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 $${\displaystyle dY_{t}=Y_{t-}\,dX_{t},\quad \quad Y_{0}=1,}$$where ${\displaystyle Y_{-}}$ 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. The 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 ${\displaystyle X}$ measures the cumulative percentage change in ${\displaystyle Y}$.