In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation d Y t = Y t − d X t , Y 0 = 1 , {\displaystyle dY_{t}=Y_{t-}\,dX_{t},\quad \quad Y_{0}=1,} where Y − {\displaystyle Y_{-}} denotes the process of left limits, i.e., Y t − = lim s ↑ t Y s {\displaystyle Y_{t-}=\lim _{s\uparrow t}Y_{s}} . The concept is named after Catherine Doléans-Dade.
In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation d Y t = Y t − d X t , Y 0 = 1 , {\displaystyle dY_{t}=Y_{t-}\,dX_{t},\quad \quad Y_{0}=1,} where Y − {\displaystyle Y_{-}} denotes the process of left limits, i.e., Y t − = lim s ↑ t Y s {\displaystyle Y_{t-}=\lim _{s\uparrow t}Y_{s}} . The concept is named after Catherine Doléans-Dade.
Source: Wikipedia "Doléans-Dade exponential" · CC BY-SA 4.0
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