In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation where Y_{-} denotes the process of left limits, i.e.,. 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. The terminology "stochastic exponential" arises from the similarity of to the natural exponential of X: If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation, whose solution is.

General formula and special cases

Properties

Useful identities

Yor's formula: for any two semimartingales U and V one has

Applications

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 Exponentiating with Y_0=1 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.