0Introduction
III Stochastic Calculus and Applications
0 Introduction
Ordinary differential equations are central in analysis. The simplest class of
equations tend to look like
˙x(t) = F (x(t)).
Stochastic differential equations are differential equations where we make the
function
F
“random”. There are many ways of doing so, and the simplest way
is to write it as
˙x(t) = F (x(t)) + η(t),
where
η
is a random function. For example, when modeling noisy physical
systems, our physical bodies will be subject to random noise. What should we
expect the function
η
to be like? We might expect that for
|t − s|
0, the
variables
η
(
t
) and
η
(
s
) are “essentially” independent. If we are interested in
physical systems, then this is a rather reasonable assumption, since random noise
is random!
In practice, we work with the idealization, where we claim that
η
(
t
) and
η
(
s
) are independent for
t 6
=
s
. Such an
η
exists, and is known as white noise.
However, it is not a function, but just a Schwartz distribution.
To understand the simplest case, we set F = 0. We then have the equation
˙x = η.
We can write this in integral form as
x(t) = x(0) +
Z
t
0
η(s) ds.
To make sense of this integral, the function
η
should at least be a signed measure.
Unfortunately, white noise isn’t. This is bad news.
We ignore this issue for a little bit, and proceed as if it made sense. If the
equation held, then for any 0 = t
0
< t
1
< ···, the increments
x(t
i
) − x(t
i−1
) =
Z
t
i
t
i−1
η(s) ds
should be independent, and moreover their variance should scale linearly with
|t
i
− t
i−1
|. So maybe this x should be a Brownian motion!
Formalizing these ideas will take up a large portion of the course, and the
work isn’t always pleasant. Then why should we be interested in this continuous
problem, as opposed to what we obtain when we discretize time? It turns out
in some sense the continuous problem is easier. When we learn measure theory,
there is a lot of work put into constructing the Lebesgue measure, as opposed
to the sum, which we can just define. However, what we end up is much easier
— it’s easier to integrate
1
x
3
than to sum
P
∞
n=1
1
n
3
. Similarly, once we have set
up the machinery of stochastic calculus, we have a powerful tool to do explicit
computations, which is usually harder in the discrete world.
Another reason to study stochastic calculus is that a lot of continuous
time processes can be described as solutions to stochastic differential equations.
Compare this with the fact that functions such as trigonometric and Bessel
functions are described as solutions to ordinary differential equations!
There are two ways to approach stochastic calculus, namely via the Itˆo
integral and the Stratonovich integral. We will mostly focus on the Itˆo integral,
which is more useful for our purposes. In particular, the Itˆo integral tends to
give us martingales, which is useful.
To give a flavour of the construction of the Itˆo integral, we consider a simpler
scenario of the Wiener integral.
Definition
(Gaussian space)
.
Let (Ω
, F, P
) be a probability space. Then a
subspace
S ⊆ L
2
(Ω
, F, P
) is called a Gaussian space if it is a closed linear
subspace and every X ∈ S is a centered Gaussian random variable.
An important construction is
Proposition.
Let
H
be any separable Hilbert space. Then there is a probability
space (Ω
, F, P
) with a Gaussian subspace
S ⊆ L
2
(Ω
, F, P
) and an isometry
I
:
H → S
. In other words, for any
f ∈ H
, there is a corresponding random
variable
I
(
f
)
∼ N
(0
,
(
f, f
)
H
). Moreover,
I
(
αf
+
βg
) =
αI
(
f
) +
βI
(
g
) and
(f, g)
H
= E[I(f)I(g)].
Proof.
By separability, we can pick a Hilbert space basis (
e
i
)
∞
i=1
of
H
. Let
(Ω
, F, P
) be any probability space that carries an infinite independent sequence
of standard Gaussian random variables
X
i
∼ N
(0
,
1). Then send
e
i
to
X
i
, extend
by linearity and continuity, and take S to be the image.
In particular, we can take H = L
2
(R
+
).
Definition
(Gaussian white noise)
.
A Gaussian white noise on
R
+
is an isometry
W N
from
L
2
(
R
+
) into some Gaussian space. For
A ⊆ R
+
, we write
W N
(
A
) =
W N(1
A
).
Proposition.
– For A ⊆ R
+
with |A| < ∞, W N (A) ∼ N(0, |A|).
–
For disjoint
A, B ⊆ R
+
, the variables
W N
(
A
) and
W N
(
B
) are indepen-
dent.
– If A =
S
∞
i=1
A
i
for disjoint sets A
i
⊆ R
+
, with |A| < ∞, |A
i
| < ∞, then
W N(A) =
∞
X
i=1
W N(A
i
) in L
2
and a.s.
Proof. Only the last point requires proof. Observe that the partial sum
M
n
=
n
X
i=1
W N(A)
is a martingale, and is bounded in L
2
as well, since
EM
2
n
=
n
X
i=1
EW N(A
i
)
2
=
n
X
i=1
|A
i
| ≤ |A|.
So we are done by the martingale convergence theorem. The limit is indeed
W N(A) because 1
A
=
P
∞
n=1
1
A
i
.
The point of the proposition is that
W N
really looks like a random measure
on
R
+
, except it is not. We only have convergence almost surely above, which
means we have convergence on a set of measure 1. However, the set depends on
which
A
and
A
i
we pick. For things to actually work out well, we must have a
fixed set of measure 1 for which convergence holds for all A and A
i
.
But perhaps we can ignore this problem, and try to proceed. We define
B
t
= W N([0, t])
for t ≥ 0.
Exercise.
This
B
t
is a standard Brownian motion, except for the continuity
requirement. In other words, for any
t
1
, t
2
, . . . , t
n
, the vector (
B
t
i
)
n
i=1
is jointly
Gaussian with
E[B
s
B
t
] = s ∧ t for s, t ≥ 0.
Moreover,
B
0
= 0 a.s. and
B
t
− B
s
is independent of
σ
(
B
r
:
r ≤ s
). Moreover,
B
t
− B
s
∼ N (0, t − s) for t ≥ s.
In fact, by picking a good basis of L
2
(R
+
), we can make B
t
continuous.
We can now try to define some stochastic integral. If
f ∈ L
2
(
R
+
) is a step
function,
f =
n
X
i=1
f
i
1
[s
i
,t
i
]
with s
i
< t
i
, then
W N(f ) =
n
X
i=1
f
i
(B
t
i
− B
s
i
)
This motivates the notation
W N(f ) =
Z
f(s) dB
S
.
However, extending this to a function that is not a step function would be
problematic.