Doléans-Dade exponential

In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation where denotes the process of left limits, i.e., .

The concept is named after Catherine Doléans-Dade.[1] Stochastic exponential plays an important role in the formulation of Girsanov's theorem and arises naturally in all applications where relative changes are important since measures the cumulative percentage change in .

Notation and terminology

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Process   obtained above is commonly denoted by  . The terminology "stochastic exponential" arises from the similarity of   to the natural exponential of  : If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation  , whose solution is  .

General formula and special cases

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  • Without any assumptions on the semimartingale  , one has  where   is the continuous part of quadratic variation of   and the product extends over the (countably many) jumps of X up to time t.
  • If   is continuous, then  In particular, if   is a Brownian motion, then the Doléans-Dade exponential is a geometric Brownian motion.
  • If   is continuous and of finite variation, then  Here   need not be differentiable with respect to time; for example,   can be the Cantor function.

Properties

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  • Stochastic exponential cannot go to zero continuously, it can only jump to zero. Hence, the stochastic exponential of a continuous semimartingale is always strictly positive.
  • Once   has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when  .
  • Unlike the natural exponential  , which depends only of the value of   at time  , the stochastic exponential   depends not only on   but on the whole history of   in the time interval  . For this reason one must write   and not  .
  • Natural exponential of a semimartingale can always be written as a stochastic exponential of another semimartingale but not the other way around.
  • Stochastic exponential of a local martingale is again a local martingale.
  • All the formulae and properties above apply also to stochastic exponential of a complex-valued  . This has application in the theory of conformal martingales and in the calculation of characteristic functions.

Useful identities

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Yor's formula:[2] for any two semimartingales   and   one has  

Applications

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Derivation of the explicit formula for continuous semimartingales

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For any continuous semimartingale X, take for granted that   is continuous and strictly positive. Then applying Itō's formula with ƒ(Y) = log(Y) gives

 

Exponentiating with   gives the solution

 

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

See also

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References

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  1. ^ Doléans-Dade, C. (1970). "Quelques applications de la formule de changement de variables pour les semimartingales". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete [Probability Theory and Related Fields] (in French). 16 (3): 181–194. doi:10.1007/BF00534595. ISSN 0044-3719. S2CID 118181229.
  2. ^ Yor, Marc (1976), "Sur les integrales stochastiques optionnelles et une suite remarquable de formules exponentielles", Séminaire de Probabilités X Université de Strasbourg, Lecture Notes in Mathematics, vol. 511, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 481–500, doi:10.1007/bfb0101123, ISBN 978-3-540-07681-0, S2CID 118228097, retrieved 2021-12-14
  • Jacod, J.; Shiryaev, A. N. (2003), Limit Theorems for Stochastic Processes (2nd ed.), Springer, pp. 58–61, ISBN 3-540-43932-3
  • Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 3-540-00313-4