Stochastic logarithm
In stochastic calculus, stochastic logarithm of a semimartingale such that  and  is the semimartingale  given byIn layperson's terms, stochastic logarithm of   measures the cumulative percentage change in .

Notation and terminology 
The process  obtained above is commonly denoted . The terminology stochastic logarithm arises from the similarity of  to the natural logarithm : If  is absolutely continuous with respect to time and , then  solves, path-by-path, the differential equation whose solution is .

General formula and special cases 

 Without any assumptions on the semimartingale  (other than ), one haswhere  is the continuous part of quadratic variation of  and the sum extends over the (countably many) jumps of  up to time .
 If  is continuous, then In particular, if  is a geometric Brownian motion, then  is a Brownian motion with a constant drift rate.
 If  is continuous and of finite variation, thenHere  need not be differentiable with respect to time; for example,  can equal 1 plus the Cantor function.

Properties 

 Stochastic logarithm is an inverse operation to stochastic exponential: If , then . Conversely, if   and , then .
 Unlike the natural logarithm , which depends only of the value of  at time , the stochastic logarithm  depends not only on  but on the whole history of  in the time interval . For this reason one must write  and not .
 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 .
 Stochastic logarithm can be defined also for processes  that are absorbed in zero after jumping to zero. Such definition is meaningful up to the first time that   reaches  continuously.

Useful identities 

 Converse of the Yor formula: If  do not vanish together with their left limits, then
 Stochastic logarithm of : If , then

Applications 

 Girsanov's theorem can be paraphrased as follows: Let  be a probability measure equivalent to another probability measure . Denote by  the uniformly integrable martingale closed by . For a semimartingale  the following are equivalent:
 Process  is special under .
 Process  is special under . 
+ If either of these conditions holds, then the -drift of  equals the -drift of .

References 

== See also ==

 Stochastic exponential

Stochastic calculus