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 $$d{Y}_t=Y_{t-}d{X}_t,Y_0=1,$$where $Y_{-}$ denotes the process of left limits, i.e., $Y_{t-}=\underset{s\uparrow t}{\lim }{Y}_s$. 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 ${ℰ}(X)$. The terminology "stochastic exponential" arises from the similarity of ${ℰ}(X)=Y$ to the natural exponential of $X$: If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation $d{Y}_t/\mathrm{d}t=Y_{t}d{X}_t/dt$, whose solution is $Y=\exp (X-X_0)$.

General formula and special cases

• Without any assumptions on the semimartingale $X$, one has $${ℰ}(X{)}_t=\exp (X_t-X_0-\frac{1}{2}[X{]}_t^c)\prod _{s\le t}(1+\mathrm{\Delta }{X}_s)\exp (-\mathrm{\Delta }{X}_s),t\ge 0,$$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 $${ℰ}(X)=\exp (X-X_0-\frac{1}{2}[X]).$$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 $${ℰ}(X)=\exp (X-X_0).$$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 ${ℰ}(X)$ has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when $\mathrm{\Delta }X=-1$.
• Unlike the natural exponential $\exp (X_t)$, which depends only of the value of $X$ at time $t$, the stochastic exponential ${ℰ}(X{)}_t$ 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 ${ℰ}(X{)}_t$ and not ${ℰ}(X_t)$.
• 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 $${ℰ}(U){ℰ}(V)={ℰ}(U+V+[U,V])$$

Applications

• Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential ${ℰ}(X)$ 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 ƒ(Y) = log(Y) gives

$\begin{array}{rl}\log (Y_t)-\log (Y_0)& =\underset{0}{\overset{t}{\int }}\frac{1}{Y_u}d{Y}_u-\underset{0}{\overset{t}{\int }}\frac{1}{2Y_u^2}d[Y{]}_u=X_t-X_0-\frac{1}{2}[X{]}_t.\end{array}$

Exponentiating with $Y_0=1$ gives the solution

$Y_t=\exp (X_t-X_0-\frac{1}{2}[X{]}_t),t\ge 0.$

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.

See also

• Stochastic logarithm

References

• Jacod, J.; Shiryaev, A. N. (2003), Limit Theorems for Stochastic Processes (2nd ed.), Springer, pp. 58–61, ISBN 3-540-43932-3
• Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 3-540-00313-4