The Doléans‑Dade exponential, also called the stochastic exponential, is a fundamental construction in stochastic analysis that associates to a semimartingale $X$ a non‑negative process $\mathcal{E}(X)$ solving the stochastic differential equation
$$ d\mathcal{E}(X)t = \mathcal{E}(X){t-},dX_t,\qquad \mathcal{E}(X)_0 = 1, $$
where the integral is interpreted in the Itô sense for continuous martingales and in the general semimartingale setting via the stochastic integral with respect to $X$. For a continuous semimartingale $X$ the Doléans‑Dade exponential admits the explicit representation
$$ \mathcal{E}(X)t = \exp!\Bigl( X_t - \tfrac{1}{2}\langle X^c\rangle_t \Bigr) \prod{0<s\le t}(1+\Delta X_s),\exp!\bigl(-\Delta X_s\bigr), $$
where $X^c$ denotes the continuous martingale part of $X$, $\langle X^c\rangle$ its quadratic variation, and $\Delta X_s = X_s - X_{s-}$ the jump at time $s$. In the purely continuous case (e.g., when $X$ is a Brownian motion with drift), the product term is absent and the formula reduces to
$$ \mathcal{E}(X)_t = \exp!\bigl( X_t - \tfrac{1}{2}[X]_t \bigr), $$
with $[X]$ the quadratic variation of $X$.
Main properties
- Positivity: $\mathcal{E}(X)_t > 0$ almost surely for all $t\ge 0$.
- Multiplicative structure: For semimartingales $X$ and $Y$,
$$ \mathcal{E}(X+Y) = \mathcal{E}(X),\mathcal{E}\bigl(Y + [X,Y]\bigr), $$
where $[X,Y]$ denotes the covariation process.
- Change of measure: If $M$ is a local martingale with $\mathcal{E}(M)$ a true martingale, then the probability measure $\mathbb{Q}$ defined by
$$ \frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_t}= \mathcal{E}(M)_t $$
is equivalent to the original measure $\mathbb{P}$ on $\mathcal{F}_t$. This is the basis of Girsanov’s theorem.
- Solution of linear SDEs: For a semimartingale $X$ and a predictable process $\theta$,
$$ dY_t = Y_{t-},\theta_t,dX_t \quad\Longleftrightarrow\quad Y_t = Y_0,\mathcal{E}!\bigl(\int_0^\cdot \theta_s,dX_s\bigr)_t . $$
Thus $\mathcal{E}$ provides the explicit solution to linear stochastic differential equations.
Historical notes
The construction is named after the French mathematicians Claude Doléans‑Dade (1935–2005) and Paul‑André Meyer, who developed the theory of stochastic integration for semimartingales in the 1970s. The exponential was introduced to generalize the classical Itô formula for exponentials of Brownian motion to the broader class of semimartingales.
Applications
- Financial mathematics: Used to model the dynamics of asset prices under risk‑neutral measures and to derive pricing formulas for derivatives.
- Stochastic control: Appears in the formulation of the stochastic Hamilton–Jacobi–Bellman equation and in the construction of optimal controls via change‑of‑measure techniques.
- Filtering theory: Provides the likelihood ratio process in nonlinear filtering problems.
- Mathematical biology: Employed in stochastic population models where birth–death processes are represented via jump semimartingales.
References
- Doléans‑Dade, C. (1970). * "Exposants stochastiques"*, Seminar on Probability, Lecture Notes in Mathematics, vol. 97, Springer.
- Protter, P. (2005). Stochastic Integration and Differential Equations, 2nd ed., Springer.
- Revuz, D., & Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed., Springer.
The Doléans‑Dade exponential remains a central tool in modern probability theory and its applications.