In stochastic calculus, stochastic logarithm of a semimartingale
is the semimartingale
{\displaystyle dX_{t}={\frac {dY_{t}}{Y_{t-}}},\quad X_{0}=0.}
In layperson's terms, stochastic logarithm of
measures the cumulative percentage change in
The process
obtained above is commonly denoted
The terminology stochastic logarithm arises from the similarity of
to the natural logarithm
log (
is absolutely continuous with respect to time and
solves, path-by-path, the differential equation
{\displaystyle {\frac {dX_{t}}{dt}}={\frac {\frac {dY_{t}}{dt}}{Y_{t}}},}
whose solution is