3Review of stochastic calculus
III Schramm--Loewner Evolutions
3 Review of stochastic calculus
Before we start studying SLE, we do some review of stochastic calculus. In
stochastic calculus, the basic object is a continuous semi-martingale. We write
these as
X
t
= M
t
+ A
t
,
where
M
is a continuous local martingale and
A
is continuous with bounded
variation.
The important concepts we will review are
(i) Stochastic integrals
(ii) Quadratic variation
(iii) Itˆo’s formula
(iv) L´evy characterization of Brownian motion
(v) Stochastic differential equations
The general setting is that we have a probability space (Ω
, F, P
) with a
continuous time filtration
F
t
, which satisfies the “usual conditions”, namely
F
0
contains all P-null sets, and F
t
is right continuous, i.e.
F
t
=
\
s>t
F
s
.
Stochastic integral
If
X
t
=
M
t
+
A
t
is a continuous semi-martingale, and
H
t
is a previsible process,
we set
Z
t
0
H
s
dX
s
=
Z
t
0
H
s
dM
s
+
Z
t
0
H
s
dA
s
,
where the first integral is the Itˆo integral, and the second is the Lebesgue–Stieltjes
integral. The first term is a continuous local martingale, and the second is a
continuous bounded variation process.
The Itˆo integral is defined in the same spirit as the Riemann integral. The
key thing that makes it work is that there is “extra cancellation” in the definition,
from the fact that we are integrating against a martingale, and so makes the
integral converge even though the process we are integrating against is not of
bounded variation.
Quadratic variation
If M is a continuous local martingale, then the quadratic variation is
[M]
t
= lim
n→∞
d2
n
te−1
X
k=0
(M
(k+1)2
−n
− M
k2
−n
)
2
,
and is the unique continuous non-decreasing process of bounded variation such
that
M
2
t
−
[
M
]
t
is a continuous local martingale. Applying the same definition
to a bounded variation process always gives zero, so if
X
is a semi-martingale, it
makes sense to define
[X]
t
= [M + A]
t
= [M]
t
.
Also, we have
"
Z
( · )
0
H
s
dM
s
#
t
=
Z
t
0
H
2
s
d[M]
s
,
where the integral on the right is the Lebesgue–Stieltjes integral.
Itˆo’s formula
Itˆo’s formula is the stochastic calculus’ analogue of the fundamental theorem of
calculus. It takes a bit of work to prove Itˆo’s formula, but it is easy to understand
what the intuition is. If f ∈ C
2
, then we have
f(t) = f(0) +
n
X
k=1
(f(t
k
) − f(t
k−1
))
for any partition 0 =
t
0
< ··· < t
n
=
t
. If we were to do a regular fundamental
theorem of calculus, then we can perform a Taylor approximation
f(t) = f(0) +
n
X
k=1
f
0
(t
k−1
)
(t
k
− t
k−1
) + o(t
k
− t
k−1
)
→ f(0) +
Z
t
0
f
0
(s) ds
as max |t
k
− t
k−1
| → 0.
In Itˆo’s formula, we want to do the same thing but with Brownian motion.
Suppose B is a Brownian motion. Then
f(B
t
) = f(B
0
) +
n
X
k=1
f(B
t
k
) − f(B
t
k−1
)
.
When we do a Taylor expansion of this, we have to be a bit more careful than
before, as we cannot ignore all the higher order terms. The point is that the
quadratic variation of B
t
is non-zero. We write this as
f(0)+
n
X
k=1
h
f
0
(B
t
k−1
)(B
t
k
−B
t
k−1
)+
1
2
f
00
(B
t
k−1
)(B
t
k
−B
t
k−1
)
2
+o((B
t
k
+B
t
k−1
)
2
)
i
.
Taking the limit, and using that E[(B
t
k
− B
t
k−1
)
2
] = t
k
− t
k−1
, we get
f(B
t
) = f(0) +
Z
t
0
f
0
(B
s
) dB
s
+
1
2
Z
t
0
f
00
(B
s
) ds.
More generally, if
X
t
=
M
t
+
A
t
is a continuous semi-martingale, and
f ∈
C
1,2
(R
t
× R), so that it is C
1
in the first variable and C
2
in the second, then
f(t, X
t
) = f(0, X
0
) +
Z
t
0
∂
s
f(s, X
s
) ds +
Z
t
0
∂
x
f(s, X
s
) dA
s
+
Z
t
0
∂
x
f(s, X
s
) dM
s
+
1
2
Z
t
0
∂
2
x
f(s, X
s
) d[M]
s
.
L´evy characterization of Brownian motion
Theorem
(L´evy characterization)
.
Let
M
t
is a continuous local martingale with
[M]
t
= t for t ≥ 0, then M
t
is a standard Brownian motion.
Proof sketch.
Use Itˆo’s formula with the exponential moment generating function
e
iθM
t
+θ
2
/2[M]
t
.
Stochastic differential equations
Suppose (Ω
, F, P
) and (
F
t
) are as before, and (
B
t
) is a Brownian motion adapted
to (
F
t
). We say a process
X
t
satisfies satisfies the stochastic differential equation
dX
t
= b(X
t
) dt + σ(X
t
) dB
t
for functions b, σ iff
X
t
=
Z
t
0
b(X
s
) ds +
Z
t
0
σ(X
s
) dB
s
+ X
0
for all
t ≥
0. At the end of the stochastic calculus course, we will see that there
is a unique solution to this equation provided b, σ are Lipschitz functions.
This in particular implies we can solve
∂
t
g
t
(z) =
2
g
t
(z) − U
t
, g
0
(z) = z
for U
t
=
√
κB
t
, where B
t
is a standard Brownian motion.