Stochastic logarithm

In stochastic calculus, stochastic logarithm of a semimartingale $$Y$$such that $$Y\neq0$$ and $$Y_-\neq0$$ is the semimartingale $$X$$ given by $$dX_t=\frac{dY_t}{Y_{t-}},\quad X_0=0.$$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
+\frac12\int_0^t\frac{d[Y]^c_s}{Y_{s-}^2} +\sum_{s\le t}\Biggl(\log\Biggl| 1 + \frac{\Delta Y_s}{Y_{s-}} \Biggr| -\frac{\Delta Y_s}{Y_{s-}}\Biggr),\qquad t\ge0,$$where $$[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$$. +\frac12\int_0^t\frac{d[Y]^c_s}{Y_{s-}^2},\qquad t\ge0.$$In particular, if $$Y$$ is a geometric Brownian motion, then $$X$$ is a Brownian motion with a constant drift rate.
 * Without any assumptions on the semimartingale $$Y$$ (other than $$Y\neq 0, Y_-\neq 0$$), one has $$\mathcal{L}(Y)_t = \log\Biggl|\frac{Y_t}{Y_0}\Biggl|
 * If $$Y$$ is continuous, then $$\mathcal{L}(Y)_t = \log\Biggl|\frac{Y_t}{Y_0}\Biggl|
 * 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
+ \mathcal{L}\bigl(Y^{(2)}\bigr) + \bigl[\mathcal{L}\bigl(Y^{(1)}\bigr),\mathcal{L}\bigl(Y^{(2)}\bigr)\bigr].$$ +\sum_{s\leq t}\frac{(\Delta X_s)^2}{1+\Delta X_s}.$$
 * Converse of the Yor formula: If $$Y^{(1)},Y^{(2)}$$ do not vanish together with their left limits, then$$\mathcal{L}\bigl(Y^{(1)}Y^{(2)}\bigr) = \mathcal{L}\bigl(Y^{(1)}\bigr)
 * Stochastic logarithm of $$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]^c_t

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 $$U+[U,\mathcal{L}(Z)]$$ is special under $$P$$.
 * + If either of these conditions holds, then the $$Q$$-drift of $$U$$ equals the $$P$$-drift of $$U+[U,\mathcal{L}(Z)]$$.