Stochastic logarithm

In stochastic calculus, stochastic logarithm of a semimartingale ${\displaystyle Y}$such that ${\displaystyle Y\neq 0}$ and ${\displaystyle Y_{-}\neq 0}$ is the semimartingale ${\displaystyle X}$ given by^[1]$${\displaystyle dX_{t}={\frac {dY_{t}}{Y_{t-}}},\quad X_{0}=0.}$$In layperson's terms, stochastic logarithm of ${\displaystyle Y}$ measures the cumulative percentage change in ${\displaystyle Y}$.

The process ${\displaystyle X}$ obtained above is commonly denoted ${\displaystyle {\mathcal {L}}(Y)}$. The terminology stochastic logarithm arises from the similarity of ${\displaystyle {\mathcal {L}}(Y)}$ to the natural logarithm ${\displaystyle \log(Y)}$: If ${\displaystyle Y}$ is absolutely continuous with respect to time and ${\displaystyle Y\neq 0}$, then ${\displaystyle X}$ solves, path-by-path, the differential equation $${\displaystyle {\frac {dX_{t}}{dt}}={\frac {\frac {dY_{t}}{dt}}{Y_{t}}},}$$whose solution is ${\displaystyle X=\log |Y|-\log |Y_{0}|}$.

• Without any assumptions on the semimartingale ${\displaystyle Y}$ (other than ${\displaystyle Y\neq 0,Y_{-}\neq 0}$), one has^[1]$${\displaystyle {\mathcal {L}}(Y)_{t}=\log {\Biggl |}{\frac {Y_{t}}{Y_{0}}}{\Biggl |}+{\frac {1}{2}}\int _{0}^{t}{\frac {d[Y]_{s}^{c}}{Y_{s-}^{2}}}+\sum _{s\leq t}{\Biggl (}\log {\Biggl |}1+{\frac {\Delta Y_{s}}{Y_{s-}}}{\Biggr |}-{\frac {\Delta Y_{s}}{Y_{s-}}}{\Biggr )},\qquad t\geq 0,}$$where ${\displaystyle [Y]^{c}}$ is the continuous part of quadratic variation of ${\displaystyle Y}$ and the sum extends over the (countably many) jumps of ${\displaystyle Y}$ up to time ${\displaystyle t}$.
• If ${\displaystyle Y}$ is continuous, then $${\displaystyle {\mathcal {L}}(Y)_{t}=\log {\Biggl |}{\frac {Y_{t}}{Y_{0}}}{\Biggl |}+{\frac {1}{2}}\int _{0}^{t}{\frac {d[Y]_{s}^{c}}{Y_{s-}^{2}}},\qquad t\geq 0.}$$In particular, if ${\displaystyle Y}$ is a geometric Brownian motion, then ${\displaystyle X}$ is a Brownian motion with a constant drift rate.
• If ${\displaystyle Y}$ is continuous and of finite variation, then$${\displaystyle {\mathcal {L}}(Y)=\log {\Biggl |}{\frac {Y}{Y_{0}}}{\Biggl |}.}$$Here ${\displaystyle Y}$ need not be differentiable with respect to time; for example, ${\displaystyle Y}$ can equal 1 plus the Cantor function.

• Stochastic logarithm is an inverse operation to stochastic exponential: If ${\displaystyle \Delta X\neq -1}$, then ${\displaystyle {\mathcal {L}}({\mathcal {E}}(X))=X-X_{0}}$. Conversely, if ${\displaystyle Y\neq 0}$ and ${\displaystyle Y_{-}\neq 0}$, then ${\displaystyle {\mathcal {E}}({\mathcal {L}}(Y))=Y/Y_{0}}$.^[1]
• Unlike the natural logarithm ${\displaystyle \log(Y_{t})}$, which depends only of the value of ${\displaystyle Y}$ at time ${\displaystyle t}$, the stochastic logarithm ${\displaystyle {\mathcal {L}}(Y)_{t}}$ depends not only on ${\displaystyle Y_{t}}$ but on the whole history of ${\displaystyle Y}$ in the time interval ${\displaystyle [0,t]}$. For this reason one must write ${\displaystyle {\mathcal {L}}(Y)_{t}}$ and not ${\displaystyle {\mathcal {L}}(Y_{t})}$.
• 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 ${\displaystyle Y}$.
• Stochastic logarithm can be defined also for processes ${\displaystyle Y}$ that are absorbed in zero after jumping to zero. Such definition is meaningful up to the first time that ${\displaystyle Y}$ reaches ${\displaystyle 0}$ continuously.^[2]

• Converse of the Yor formula:^[1] If ${\displaystyle Y^{(1)},Y^{(2)}}$ do not vanish together with their left limits, then$${\displaystyle {\mathcal {L}}{\bigl (}Y^{(1)}Y^{(2)}{\bigr )}={\mathcal {L}}{\bigl (}Y^{(1)}{\bigr )}+{\mathcal {L}}{\bigl (}Y^{(2)}{\bigr )}+{\bigl [}{\mathcal {L}}{\bigl (}Y^{(1)}{\bigr )},{\mathcal {L}}{\bigl (}Y^{(2)}{\bigr )}{\bigr ]}.}$$
• Stochastic logarithm of ${\displaystyle 1/{\mathcal {E}}(X)}$:^[2] If ${\displaystyle \Delta X\neq -1}$, then$${\displaystyle {\mathcal {L}}{\biggl (}{\frac {1}{{\mathcal {E}}(X)}}{\biggr )}_{t}=X_{0}-X_{t}-[X]_{t}^{c}+\sum _{s\leq t}{\frac {(\Delta X_{s})^{2}}{1+\Delta X_{s}}}.}$$

• Girsanov's theorem can be paraphrased as follows: Let ${\displaystyle Q}$ be a probability measure equivalent to another probability measure ${\displaystyle P}$. Denote by ${\displaystyle Z}$ the uniformly integrable martingale closed by ${\displaystyle Z_{\infty }=dQ/dP}$. For a semimartingale ${\displaystyle U}$ the following are equivalent:
  1. Process ${\displaystyle U}$ is special under ${\displaystyle Q}$.
  2. Process ${\displaystyle U+[U,{\mathcal {L}}(Z)]}$ is special under ${\displaystyle P}$.
• + If either of these conditions holds, then the ${\displaystyle Q}$-drift of ${\displaystyle U}$ equals the ${\displaystyle P}$-drift of ${\displaystyle U+[U,{\mathcal {L}}(Z)]}$.

• Stochastic exponential