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FinM 345/Stat 390 Stochastic Calculus,
Autumn 2009
Floyd B. Hanson
, Visiting Professor
Email: [email protected]
Master of Science in Financial Mathematics Program
University of Chicago
Lecture 4 (from Singapore)
Jump & JumpDiffusion Stochastic Calculus
6:309:30 pm, 19 October 2009, Kent 120 in Chicago
7:3010:30 pm, 19 October 2009 at UBS Stamford
7:3010:30 am, 20 October 2009 at Spring in Singapore
Copyright
c
±
2009 by the Society for Industrial and Applied Mathematics, and
Floyd B. Hanson.
FINM 345/Stat 390 Stochastic Calculus
—
Lecture4–page1
—
Floyd B. Hanson
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View Full Document FinM 345 Stochastic Calculus:
4. Jump & JumpDiffusion Stochastic Calculus:
4.1. Poisson Jump Calculus Basic Chain Rules:
The Poisson process is quite different from the continuous
diffusion process, primarily because of the discontinuity
property of the Poisson process and the property that multiple
jumps are highly unlikely during small increments in time
Δ
t
.
•
4.1.1. Jump Calculus Rule for
h
(
dP
(
t
))
: Thus, the
most basic rule is the
zeroone law (ZOL) for jumps for
dP
(
t
)
in precision
dt
compact differential form,
(
dP
)
m
(
t
)
dt
=
zol
dP
(
t
)
,
(4.1)
provided the integer
m
≥
1
, the case
m
=0
being trivial.
FINM 345/Stat 390 Stochastic Calculus
—
Lecture4–page2
—
Floyd B. Hanson
An immediate generalization of this law is the following
corollary.
Corollary 4.1 ZeroOne Jump Law for
h
(
dP
(
t
))
:
h
(
dP
(
t
))
dt
=
zol
h
(1)
dP
(
t
) +
h
(0)(1

dP
(
t
))
,
(4.2)
with probability one, provided the function
h
(
p
)
is
rightcontinuous, such that values
h
(0)
and
h
(1)
exist
and are bounded.
Proof
: This follows by simple substitution of the
zeroone jump law
,
h
(
dP
(
t
))
dt
=
zol
h
(1)
, dP
(
t
)=1
h
(0)
, dP
(
t
)=0
dt
=
zol
h
(1)
dP
(
t
)+
h
(0)(1

dP
(
t
))
,
dP
(
t
) = 0
or
dP
(
t
) = 1
with probability one to
precision
dt
.
±
FINM 345/Stat 390 Stochastic Calculus
—
Lecture4–page3
—
Floyd B. Hanson
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View Full Document Formally, the differential
dP
(
t
)
can be treated as a
condition to test whether there has been a jump
. This
form (4.2) of the zeroone law suggests another extension
of the
jump function deﬁnitions
(B.178B.179). For
example, recall in (B.185) for a jump
in a
transformation
F
of the state process
X
at
t
1
,
[
F
](
X
(
t
1
)
,t
1
) =
F
(
X
(
t
+
1
)
+
1
)

F
(
X
(
t

1
)

1
)
.
Deﬁnition 4.1. Jump Function
[
h
](
dP
(
t
))
:
[
h
](
dP
(
t
))
dt
=
zol
h
(
dP
(
t
))

h
(0)
(4.3)
to precision
dt
, provided
h
(
p
)
is rightcontinuous, such
that values
h
(0)
and
h
(
dP
(
t
))
exist and are bounded.
FINM 345/Stat 390 Stochastic Calculus
—
Lecture4–page4
—
Floyd B. Hanson
With this deﬁnition, version (4.2) of the
zeroone law
can
immediately be written.
Corollary 4.2 ZeroOne Jump Law for
h
(
dP
(
t
))
with Jump Function
:
h
(
dP
(
t
))
dt
=
zol
h
(0) + [
h
](
dP
(
t
))
(4.4)
in terms of the jump function
[
h
](
dP
(
t
))
. Alternatively,
the jump function is written as
[
h
](
dP
(
t
))
dt
=
zol
(
h
(1)

h
(0))
dP
(
t
)
.
(4.5)
FINM 345/Stat 390 Stochastic Calculus
—
Lecture4–page5
—
Floyd B. Hanson
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View Full Document •
4.1.2. Jump Calculus Rule for
H
(
P
(
t
)
,t
)
:
Equations (4.4 – 4.5) are a primitive differential chain
rule for functions of only the Poisson differential
dP
(
t
)
.
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This note was uploaded on 04/19/2010 for the course FIN 390 taught by Professor Hansen during the Fall '09 term at University of Illinois, Urbana Champaign.
 Fall '09
 Hansen

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