Doléans-Dade exponential

In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale ''X'' is the unique strong solution of the stochastic differential equation $dY_t = Y_\,dX_t,\quad\quad Y_0=1,$where $Y_$ denotes the process of left limits, i.e., $Y_=\lim_Y_s$.

The concept is named after Catherine Doléans-Dade. Stochastic exponential plays an important role in the formulation of Girsanov's theorem and arises naturally in all applications where relative changes are important since $X$ measures the cumulative percentage change in $Y$.

Notation and terminology

Process $Y$ obtained above is commonly denoted by $\mathcal(X)$. The terminology "stochastic exponential" arises from the similarity of $\mathcal(X)=Y$ to the natural exponential of $X$: If ''X'' is absolutely continuous with respect to time, then ''Y'' solves, path-by-path, the differential equation $dY_t/\mathrmt = Y_tdX_t/dt$, whose solution is $Y=\exp(X-X_0)$.

General formula and special cases

* Without any assumptions on the semimartingale $X$, one has \mathcal(X)_t = \exp\Bigl(X_t-X_0-\frac12 c_t\Bigr)\prod_(1+\Delta X_s) \exp (-\Delta X_s),\qquad t\ge0,where c is the continuous part of quadratic variation of $X$ and the product extends over the (countably many) jumps of ''X'' up to time ''t''.
* If $X$ is continuous, then \mathcal(X) = \exp\Bigl(X-X_0-\frac12 Bigr).In particular, if $X$ is a Brownian motion, then the Doléans-Dade exponential is a geometric Brownian motion.
* If $X$ is continuous and of finite variation, then $\mathcal(X)=\exp(X-X_0).$Here $X$ need not be differentiable with respect to time; for example, $X$ can be the Cantor function.

Properties

* Stochastic exponential cannot go to zero continuously, it can only jump to zero. Hence, the stochastic exponential of a continuous semimartingale is always strictly positive.
* Once $\mathcal(X)$ has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when $\Delta X=-1$.
*Unlike the natural exponential $\exp(X_t)$, which depends only of the value of $X$ at time $t$, the stochastic exponential $\mathcal(X)_t$ depends not only on $X_t$ but on the whole history of $X$ in the time interval ,t/math>. For this reason one must write $\mathcal(X)_t$ and not $\mathcal(X_t)$.
* Natural exponential of a semimartingale can always be written as a stochastic exponential of another semimartingale but not the other way around.
* Stochastic exponential of a local martingale is again a local martingale.
* All the formulae and properties above apply also to stochastic exponential of a complex-valued $X$. This has application in the theory of conformal martingales and in the calculation of characteristic functions.

Useful identities

Yor's formula: for any two semimartingales $U$ and $V$ one has \mathcal(U)\mathcal(V) = \mathcal(U+V+ ,V

Applications

* Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential $\mathcal(X)$ of a continuous local martingale $X$ is a martingale are given by Kazamaki's condition, Novikov's condition, and Beneš's condition.

Derivation of the explicit formula for continuous semimartingales

For any continuous semimartingale ''X'', take for granted that $Y$ is continuous and strictly positive. Then applying Itō's formula with gives

:
\begin
\log(Y_t)-\log(Y_0) &= \int_0^t\frac\,dY_u -\int_0^t\frac\,d u
= X_t-X_0 - \frac t.
\end

Exponentiating with $Y_0=1$ gives the solution

:Y_t = \exp\Bigl(X_t-X_0-\frac12 t\Bigr),\qquad t\ge0.

This differs from what might be expected by comparison with the case where ''X'' has finite variation due to the existence of the quadratic variation term /nowiki>''X''/nowiki> in the solution.

See also

* Stochastic logarithm

References

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{{DEFAULTSORT:Doleans-Dade exponential
Martingale theory
Stochastic differential equations