In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation
The concept is named after Catherine Doléans-Dade.[1] 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 measures the cumulative percentage change in .
Notation and terminology
Process obtained above is commonly denoted by . The terminology "stochastic exponential" arises from the similarity of to the natural exponential of : If X is absolutely continuous with respect to time[clarification needed], then Y solves, path-by-path, the differential equation , whose solution is .
General formula and special cases
- Without any assumptions on the semimartingale , one has
- If is continuous, then
- If is continuous and of finite variation, then
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 has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when .
- Unlike the natural exponential , which depends only of the value of at time , the stochastic exponential depends not only on but on the whole history of in the time interval . For this reason one must write and not .
- 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 . This has application in the theory of conformal martingales and in the calculation of characteristic functions.
Useful identities
Yor's formula:[2] for any two semimartingales and one has
Applications
- Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential of a continuous local martingale 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 is continuous and strictly positive. Then applying Itō's formula with ƒ(Y) = log(Y) gives
Exponentiating with gives the solution
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
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