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Difficulty: Hard
Category: Stochastic Calculus
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Topics: stochastic-calculus, martingale, novikov-condition, brownian-motion
Consider a stochastic process defined by the stochastic exponential $ \mathcal{E}(\theta \cdot W)_t = \exp(\theta W_t - \frac{1}{2} \int_0^t \theta_s^2 ds) $, where $W_t$ is a standard Brownian motion and $ \theta_t $ is a stochastic process. Under what condition is $ \mathcal{E}(\theta \cdot W)_t $ a true martingale?
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