Novikov's condition
/ noh-VEE-kov /
Novikov's condition is the standard, easy-to-check sufficient condition guaranteeing that a stochastic exponential is a TRUE martingale (not merely a local martingale). It is the safety check you run before applying Girsanov's theorem: the change of measure dQ/dP = E(integral H dB) is legitimate only if that exponential has expectation 1, i.e. is a genuine martingale, and Novikov's condition is the most commonly invoked guarantee of exactly that.
The condition states: if H is a predictable process and E[ exp( (1/2) integral_0^T H_s^2 ds ) ] < infinity, then the stochastic exponential Z_t = E(integral H dB)_t = exp( integral_0^t H_s dB_s - (1/2) integral_0^t H_s^2 ds ) is a true (uniformly integrable) martingale on [0, T], so in particular E[Z_T] = 1 = E[Z_0]. The intuition is that the dangerous behaviour of a stochastic exponential — the route by which a nonnegative local martingale can leak mass and become a strict supermartingale with E[Z_t] < 1 — is driven by the exponential of the accumulated quadratic variation integral H^2 ds; Novikov controls precisely the exponential half-moment of that quantity, which is enough to rule out the leakage. The proof bounds the exponential supermartingale from above using the finite half-exponential moment, yielding uniform integrability and hence the martingale property. A localized version applies on [0, T] piecewise, and a weaker companion (Kazamaki's condition, E[exp((1/2) integral H dB)] < infinity on the martingale itself) sometimes works when Novikov's does not.
Where it bites. Novikov is the workhorse hypothesis in mathematical finance (verifying that a proposed risk-neutral density is a true density), in filtering, and in any application of Girsanov. The honest caveats are important: (1) Novikov's condition is SUFFICIENT, not necessary — many stochastic exponentials are true martingales while violating it; failing Novikov does not prove E(integral H dB) is a strict local martingale, it just means this particular test is inconclusive. (2) The condition is global in the sense that the half-exponential moment of the WHOLE accumulated integral H^2 must be finite; for unbounded H or long horizons this can fail, which is the genuine technical obstruction in, e.g., quadratic or exponentially-growing drifts. (3) Without a guarantee like Novikov, the candidate density may satisfy E[Z_T] < 1, the 'probability measure' Q has total mass below 1, and Girsanov's conclusion is simply false — this is not a pedantic worry but a real failure mode in models with explosive drift.
If H is bounded, say |H_s| <= K, then integral_0^T H^2 ds <= K^2 T is bounded, so E[exp((1/2) integral H^2 ds)] <= exp(K^2 T / 2) < infinity and Novikov holds trivially — bounded market prices of risk always give a legitimate Girsanov change of measure. The classic failure case is a drift that grows fast enough that integral_0^T H^2 ds has infinite exponential half-moment (e.g. certain Bessel-type or quadratically-growing H over a long horizon), where Novikov is inconclusive and the exponential can be a strict local martingale.
Bounded H makes Novikov automatic; fast-growing H can fail it, and then the exponential may not be a true martingale.
Novikov's condition is sufficient but not necessary: violating it does not prove E(integral H dB) is a strict local martingale, only that this test fails — try Kazamaki's condition or a direct argument. Always verify the true-martingale property before invoking Girsanov.