In stochastic calculus

Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes...

, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale

Semimartingale

In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

 X is defined to be the solution to the stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

  with initial condition . The concept is named after Catherine Doléans-Dade. It is sometimes denoted by (X).
In the case where X is differentiable, then Y is given by the differential equation to which the solution is .
Alternatively, if for a Brownian motion

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

 B, then the Doléans-Dade exponential is a geometric Brownian motion

Geometric Brownian motion

A geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, also called a Wiener process...

. For any continuous semimartingale X, applying Itō's lemma

Ito's lemma

In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...

 with gives


Exponentiating gives the solution


This differs from what might be expected by comparison with the case where X is differentiable due to the existence of the quadratic variation

Quadratic variation

In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and martingales. Quadratic variation is just one kind of variation of a process.- Definition :...

 term [X] in the solution.

The Doléans-Dade exponential is useful in the case when X is a local martingale

Local martingale

In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; however, in general a local martingale is not a martingale, because its...

. Then, (X) will also be a local martingale whereas the normal exponential exp(X) is not. This is used in the Girsanov theorem

Girsanov theorem

In probability theory, the Girsanov theorem describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure...

. Criteria for a continuous local martingale X to ensure that its stochastic exponential (X) is actually a (uniformly integrable) martingale

Martingale (probability theory)

In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

 are given by Kazamaki's condition

Kazamaki's condition

Kazamaki's condition gives a sufficient criterion ensuring that the Doléans-Dade exponential of a local martingale is a true martingale. In mathematics, this is particularly important if one wishes to apply Girsanov's theorem in order to perform a change of measure...

, Novikov's condition

Novikov's condition

In probability theory, Novikov's condition is the sufficient condition for a stochastic process which takes the form of the Radon-Nikodym derivative in Girsanov's theorem to be a martingale...

 and Beneš' condition.

It is possible to apply Itō's lemma for non-continuous semimartingales in a similar way to show that the Doléans-Dade exponential of any semimartingale X is


where the product extents over the (countable many) jumps of X up to time t.