In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation
denotes the process of left limits, i.e.,
{\displaystyle Y_{t-}=\lim _{s\uparrow t}Y_{s}}
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
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, then Y solves, path-by-path, the differential equation
Yor's formula:[2] for any two 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.