State Novikov's condition. For theta_s = W_s, determine the largest T for which the stochastic exponential of ∫ theta dW is guaranteed a true martingale on [0,T] by Novikov, and say what goes wrong when the exponential is only a supermartingale.

State Novikov's condition. For theta_s = W_s, determine the largest T for which the stochastic exponential of ∫ theta dW is guaranteed a true martingale on [0,T] by Novikov, and say what goes wrong when the exponential is only a supermartingale.

Approach: Write Novikov's integrability requirement for the given integrand, then use the known Laplace transform of the time integral of a squared Brownian path to locate the horizon where it first fails.

Novikov's condition is E[exp((1/2) ∫_0^T theta_s^2 ds)] < ∞, and for theta_s = W_s that expectation equals (cos T)^{-1/2}, which is finite exactly for T < pi/2, so pi/2 is the horizon at which the criterion fails. The quantity Z_T = exp(∫_0^T theta dW - (1/2)∫_0^T theta^2 ds) is always a positive local martingale, hence a supermartingale by Fatou, so E[Z_T] ≤ 1 with equality precisely when Z is a true martingale. If the inequality is strict, the candidate change of measure dQ = Z_T dP assigns total mass below 1, so Q is not a probability measure and Girsanov's conclusion is unavailable. The Laplace transform E[exp(-(lambda^2/2) ∫_0^T W_s^2 ds)] = (cosh(lambda T))^{-1/2} continues analytically to the required positive exponent and produces the cosine, which explains why the failure is at a finite horizon rather than for all T.

Follow-up: Which weaker condition than Novikov still guarantees the true martingale property, and what does it give on this example?

Key concepts: Novikov condition, local martingale, supermartingale, change of measure.