Vt (Π) = hXit .
∥Π∥→0

Sometimes the quadratic variation hXit is defined as the unique, adapted, natural, increasing,
process, for which hXi0 = 0 and X 2 −hXi is a martingale, but these two definitions are equivalent.
So that’s why sometimes Itô’s formula is stated as follows (see [KS91, Theorem 3.3.3]): If Xt
is a continuous semimartingale with decomposition Xt = X0 + Mt + Wt , then
Z t Z t Z
′ ′ 1 t ′′
F (Xt ) = F (X0 ) + F (Xs ) dMs + F (Xs ) dWs + F (Xs ) dhM is .
0 0 2 0
Thus, if Mt = Wt , hM is = ds.
If we consider fractional Brownian motion with Hurst parameter H ∈ (0, 1), WtH , (recall that
classical Brownian motion has H = 1/2), then fixing t > 0 and letting tnk = kt/n,


+∞ , p < 1/H;
X
n−1
 H 1/H 
lim |Wtn − Wtn | = tE |W1 |
H H p
, p = 1/H;
n→∞ k+1 k 

k=0 0, p > 1/H.
Setting H = 1/2 we get the previous result for classical Brownian motion.
 NOTES ON BROWNIAN MOTION AND STOCHASTIC PROCESSES 11

4.6. Examples.
Example 4.7. Going back to Section 4.1 we can take Xt = Wt by setting f = 0 and g = 1 in the
definition of the Itô process, and F (t, x) = x2 /2 (because it’s convenient). Applying Itô’s formula
we get
 W 2  ∂F 1 ∂2F
t
d = (t, Wt ) dWt + (t, Wt ) dt.
2 ∂x 2 ∂x2
Here we can see the convenience of taking F (t, x) = x2 /2, the first term in the right-hand side is
the one we want to calculate. Isolating it and changing to integral notation gives
Z Z  2 Z
Wt 1 W2 t
Wt dWt = d − dt = t − .
2 2 2 2
As we wanted to show.
Rb
Example 4.8. Let’s now calculate a Wtn dWt . For this, we will use Theorem 4.6 with Xt = Wt ,
f = 0, g = 1 and F (t, x) = xn+1 . And
Z b Z b
1
Wbn+1
− Wan+1
= (n + 1)n n−1
Wt dt + (n + 1) Wtn dWt .
2 a a
Isolating the third term gives
Z b Z b
1 n
Wtn dWt = (Wbn+1 − Wan+1 ) − Wtn−1 dt
a n+1 2 a

Example 4.9 (Integration by parts). We will now prove that

Z t Z t
tWt = Ws dWs + s dWs .
0 0

Let Xt = Wt and F (t, x) = tx, then by Itô’s formula (Theorem 4.6),

Z t Z t
tWt = Ws ds + s dWs ,
0 0
as we wanted to show.
a2
Example 4.10. Let F (t, x) = eax− 2
t
, we will apply Itô’s formula to show that F (t, Wt ) solves
the system
dXt = 1 + aXt dWt .
Set Xt = Wt in Theorem 4.6, then f = 0 and g = 1, and
Z t Z t
∂F 1 ∂2F ∂F
F (t, Wt ) = F (0, W0 ) + (x, Xs ) + 2
(s, X s ) ds + (s, Xs ) dWs .
0 ∂t 2 ∂x 0 ∂x
Z Z Z t
a2 t aWs − a2 s a2 t aWs − a2 s a2
=1− e 2 ds + e 2 ds + a eaWs − 2 s dWs
2 0 2 0 0
Z t 2
eaWs − 2 s dWs
a
=1+a
0
Z t
=1+a F (s, Ws ) dWs ,
0
as we wanted to show.
12 ANA E. DE ORELLANA

Example 4.11. Let Wt be standard Brownian motion. Define the process Y by Yt = t2 Wt3 for
t ⩾ 0. Then Y satisfies the stochastic differential equation
 2Y 
t
dYt = + 3(t4 Yt )1/3 dt + 3(tYt )2/3 dWt , Y0 = 0.
t
That Y satisfies Y0 = 0 is trivial. Now to verify that Y satisfies that equation note that
Y = F (t, Wt ) with F (t, x) = t2 x3 , which is continuous and with continuous second derivatives.
Thus, we can use Itô’s formula to get (with f = 0 and g = 1)
∂F 1 ∂2F ∂F
dYt = dF (t, Wt ) = (t, Wt ) dt + 2
(t, Wt ) dt + (t, Wt ) dWt
∂t 2 ∂x ∂x
= 2tWt3 dt + 3t2 Wt dt + 3t2 Wt2 dWt
 2Y 
t
= + 3(t4 Yt )1 /3 dt + 3(tYt )2/3 .
t
4.6.1. Hermite polynomials. We will see that Hermite polynomials play the same role that tn /n!
plays in classic calculus.
For n ∈ N0 the n-th Hermite polynomial is defined as
(−t)n x2 dn  − x2 
hn (x, t) := e 2t n e 2t .
n! dx
Then 

h0 (x, t) = 1, h1 (x, t) = x,
2 3
h2 (x, t) = x2 − 2t , h3 (x, t) = x6 − tx
2,

 x4 tx2 t2
h4 (x, t) = 24 − 4 + 8 , etc...
For t ⩾ 0 and n ∈ N0 ,
Z t
hn (Ws , s) dWs = hn+1 (Wt , t).
0
Equivalently,
dhn+1 (Wt , t) = hn (Wt , t) dWt .
First note that
dn  λx− λ2 t  n x2t d
2 n 2

− x2t
e 2 | λ=0 = (−t) e e = n!hn (x, t).
dλn dxn
λ2 t P∞
Thus, eλx− 2 = n=0 λ
nh
n (x, t). Define the stochastic process Yt as
2
∞
X
λWt − λ2 t
Yt = e = λn hn (Wt , t).
n=0
From Example 4.10 we know that Yt solves the equation
Z t
Yt = 1 + λ Ys dWs .
0
That is,
∞
X Z tX
∞
n
λ hn (Wt , t) = 1 + λ λn hn (Ws , s) dWs
n=0 0 n=0
X Z t
=1+ λn hn−1 (Ws , s) dWs .
n=1 0
 NOTES ON BROWNIAN MOTION AND STOCHASTIC PROCESSES 13

Since this holds for any λ, then

Z t
hn (Wt , t) = hn−1 (Ws , s) dWs .
0

5. d-dimensional Brownian motion and higher dimensional Itô calculus

The standard d-dimensional Brownian motion W t = (Wt1 , . . . , Wtd ), defined for t ⩾ 0, has Rd as
its state space, where for each ℓ = 1, . . . , d, Wtℓ is an independent scalar standard Brownian motion
as defined previously. That is, each Wtℓ for ℓ = 1, . . . , d is a scalar process with independent,
stationary, N (0, |t − s|)-distributed increments Wtℓ − Wsℓ , and satisfying that Wt 0ℓ = 0 almost
surely.
Definition 5.1. We say that an Rn valued stochastic process X t is in L2n (0, T ) if X t = (Xti )ni=1
and for each i = 1, . . . , n,
Z T 
E (Xt ) dt < ∞.
i 2
0
That is, if each Xti ∈ L2 .

Given an n-dimensional random variable X 0 independent of W t , we will take Ft := F (W s (0 ⩽

s ⩽ t), X 0 ), the σ-algebra generated by the history of W t and X 0 .
Let f : Rn × [0, T ] → Rn and g : Rn × [0, T ] → Rd be two deterministic functions. We say that
X t is a solution of the Itô stochastic differential equation
dX t = f dt + g dW t , (5.1)
if Z Z
t t
Xt = Xs + f (u, X u ) du + g(u, X u ) dW u ,
s s
which means that for each i = 1, . . . , n
X
d
dXti = f i dt + g iℓ dWtℓ .
ℓ=1

Theorem 5.2 (Itô’s chain rule in n dimensions). Let X t be an n-dimensional continuous mar-
tingale. Let F : [0, T ] × Rn → R be a continuous function with continuous partial derivatives
∂F ∂F ∂2F
∂t , ∂xi , ∂xi ∂xj for i, j = 1, . . . , n. Then the process

∂F X
n
∂F 1 X ∂2F
n X
d
dF (t, X t ) = (t, X t ) dt + (t, X t )dXti + (t, X t ) g iℓ g jℓ dt.
∂t ∂xi 2 ∂xi ∂xj
i=1 i,j=1 ℓ=1

In particular, if X t has stochastic differential (5.1), then

∂F X n
∂F
dF (t, X t ) = (t, X t ) dt + f i (t, X t ) (t, X t ) dt
∂t ∂xi
i=1

1 X
n
∂2F X
d
+ (t, X t ) g iℓ (t, X t )g jℓ (t, X t ) dt
2 ∂xi ∂xj
i,j=1 ℓ=1

X
n X
d
∂F
+ g iℓ (t, X t ) (t, X t ) dWtℓ ,
∂xi
i=1 ℓ=1
14 ANA E. DE ORELLANA

and
Z t X n
∂F ∂F
F (t, X t ) − F (0, X 0 ) = (s, X s ) + f i (s, X s ) (s, X s )
0 ∂t ∂xi
i=1

1 X
n
∂2F X d
+ (s, X s ) g iℓ (s, X s )g jℓ (s, X s ) ds
2 ∂xi ∂xj
i,j=1 ℓ=1
d Z t
n X
X ∂F
+ g iℓ (s, X s ) (s, X s ) dWsℓ .
0 ∂xi
i=1 ℓ=1

References
[BDG72] D. L. Burkholder, B. J. Davis, and R. F. Gundy. Integral inequalities for convex
functions of operators on martingales. Berkely Symp. on Math. Statist. and Prob.,
6.2:223–240, 1972.
[Eva12] L.C. Evans. An Introduction to Stochastic Differential Equations. American Mathe-
matical Society, 2012.
[Gar88] T.C. Gard. Introduction to Stochastic Differential Equations. Monographs and Text-
books in Pure and Applied Mathematics. New York, NY, 1988.
[KS91] I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus., volume 113 of
Grad. Texts Math. New York etc.: Springer-Verlag, 2nd ed. edition, 1991.
[Peš96] G. Peškir. On the exponential Orlicz norms of stopped Brownian motion. Studia
Mathematica, 117(3):253–273, 1996.
A. E. de Orellana, University of St Andrews, Scotland
Email address: aedo1@st-andrews.ac.uk