Brownian Bridge™ logoBrownian Bridge™
L2

Stochastic Exponential & Novikov

Senior Quant · Stochastic Calculus

Question

Define the stochastic exponential E(X)t=exp⁡(Xt−12⟨X⟩t)\mathcal{E}(X)_t = \exp(X_t - \tfrac{1}{2}\langle X \rangle_t). Show it satisfies dE=E dXd\mathcal{E} = \mathcal{E}\,dX. State Novikov's condition and explain what it guarantees about E(X)\mathcal{E}(X) as a martingale versus a strict local martingale.
←Stopped Martingale
5 / 10
Lévy's Characterisation→