“Doléans-Dade exponential”的意思、由来-开放百科全书

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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 {{nowrap|dY_t {{=}} Y_t dX_t}} with initial condition {{nowrap|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 {{nowrap|dY/dt {{=}} Y dX/dt}} to which the solution is {{nowrap|Y {{=}} exp(X − X₀)}}.

Alternatively, if {{nowrap|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 {{nowrap|ƒ(Y) {{=}} log(Y)}} 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 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

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

References

{{Citation |last=Protter |first=Philip E. |year=2004 |title=Stochastic Integration and Differential Equations |publisher=Springer |edition=2nd |isbn=3-540-00313-4 }}

2 : Martingale theory|Stochastic differential equations

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