Logarithmic convolution

Results

The logarithmic convolution can be related to the ordinary convolution by changing the variable from t to v = \log t:

: \begin{align} 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{align}

Define f(v) = s(e^v) and g(v) = r(e^v) and let v = \log t, then

: s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v).

References

References

1. (2012). "An Introduction to Exotic Option Pricing". CRC Press.
2. (22 March 2013). "logarithmic convolution". Planet Math.