the stochastic exponential (Doleans-Dade)
/ doh-lay-AHN dahd (Doléans-Dade) /
The stochastic exponential is the right analogue of e^x in the world of semimartingales — the process that solves the multiplicative equation dZ = Z dX, the stochastic version of 'the exponential is its own derivative'. It is the central object behind change-of-measure (Girsanov), the geometric Brownian models of finance, and the construction of likelihood ratios for diffusions. Because of the Ito correction, it is NOT simply e^X; the formula carries a quadratic-variation term that is the whole point.
For a continuous semimartingale X with X_0 = 0, the stochastic exponential E(X), also written Exp(X), is the unique solution of the linear SDE dZ_t = Z_t dX_t, Z_0 = 1. Solving it by Ito's formula gives the closed form E(X)_t = exp(X_t - (1/2) [X]_t), where [X] is the quadratic variation of X. The -(1/2)[X] term is exactly the Ito correction that makes dZ = Z dX hold (apply Ito to exp(X - [X]/2): the second-order term -(1/2) d[X] cancels the (1/2) d[X] from differentiating the exponential). The key property follows immediately: if X is a continuous local martingale, then E(X) is also a continuous local martingale (because dZ = Z dX has no drift). For X = integral H dB this reads E(integral H dB)_t = exp(integral_0^t H dB - (1/2) integral_0^t H^2 ds). For semimartingales with jumps there is a more elaborate product formula (the cadlag Doleans-Dade exponential includes a product over jumps), but the continuous case is the workhorse.
Its role is to convert additive martingales into multiplicative ones and to define new measures. In Girsanov's theorem the Radon-Nikodym density dQ/dP is precisely a stochastic exponential E(integral H dB), and under Q the process B - integral H ds becomes a Brownian motion. The honest and crucial caveat: E(X) is always a local martingale when X is, but it need NOT be a true martingale — and whether it is is exactly the technical heart of Girsanov's theorem. A local martingale that is also nonnegative (which E(X) is, being an exponential) is automatically a supermartingale, hence E[E(X)_t] <= 1, with EQUALITY (true martingale) only under additional conditions. This is why one needs sufficient conditions like Novikov's E[exp((1/2) integral_0^T H^2 ds)] < infinity or Kazamaki's to guarantee E(integral H dB) is a genuine martingale; without them the candidate density may have total mass strictly less than 1 and the change of measure fails.
Take X_t = sigma B_t, a continuous martingale with [X]_t = sigma^2 t. Its stochastic exponential is E(X)_t = exp(sigma B_t - (1/2) sigma^2 t), which satisfies dZ = sigma Z dB and is the classic exponential (geometric Brownian) martingale with E[Z_t] = 1 — here Novikov's condition holds trivially for constant sigma. By contrast, taking H_s = (something growing fast enough) can make E(integral H dB) a strict local martingale with E < 1, which is exactly the situation Novikov's condition is designed to exclude.
E(X) = exp(X - [X]/2) solves dZ = Z dX; it is always a local martingale but a true martingale only under conditions like Novikov's.
Do not confuse E(X) with e^X: the stochastic exponential is exp(X - [X]/2), and the -[X]/2 is essential. As a nonnegative local martingale E(X) is always a supermartingale (E[E(X)_t] <= 1); equality, i.e. the true-martingale property needed for Girsanov, requires extra conditions such as Novikov's.