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

$$dY_{t}=Y_{t-}\,dX_{t},\quad \quad Y_{0}=1,$$

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

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

General formula and special cases

• Without any assumptions on the semimartingale $X$, one has
  $${\mathcal {E}}(X)_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}^{c}{\Bigr )}\prod _{s\leq t}(1+\Delta X_{s})\exp(-\Delta X_{s}),\qquad t\geq 0,$$
  where $[X]^{c}$ 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
  $${\mathcal {E}}(X)=\exp {\Bigl (}X-X_{0}-{\frac {1}{2}}[X]{\Bigr )}.$$
  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
  $${\mathcal {E}}(X)=\exp(X-X_{0}).$$
  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 ${\mathcal {E}}(X)$ has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when $\Delta X=-1$.
• Unlike the natural exponential $\exp(X_{t})$, which depends only of the value of $X$ at time $t$, the stochastic exponential ${\mathcal {E}}(X)_{t}$ 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 ${\mathcal {E}}(X)_{t}$ and not ${\mathcal {E}}(X_{t})$.
• 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 $X$. 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

$${\mathcal {E}}(U){\mathcal {E}}(V)={\mathcal {E}}(U+V+[U,V])$$

Applications

• Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential ${\mathcal {E}}(X)$ 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 ƒ(Y) = log(Y) gives

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

Exponentiating with $Y_{0}=1$ gives the solution

$Y_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}{\Bigr )},\qquad t\geq 0.$

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

• 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