Doléans-Dade exponential

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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 dYt = Yt dXt 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 Xt = σBt + 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

{\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

Y_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac 12}[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' 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

Y_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac 12}[X]_{t}{\Bigr )}\prod _{{s\leq t}}(1+\Delta X_{s})\exp {\Bigl (}-\Delta X_{s}+{\frac 12}\Delta X_{s}^{2}{\Bigr )},\qquad t\geq 0,

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

See also

  • Stochastic logarithm

References

  • Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 3-540-00313-4 
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