---
tag: 3D88622PBQ
kind: concept
name: Doléans-Dade exponential
msc: [60H05, 60G44]
sources:
  - url: https://en.wikipedia.org/wiki/Dol%C3%A9ans-Dade_exponential
same_as:
  wikidata: Q5289708
  wikipedia: Doléans-Dade exponential
---

## Summary

The Doléans-Dade exponential, or stochastic exponential, $\mathcal{E}(X)$ of a [[semimartingale|4EJB6E2AZK]] $X$ is the solution of the equation $dZ = Z \, dX$ with $Z_0 = 1$.
It plays the role of the exponential function in [[Itô calculus|JA95NH206E]].
For a continuous semimartingale with $X_0 = 0$ it equals $\exp(X - \tfrac12 [X])$.
It is the density process in the [[Girsanov theorem|Z76HD1BZ29]].

## Explanations

### L1 human

In ordinary calculus, the solution of $dz = z \, dx$ with $z(0) = 1$ is $z = e^{x}$.
For a random driver such as Brownian motion, $e^{X}$ grows a little too fast, because Itô's lemma adds a drift.
The stochastic exponential removes this drift by the factor $e^{-[X]/2}$.
For $X = \sigma B$ with a constant $\sigma$, it is $\exp(\sigma B_t - \tfrac12 \sigma^2 t)$, a positive martingale with mean $1$.