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 $${\displaystyle dY_{t}=Y_{t-}\,dX_{t},\quad \quad Y_{0}=1,}$$where ${\displaystyle Y_{-}}$ denotes the process of left limits, i.e., ${\displaystyle Y_{t-}=\lim _{s\uparrow t}Y_{s}}$.

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 ${\displaystyle X}$ measures the cumulative percentage change in ${\displaystyle Y}$.

Notation and terminology

Process ${\displaystyle Y}$ obtained above is commonly denoted by ${\displaystyle {\mathcal {E}}(X)}$. The terminology "stochastic exponential" arises from the similarity of ${\displaystyle {\mathcal {E}}(X)=Y}$ to the natural exponential of ${\displaystyle X}$: If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation ${\displaystyle dY_{t}/\mathrm {d} t=Y_{t}dX_{t}/dt}$, whose solution is ${\displaystyle Y=\exp(X-X_{0})}$.

General formula and special cases

• Without any assumptions on the semimartingale ${\displaystyle X}$, one has $${\displaystyle {\mathcal {E}}(X)_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}^{c}{\Bigr )}\prod _{s\leq t}(1+\Delta X_{s})\exp(-\Delta X_{s}),\qquad t\geq 0,}$$where ${\displaystyle [X]^{c}}$ is the continuous part of quadratic variation of ${\displaystyle X}$ and the product extends over the (countably many) jumps of X up to time t.
• If ${\displaystyle X}$ is continuous, then $${\displaystyle {\mathcal {E}}(X)=\exp {\Bigl (}X-X_{0}-{\frac {1}{2}}[X]{\Bigr )}.}$$In particular, if ${\displaystyle X}$ is a Brownian motion, then the Doléans-Dade exponential is a geometric Brownian motion.
• If ${\displaystyle X}$ is continuous and of finite variation, then $${\displaystyle {\mathcal {E}}(X)=\exp(X-X_{0}).}$$Here ${\displaystyle X}$ need not be differentiable with respect to time; for example, ${\displaystyle X}$ 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 ${\displaystyle {\mathcal {E}}(X)}$ has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when ${\displaystyle \Delta X=-1}$.
• Unlike the natural exponential ${\displaystyle \exp(X_{t})}$, which depends only of the value of ${\displaystyle X}$ at time ${\displaystyle t}$, the stochastic exponential ${\displaystyle {\mathcal {E}}(X)_{t}}$ depends not only on ${\displaystyle X_{t}}$ but on the whole history of ${\displaystyle X}$ in the time interval ${\displaystyle [0,t]}$. For this reason one must write ${\displaystyle {\mathcal {E}}(X)_{t}}$ and not ${\displaystyle {\mathcal {E}}(X_{t})}$.
• 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 ${\displaystyle X}$. This has application in the theory of conformal martingales and in the calculation of characteristic functions.

Useful identities

Yor's formula:^[2] for any two semimartingales ${\displaystyle U}$ and ${\displaystyle V}$ one has $${\displaystyle {\mathcal {E}}(U){\mathcal {E}}(V)={\mathcal {E}}(U+V+[U,V])}$$

Applications

• Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential ${\displaystyle {\mathcal {E}}(X)}$ of a continuous local martingale ${\displaystyle X}$ 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 ${\displaystyle Y}$ is continuous and strictly positive. Then applying Itō's formula with ƒ(Y) = log(Y) gives

${\displaystyle {\begin{aligned}\log(Y_{t})-\log(Y_{0})&=\int _{0}^{t}{\frac {1}{Y_{u}}}\,dY_{u}-\int _{0}^{t}{\frac {1}{2Y_{u}^{2}}}\,d[Y]_{u}=X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}.\end{aligned}}}$

Exponentiating with ${\displaystyle Y_{0}=1}$ 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 has finite variation due to the existence of the quadratic variation term [X] in the solution.

See also

• Stochastic logarithm

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] (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