Stochastic logarithm

In stochastic calculus, stochastic logarithm of a semimartingale $Y$such that $Y\neq 0$ and $Y_{-}\neq 0$ is the semimartingale $X$ given by

$$dX_{t}={\frac {dY_{t}}{Y_{t-}}},\quad X_{0}=0.$$

In layperson's terms, stochastic logarithm of $Y$ measures the cumulative percentage change in $Y$.

Notation and terminology

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

$${\frac {dX_{t}}{dt}}={\frac {\frac {dY_{t}}{dt}}{Y_{t}}},$$

whose solution is $X=\log |Y|-\log |Y_{0}|$.

General formula and special cases

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

Properties

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

Useful identities

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

Applications

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

See also

• Stochastic exponential