In stochastic calculus, stochastic logarithm of a semimartingale such that and is the semimartingale given by[1]In 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 has[1]where 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 .[1]
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.[2]
Useful identities
Converse of the Yor formula:[1] If do not vanish together with their left limits, then
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 .