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Stochastic logarithm
In stochastic calculus, stochastic logarithm of a semimartingale
Y
{\displaystyle Y}
such that
Y
≠
0
{\displaystyle Y\neq 0}
and
Y
−
≠
0
{\displaystyle Y_{-}\neq 0}
is the semimartingale
X
{\displaystyle X}
given by[1]
d
X
t
=
d
Y
t
Y
t
−
,
X
0
=
0.
{\displaystyle dX_{t}={\frac {dY_{t}}{Y_{t-}}},\quad X_{0}=0.}
In layperson's terms, stochastic logarithm of
Y
{\displaystyle Y}
measures the cumulative percentage change in
Y
{\displaystyle Y}
.
Notation and terminology [ edit ]
The process
X
{\displaystyle X}
obtained above is commonly denoted
L
(
Y
)
{\displaystyle {\mathcal {L}}(Y)}
. The terminology stochastic logarithm arises from the similarity of
L
(
Y
)
{\displaystyle {\mathcal {L}}(Y)}
to the natural logarithm
log
(
Y
)
{\displaystyle \log(Y)}
: If
Y
{\displaystyle Y}
is absolutely continuous with respect to time and
Y
≠
0
{\displaystyle Y\neq 0}
, then
X
{\displaystyle X}
solves, path-by-path, the differential equation
d
X
t
d
t
=
d
Y
t
d
t
Y
t
,
{\displaystyle {\frac {dX_{t}}{dt}}={\frac {\frac {dY_{t}}{dt}}{Y_{t}}},}
whose solution is
X
=
log
|
Y
|
−
log
|
Y
0
|
{\displaystyle X=\log |Y|-\log |Y_{0}|}
.
General formula and special cases [ edit ]
Without any assumptions on the semimartingale
Y
{\displaystyle Y}
(other than
Y
≠
0
,
Y
−
≠
0
{\displaystyle Y\neq 0,Y_{-}\neq 0}
), one has[1]
L
(
Y
)
t
=
log
|
Y
t
Y
0
|
+
1
2
∫
0
t
d
[
Y
]
s
c
Y
s
−
2
+
∑
s
≤
t
(
log
|
1
+
Δ
Y
s
Y
s
−
|
−
Δ
Y
s
Y
s
−
)
,
t
≥
0
,
{\displaystyle {\mathcal {L}}(Y)_{t}=\log {\Biggl |}{\frac {Y_{t}}{Y_{0}}}{\Biggl |}+{\frac {1}{2}}\int _{0}^{t}{\frac {d[Y]_{s}^{c}}{Y_{s-}^{2}}}+\sum _{s\leq t}{\Biggl (}\log {\Biggl |}1+{\frac {\Delta Y_{s}}{Y_{s-}}}{\Biggr |}-{\frac {\Delta Y_{s}}{Y_{s-}}}{\Biggr )},\qquad t\geq 0,}
where
[
Y
]
c
{\displaystyle [Y]^{c}}
is the continuous part of quadratic variation of
Y
{\displaystyle Y}
and the sum extends over the (countably many) jumps of
Y
{\displaystyle Y}
up to time
t
{\displaystyle t}
.
If
Y
{\displaystyle Y}
is continuous, then
L
(
Y
)
t
=
log
|
Y
t
Y
0
|
+
1
2
∫
0
t
d
[
Y
]
s
c
Y
s
−
2
,
t
≥
0.
{\displaystyle {\mathcal {L}}(Y)_{t}=\log {\Biggl |}{\frac {Y_{t}}{Y_{0}}}{\Biggl |}+{\frac {1}{2}}\int _{0}^{t}{\frac {d[Y]_{s}^{c}}{Y_{s-}^{2}}},\qquad t\geq 0.}
In particular, if
Y
{\displaystyle Y}
is a geometric Brownian motion, then
X
{\displaystyle X}
is a Brownian motion with a constant drift rate.
If
Y
{\displaystyle Y}
is continuous and of finite variation, then
L
(
Y
)
=
log
|
Y
Y
0
|
.
{\displaystyle {\mathcal {L}}(Y)=\log {\Biggl |}{\frac {Y}{Y_{0}}}{\Biggl |}.}
Here
Y
{\displaystyle Y}
need not be differentiable with respect to time; for example,
Y
{\displaystyle Y}
can equal 1 plus the Cantor function .
Properties [ edit ]
Stochastic logarithm is an inverse operation to stochastic exponential : If
Δ
X
≠
−
1
{\displaystyle \Delta X\neq -1}
, then
L
(
E
(
X
)
)
=
X
−
X
0
{\displaystyle {\mathcal {L}}({\mathcal {E}}(X))=X-X_{0}}
. Conversely, if
Y
≠
0
{\displaystyle Y\neq 0}
and
Y
−
≠
0
{\displaystyle Y_{-}\neq 0}
, then
E
(
L
(
Y
)
)
=
Y
/
Y
0
{\displaystyle {\mathcal {E}}({\mathcal {L}}(Y))=Y/Y_{0}}
.[1]
Unlike the natural logarithm
log
(
Y
t
)
{\displaystyle \log(Y_{t})}
, which depends only of the value of
Y
{\displaystyle Y}
at time
t
{\displaystyle t}
, the stochastic logarithm
L
(
Y
)
t
{\displaystyle {\mathcal {L}}(Y)_{t}}
depends not only on
Y
t
{\displaystyle Y_{t}}
but on the whole history of
Y
{\displaystyle Y}
in the time interval
[
0
,
t
]
{\displaystyle [0,t]}
. For this reason one must write
L
(
Y
)
t
{\displaystyle {\mathcal {L}}(Y)_{t}}
and not
L
(
Y
t
)
{\displaystyle {\mathcal {L}}(Y_{t})}
.
Stochastic logarithm of a local martingale that does not vanish together with its left limit is again a local martingale.
All the formulae and properties above apply also to stochastic logarithm of a complex -valued
Y
{\displaystyle Y}
.
Stochastic logarithm can be defined also for processes
Y
{\displaystyle Y}
that are absorbed in zero after jumping to zero. Such definition is meaningful up to the first time that
Y
{\displaystyle Y}
reaches
0
{\displaystyle 0}
continuously.[2]
Useful identities [ edit ]
Converse of the Yor formula:[1] If
Y
(
1
)
,
Y
(
2
)
{\displaystyle Y^{(1)},Y^{(2)}}
do not vanish together with their left limits, then
L
(
Y
(
1
)
Y
(
2
)
)
=
L
(
Y
(
1
)
)
+
L
(
Y
(
2
)
)
+
[
L
(
Y
(
1
)
)
,
L
(
Y
(
2
)
)
]
.
{\displaystyle {\mathcal {L}}{\bigl (}Y^{(1)}Y^{(2)}{\bigr )}={\mathcal {L}}{\bigl (}Y^{(1)}{\bigr )}+{\mathcal {L}}{\bigl (}Y^{(2)}{\bigr )}+{\bigl [}{\mathcal {L}}{\bigl (}Y^{(1)}{\bigr )},{\mathcal {L}}{\bigl (}Y^{(2)}{\bigr )}{\bigr ]}.}
Stochastic logarithm of
1
/
E
(
X
)
{\displaystyle 1/{\mathcal {E}}(X)}
:[2] If
Δ
X
≠
−
1
{\displaystyle \Delta X\neq -1}
, then
L
(
1
E
(
X
)
)
t
=
X
0
−
X
t
−
[
X
]
t
c
+
∑
s
≤
t
(
Δ
X
s
)
2
1
+
Δ
X
s
.
{\displaystyle {\mathcal {L}}{\biggl (}{\frac {1}{{\mathcal {E}}(X)}}{\biggr )}_{t}=X_{0}-X_{t}-[X]_{t}^{c}+\sum _{s\leq t}{\frac {(\Delta X_{s})^{2}}{1+\Delta X_{s}}}.}
Applications [ edit ]
Girsanov's theorem can be paraphrased as follows: Let
Q
{\displaystyle Q}
be a probability measure equivalent to another probability measure
P
{\displaystyle P}
. Denote by
Z
{\displaystyle Z}
the uniformly integrable martingale closed by
Z
∞
=
d
Q
/
d
P
{\displaystyle Z_{\infty }=dQ/dP}
. For a semimartingale
U
{\displaystyle U}
the following are equivalent:
Process
U
{\displaystyle U}
is special under
Q
{\displaystyle Q}
.
Process
U
+
[
U
,
L
(
Z
)
]
{\displaystyle U+[U,{\mathcal {L}}(Z)]}
is special under
P
{\displaystyle P}
.
+ If either of these conditions holds, then the
Q
{\displaystyle Q}
-drift of
U
{\displaystyle U}
equals the
P
{\displaystyle P}
-drift of
U
+
[
U
,
L
(
Z
)
]
{\displaystyle U+[U,{\mathcal {L}}(Z)]}
.
References [ edit ]
See also