IEOR 4703: Homework 8
Given a stochastic differential equation (SDE) for a diffusion,
dX
(
t
) =
a
(
X
(
t
))
dt
+
b
(
X
(
t
))
dB
(
t
)
, X
(0) =
X
0
,
where
{
B
(
t
) :
t
≥
0
}
denotes a standard BM, the
Euler method
for approximating the sample
paths of
X
=
{
X
(
t
) : 0
≤
t
≤
T
}
is given by choosing a small
h >
0, setting
N
=
b
T/h
c
,
the integer part of
T/h
, then generating
N
iid
N
(0
,
1) rvs,
Z
1
, . . . , Z
N
, and using the recursion
(with
X
h
0
def
=
X
0
)
X
h
1
=
X
0
+
a
(
X
0
)
h
+
b
(
X
0
)
√
hZ
1
X
h
2
=
X
h
1
+
a
(
X
h
1
)
h
+
b
(
X
h
1
)
√
hZ
2
.
.
.
X
h
N
=
X
h
N

1
+
a
(
X
h
N

1
)
h
+
b
(
X
h
N

1
)
√
hZ
N
.
This approximation yields
N
values,
{
X
h
i
: 1
≤
i
≤
N
}
, which approximate
X
(
t
) only at the
time points
t
∈ {
h,
2
h, . . . , Nh
}
, that is
X
(
ih
)
≈
X
h
i
. To get an approximation over the entire
time interval [0
, T
] (denoted by
{
X
h
(
t
) : 0
≤
t
≤
T
}
) we can either linearly interpolate between
time points (yielding continuous sample paths), or use the piecewise constant approach via
X
h
(
t
)
def
=
X
h
i
, ih
≤
t <
(
i
+ 1)
h,
0
≤
i
≤
N.
(1)
In what follows, we shall use the piecewise constant approach for simplicity.
For example, if
T
= 1, and we choose
h
= 1
/
100, then
N
= 100 (we have partitioned
the interval [0
,
100] into 100 subintervals of length 1
/
100) and we have approximated the path
{
X
(
t
) : 0
≤
t
≤
1
}
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 sigman
 Stochastic differential equation, Itō calculus, Langevin equation, New York.

Click to edit the document details