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Logarithmic convolution

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In mathematics, the scale convolution of two functions s ( t ) {\displaystyle s(t)} and r ( t ) {\displaystyle r(t)} , also known as their logarithmic convolution or log-volution is defined as the function

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} to v = log ⁡ t {\displaystyle v=\log t} :

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})} and g ( v ) = r ( e v ) {\displaystyle g(v)=r(e^{v})} and let v = log ⁡ t {\displaystyle v=\log t} , then

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

  1. Peter Buchen (2012). An Introduction to Exotic Option Pricing. Chapman and Hall/CRC Financial Mathematics Series. CRC Press. ISBN 9781420091021.
  2. ^ "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.


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