Stochastic Logarithm

In stochastic calculus, stochastic logarithm of a semimartingale $Y$such that $Y\neq0$ and $Y_-\neq0$ is the semimartingale $X$ given by$dX_t=\frac,\quad X_0=0.$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 $\mathcal(Y)$. The terminology ''stochastic logarithm'' arises from the similarity of $\mathcal(Y)$ 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 $\frac = \frac,$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$\mathcal(Y)_t = \log\Biggl, \frac\Biggl, +\frac12\int_0^t\frac +\sum_\Biggl(\log\Biggl, 1 + \frac \Biggr, -\frac\Biggr),\qquad t\ge0,$where 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 $\mathcal(Y)_t = \log\Biggl, \frac\Biggl, +\frac12\int_0^t\frac,\qquad t\ge0.$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$\mathcal(Y) = \log\Biggl, \frac\Biggl, .$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 $\mathcal(\mathcal(X)) = X-X_0$. Conversely, if $Y\neq 0$ and $Y_-\neq 0$, then $\mathcal(\mathcal(Y)) = Y/Y_0$.
* Unlike the natural logarithm $\log(Y_t)$, which depends only of the value of $Y$ at time $t$, the stochastic logarithm $\mathcal(Y)_t$ depends not only on $Y_t$ but on the whole history of $Y$ in the time interval ,t/math>. For this reason one must write $\mathcal(Y)_t$ and not $\mathcal(Y_t)$.
* 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 $Y^,Y^$ do not vanish together with their left limits, then\mathcal\bigl(Y^Y^\bigr) = \mathcal\bigl(Y^\bigr)
+ \mathcal\bigl(Y^\bigr)
+ \bigl mathcal\bigl(Y^\bigr),\mathcal\bigl(Y^\bigr)\bigr
* Stochastic logarithm of $1/\mathcal(X)$: If $\Delta X\neq -1$, then\mathcal\biggl(\frac\biggr)_t = X_0-X_t- c_t
+\sum_\frac.

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:
*# Process $U$ is special under $Q$.
*# Process U+ ,\mathcal(Z)/math> is special under $P$.
*+ If either of these conditions holds, then the $Q$-drift of $U$ equals the $P$-drift of U+ ,\mathcal(Z)/math>.

References

{{Reflist

See also

* Stochastic exponential

Stochastic calculus