Doléans-Dade exponential explained

In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation $$dY\_t\; =\; Y\_\backslash ,dX\_t,\backslash quad\backslash quad\; Y\_0=1,$$where

Y _{-} denotes the process of left limits, i.e., Y _{t-} = \lim _{s\uparrow} Y _{s} .

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

X measures the cumulative percentage change in Y .

Notation and terminology

Process

Y obtained above is commonly denoted by l{E}(X) . The terminology "stochastic exponential" arises from the similarity of l{E}(X)=Y to the natural exponential of X : If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation dY _{t/dt} = Y _{tdX} _{t/dt} , whose solution is Y = \exp(X - X _{0)} .

General formula and special cases

• Without any assumptions on the semimartingale

X , one has $$\backslash mathcal(X)\_t\; =\; \backslash exp\backslash Bigl(X\_t-X\_0-\backslash frac12[X]^c\_t\backslash Bigr)\backslash prod\_(1+\backslash Delta\; X\_s)\; \backslash exp\; (-\backslash Delta\; X\_s),\backslash qquad\; t\backslash ge0,$$where [X] ^{c} is the continuous part of quadratic variation of X and the product extends over the (countably many) jumps of X up to time t.

• If

X is continuous, then $$\backslash mathcal(X)\; =\; \backslash exp\backslash Bigl(X-X\_0-\backslash frac12[X]\backslash Bigr).$$In particular, if X is a Brownian motion, then the Doléans-Dade exponential is a geometric Brownian motion.

• If

X is continuous and of finite variation, then $$\backslash mathcal(X)=\backslash exp(X-X\_0).$$Here X need not be differentiable with respect to time; for example, X can be the Cantor function.

Properties

• 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

l{E}(X) has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when \Delta X =-1 .

• Unlike the natural exponential

\exp(X _{t)} , which depends only of the value of X at time t , the stochastic exponential l{E}(X) _{t} depends not only on X _{t} but on the whole history of X in the time interval [ 0 ,t] . For this reason one must write l{E}(X) _{t} and not l{E}(X _{t)} .

• 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

X . This has application in the theory of conformal martingales and in the calculation of characteristic functions.

Useful identities

Yor's formula: for any two semimartingales

U and V one has $$\backslash mathcal(U)\backslash mathcal(V)\; =\; \backslash mathcal(U+V+[U,V])$$

Applications

• Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential

l{E}(X) of a continuous local martingale X is a martingale are given by Kazamaki's condition, Novikov's condition, and Beneš's condition.

Derivation of the explicit formula for continuous semimartingales

For any continuous semimartingale X, take for granted that

Y is continuous and strictly positive. Then applying Itō's formula with gives\begin{align} log (Y _{t)-log(Y} _{0)} & = \int ^{t (1)/(Y _{u})} _{0} dY _{u} - \int ^{t (1)/(2 Y ^{2} _{u})} _{0} d[Y] _{u =} X _{t-X} _{0} - (1)/(2) [X] _{t. \end{align}}

Exponentiating with

Y _{0=1} gives the solutionY _{t} = \exp l (X _{t-X} ^{0- (12 [X])/(_{tr),})} t\ge 0 .

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

• Stochastic logarithm

Notes and References

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. fr. 16. 3. 181–194. 10.1007/BF00534595. 118181229 . 0044-3719. free.