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 $$dY_t = Y_{t-}\,dX_t,\quad\quad Y_0=1,$$where $$Y_{-}$$ denotes the process of left limits, i.e., $$Y_{t-}=\lim_{s\uparrow t}Y_s$$.

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 $$\mathcal{E}(X)$$. The terminology "stochastic exponential" arises from the similarity of $$\mathcal{E}(X)=Y$$ to the natural exponential of $$X$$: If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation $$dY_t/\mathrm{d}t = Y_tdX_t/dt$$, whose solution is $$Y=\exp(X-X_0)$$.

General formula and special cases

 * Without any assumptions on the semimartingale $$X$$, one has $$\mathcal{E}(X)_t = \exp\Bigl(X_t-X_0-\frac12[X]^c_t\Bigr)\prod_{s\le t}(1+\Delta X_s) \exp (-\Delta X_s),\qquad t\ge0,$$where $$[X]^c$$ 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 $$\mathcal{E}(X) = \exp\Bigl(X-X_0-\frac12[X]\Bigr).$$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 $$\mathcal{E}(X)=\exp(X-X_0).$$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 $$\mathcal{E}(X)$$ 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 $$\mathcal{E}(X)_t$$ depends not only on $$X_t$$ but on the whole history of $$X$$ in the time interval $$[0,t]$$. For this reason one must write $$\mathcal{E}(X)_t$$ and not $$\mathcal{E}(X_t)$$.
 * 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 $$\mathcal{E}(U)\mathcal{E}(V) = \mathcal{E}(U+V+[U,V])$$

Applications

 * Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential $$\mathcal{E}(X)$$ 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{align} \log(Y_t)-\log(Y_0) &= \int_0^t\frac{1}{Y_u}\,dY_u -\int_0^t\frac{1}{2Y_u^2}\,d[Y]_u = X_t-X_0 - \frac{1}{2}[X]_t. \end{align} $$

Exponentiating with $$ Y_0=1 $$ gives the solution


 * $$Y_t = \exp\Bigl(X_t-X_0-\frac12[X]_t\Bigr),\qquad t\ge0.$$

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 [ X ] in the solution.