Chapter 20
More on Stochastic
Di
ff
erential Equations
Section 20.1 shows that the solutions of SDEs are di
ff
usions, and
how to find their generators. Our previous work on Feller processes
and martingale problems pays o
ff
here. Some other basic properties
of solutions are sketched, too.
Section 20.2 explains the “forward” and “backward” equations
associated with a di
ff
usion (or other Feller process).
We get our
first taste of finding invariant distributions by looking for stationary
solutions of the forward equation.
Section 20.3 makes sense of the idea of white noise. This topic
will be continued in the next lecture, forming one of the bridges to
ergodic theory.
For the rest of this lecture, whenever I say “an SDE”, I mean “an SDE
satisfying the requirements of the existence and uniqueness theorem”, either
Theorem 215 (in one dimension) or Theorem 216 (in multiple dimensions). And
when I say “a solution”, I mean “a strong solution”. If you are really curious
about what has to be changed to accommodate weak solutions, see Rogers and
Williams (2000, ch. V, sec. 16–18).
20.1
Solutions of SDEs are Di
ff
usions
Solutions of SDEs are di
ff
usions: i.e., continuous, homogeneous strong Markov
processes.
Theorem 217
The solution of an SDE is nonanticipating, and has a version
with continuous sample paths. If
X
(0) =
x
is fixed, then
X
(
t
)
is
F
W
t
adapted.
Proof:
Every solution is an Itˆ
o process, so it is nonanticipating by Lemma
198. The adaptation for nonrandom initial conditions follows similarly. (Infor
123
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CHAPTER 20.
MORE ON SDES
124
mally: there’s nothing else for it to depend on.) In the proof of the existence
of solutions, each of the successive approximations is continuous, and we bound
the maximum deviation over time, so the solution must be continuous too.
Theorem 218
Let
X
x
be the process solving a onedimensional SDE with non
random initial condition
X
(0) =
x
.
Then
X
x
forms a homogeneous strong
Markov family.
Proof:
By Exercise 19.4, for every
C
2
function
f
,
f
(
X
(
t
))

f
(
X
(0))

t
0
a
(
X
(
s
))
∂
f
∂
x
(
X
(
s
)) +
1
2
b
2
(
X
(
s
))
∂
2
f
∂
x
2
(
X
(
s
))
ds
(20.1)
is a martingale. Hence, for every
x
0
, there is a unique, continuous solution to
the martingale problem with operator
G
=
a
(
x
)
∂
∂
x
+
1
2
b
2
(
x
)
∂
2
∂
x
2
and function
class
D
=
C
2
.
Since the process is continuous, it is also cadlag.
Therefore,
by Theorem 137,
X
is a homogeneous strong Markov family, whose generator
equals
G
on
C
2
.
Similarly, for a multidimensional SDE, where
a
is a vector and
b
is a matrix,
the generator extends
1
a
i
(
x
)
∂
i
+
1
2
(
bb
T
)
ij
(
x
)
∂
2
ij
. Notice that the coe
ffi
cients are
outside
the di
ff
erential operators.
Corollary 219
Solutions of SDEs are di
ff
usions.
Proof:
Obvious from Theorem 218, continuity, and Definition 177.
Remark:
To see what it is like to try to prove this without using our more
general approach, read pp. 103–114 in Øksendal (1995).
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 Spring '06
 Schalizi
 Equations, Wiener process, Sdes, Prop osition

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