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 [math]\displaystyle{ dY_t = Y_{t-}\,dX_t,\quad\quad Y_0=1, }[/math]where [math]\displaystyle{ Y_{-} }[/math] denotes the process of left limits, i.e., [math]\displaystyle{ Y_{t-}=\lim_{s\uparrow t}Y_s }[/math]. 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 [math]\displaystyle{ X }[/math] measures the cumulative percentage change in [math]\displaystyle{ Y }[/math].

Notation and terminology

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

General formula and special cases

• Without any assumptions on the semimartingale [math]\displaystyle{ X }[/math], one has [math]\displaystyle{ \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, }[/math]where [math]\displaystyle{ [X]^c }[/math] is the continuous part of quadratic variation of [math]\displaystyle{ X }[/math] and the product extends over the (countably many) jumps of X up to time t.
• If [math]\displaystyle{ X }[/math] is continuous, then [math]\displaystyle{ \mathcal{E}(X) = \exp\Bigl(X-X_0-\frac12[X]\Bigr). }[/math]In particular, if [math]\displaystyle{ X }[/math] is a Brownian motion, then the Doléans-Dade exponential is a geometric Brownian motion.
• If [math]\displaystyle{ X }[/math] is continuous and of finite variation, then [math]\displaystyle{ \mathcal{E}(X)=\exp(X-X_0). }[/math]Here [math]\displaystyle{ X }[/math] need not be differentiable with respect to time; for example, [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ \mathcal{E}(X) }[/math] has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when [math]\displaystyle{ \Delta X=-1 }[/math].
• Unlike the natural exponential [math]\displaystyle{ \exp(X_t) }[/math], which depends only of the value of [math]\displaystyle{ X }[/math] at time [math]\displaystyle{ t }[/math], the stochastic exponential [math]\displaystyle{ \mathcal{E}(X)_t }[/math] depends not only on [math]\displaystyle{ X_t }[/math] but on the whole history of [math]\displaystyle{ X }[/math] in the time interval [math]\displaystyle{ [0,t] }[/math]. For this reason one must write [math]\displaystyle{ \mathcal{E}(X)_t }[/math] and not [math]\displaystyle{ \mathcal{E}(X_t) }[/math].
• 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 [math]\displaystyle{ X }[/math]. 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 [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] one has [math]\displaystyle{ \mathcal{E}(U)\mathcal{E}(V) = \mathcal{E}(U+V+[U,V]) }[/math]

Applications

• Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential [math]\displaystyle{ \mathcal{E}(X) }[/math] of a continuous local martingale [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ Y }[/math] is continuous and strictly positive. Then applying Itō's formula with ƒ(Y) = log(Y) gives

[math]\displaystyle{ \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} }[/math]

Exponentiating with [math]\displaystyle{ Y_0=1 }[/math] gives the solution

[math]\displaystyle{ Y_t = \exp\Bigl(X_t-X_0-\frac12[X]_t\Bigr),\qquad t\ge0. }[/math]

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