Stochastic logarithm

In stochastic calculus, stochastic logarithm of a semimartingale $Y$such that $Y\ne 0$ and $Y_{-}\ne 0$ is the semimartingale $X$ given by$$d{X}_t=\frac{d{Y}_t}{Y_{t-}},X_0=0.$$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 $ℒ(Y)$. The terminology stochastic logarithm arises from the similarity of $ℒ(Y)$ to the natural logarithm $\log (Y)$: If $Y$ is absolutely continuous with respect to time and $Y\ne 0$, then $X$ solves, path-by-path, the differential equation $$\frac{d{X}_t}{dt}=\frac{\frac{d{Y}_t}{dt}}{Y_t},$$whose solution is $X=\log |Y|-\log |Y_0|$.

General formula and special cases

• Without any assumptions on the semimartingale $Y$ (other than $Y\ne 0,Y_{-}\ne 0$), one has$$ℒ(Y{)}_t=\log |\frac{Y_t}{Y_0}|+\frac{1}{2}\underset{0}{\overset{t}{\int }}\frac{d[Y{]}_s^c}{Y_{s-}^2}+\sum _{s\le t}(\log |1+\frac{\mathrm{\Delta }{Y}_s}{Y_{s-}}|-\frac{\mathrm{\Delta }{Y}_s}{Y_{s-}}),t\ge 0,$$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 $$ℒ(Y{)}_t=\log |\frac{Y_t}{Y_0}|+\frac{1}{2}\underset{0}{\overset{t}{\int }}\frac{d[Y{]}_s^c}{Y_{s-}^2},t\ge 0.$$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$$ℒ(Y)=\log \left|\frac{Y}{Y_0}\right|.$$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 $\mathrm{\Delta }X\ne -1$, then $ℒ(ℰ(X))=X-X_0$. Conversely, if $Y\ne 0$ and $Y_{-}\ne 0$, then $ℰ(ℒ(Y))=Y/Y_0$.
• Unlike the natural logarithm $\log (Y_t)$, which depends only of the value of $Y$ at time $t$, the stochastic logarithm $ℒ(Y{)}_t$ 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 $ℒ(Y{)}_t$ and not $ℒ(Y_t)$.
• 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$$ℒ\left(Y^{(1)}{Y}^{(2)}\right)=ℒ\left(Y^{(1)}\right)+ℒ\left(Y^{(2)}\right)+\left[ℒ\right(Y^{(1)}),ℒ(Y^{(2)}\left)\right].$$
• Stochastic logarithm of $1/ℰ(X)$: If $\mathrm{\Delta }X\ne -1$, then$$ℒ(\frac{1}{ℰ(X)}{)}_t=X_0-X_t-[X{]}_t^c+\sum _{s\le t}\frac{(\mathrm{\Delta }{X}_s{)}^2}{1+\mathrm{\Delta }{X}_s}.$$

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 $Z_{\mathrm{\infty }}=dQ/dP$. For a semimartingale $U$ the following are equivalent:
  1. Process $U$ is special under $Q$.
  2. Process $U+[U,ℒ(Z)]$ is special under $P$.
• + If either of these conditions holds, then the $Q$-drift of $U$ equals the $P$-drift of $U+[U,ℒ(Z)]$.

See also

• Stochastic exponential