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Stochastic logarithm

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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

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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

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  • 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

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  • 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

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  • Converse of the Yor formula:[1] If do not vanish together with their left limits, then
  • Stochastic logarithm of :[2] If , then

Applications

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  • 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:
    1. Process is special under .
    2. Process is special under .
  • + If either of these conditions holds, then the -drift of equals the -drift of .

References

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  1. ^ a b c d Jacod, Jean; Shiryaev, Albert Nikolaevich (2003). Limit theorems for stochastic processes (2nd ed.). Berlin: Springer. pp. 134–138. ISBN 3-540-43932-3. OCLC 50554399.
  2. ^ a b Larsson, Martin; Ruf, Johannes (2019). "Stochastic exponentials and logarithms on stochastic intervals — A survey". Journal of Mathematical Analysis and Applications. 476 (1): 2–12. arXiv:1702.03573. doi:10.1016/j.jmaa.2018.11.040. S2CID 119148331.

See also