Doléans-Dade exponential

In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale X is defined to be the solution to the stochastic differential equation dY_t = Y_t dX_t with initial condition Y₀ = 1. The concept is named after Catherine Doléans-Dade. It is sometimes denoted by Ɛ(X). In the case where X is differentiable, then Y is given by the differential equation dY/dt = Y dX/dt to which the solution is Y = exp(X − X₀). Alternatively, if X_t = σB_t + μt for a Brownian motion B, then the Doléans-Dade exponential is a geometric Brownian motion. For any continuous semimartingale X, applying Itō's lemma with ƒ(Y) = log(Y) gives

${\displaystyle {\begin{aligned}d\log(Y)&={\frac {1}{Y}}\,dY-{\frac {1}{2Y^{2}}}\,d[Y]\\&=dX-{\frac {1}{2}}\,d[X].\end{aligned}}}$

Exponentiating gives the solution

${\displaystyle Y_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}{\Bigr )},\qquad t\geq 0.}$

This differs from what might be expected by comparison with the case where X is differentiable due to the existence of the quadratic variation term [X] in the solution.

The Doléans-Dade exponential is useful in the case when X is a local martingale. Then, Ɛ(X) will also be a local martingale whereas the normal exponential exp(X) is not. This is used in the Girsanov theorem. Criteria for a continuous local martingale X to ensure that its stochastic exponential Ɛ(X) is actually a martingale are given by Kazamaki's condition, Novikov's condition, and Beneš's 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

${\displaystyle Y_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}{\Bigr )}\prod _{s\leq t}(1+\Delta X_{s})\exp {\Bigl (}-\Delta X_{s}+{\frac {1}{2}}\Delta X_{s}^{2}{\Bigr )},\qquad t\geq 0,}$

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