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In mathematics , the scale convolution of two functions
s
(
t
)
{\displaystyle s(t)}
r
(
t
)
{\displaystyle r(t)}
logarithmic convolution or log-volution [ 1] [ 2]
s
∗
l
r
(
t
)
=
r
∗
l
s
(
t
)
=
∫
0
∞
s
(
t
a
)
r
(
a
)
d
a
a
{\displaystyle s*_{l}r(t)=r*_{l}s(t)=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}}
when this quantity exists.
Results
The logarithmic convolution can be related to the ordinary convolution by changing the variable from
t
{\displaystyle t}
v
=
log
t
{\displaystyle v=\log t}
[ 2]
s
∗
l
r
(
t
)
=
∫
0
∞
s
(
t
a
)
r
(
a
)
d
a
a
=
∫
−
∞
∞
s
(
t
e
u
)
r
(
e
u
)
d
u
=
∫
−
∞
∞
s
(
e
log
t
−
u
)
r
(
e
u
)
d
u
.
{\displaystyle {\begin{aligned}s*_{l}r(t)&=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}\\&=\int _{-\infty }^{\infty }s\left({\frac {t}{e^{u}}}\right)r(e^{u})\,du\\&=\int _{-\infty }^{\infty }s\left(e^{\log t-u}\right)r(e^{u})\,du.\end{aligned}}}
Define
f
(
v
)
=
s
(
e
v
)
{\displaystyle f(v)=s(e^{v})}
g
(
v
)
=
r
(
e
v
)
{\displaystyle g(v)=r(e^{v})}
v
=
log
t
{\displaystyle v=\log t}
s
∗
l
r
(
v
)
=
f
∗
g
(
v
)
=
g
∗
f
(
v
)
=
r
∗
l
s
(
v
)
.
{\displaystyle s*_{l}r(v)=f*g(v)=g*f(v)=r*_{l}s(v).}
See also
References
^ Peter Buchen (2012). An Introduction to Exotic Option Pricing . Chapman and Hall/CRC Financial Mathematics Series. CRC Press. ISBN 9781420091021 ^ a b "logarithmic convolution" . Planet Math . 22 March 2013. Retrieved 15 September 2024 .
External links
This article incorporates material from logarithmic convolution on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
