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Difficulty: Hard
Category: Stochastic Calculus
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Topics: stochastic-calculus, girsanov, change-of-measure, brownian-motion, risk-neutral
Suppose you are modeling the price process $X_t$ of an asset under the real-world probability measure $P$. You have determined that $X_t$ follows the stochastic differential equation: $dX_t = \mu dt + \sigma dW_t$, where $W_t$ is a standard Brownian motion under $P$, $\mu$ is the drift, and $\sigma$ is the volatility. Now, you want to price a derivative on $X_t$ using a risk-neutral measure $Q$ equivalent to $P$. Under this new measure $Q$, what stochastic process does $X_t$ follow?
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