AdvancedMultiple choice

Under $\mathbb{P}$ , $W$ is standard Brownian motion. Define $\mathbb{Q}$ by the density $Z_T = \exp(-\lambda W_T - \tfrac{1}{2}\lambda^2 T)$ . What is $W$ under $\mathbb{Q}$ ?

AStill a standard Brownian motion, since changing measure never changes a process
BA Brownian motion with drift $-\lambda$
CA Brownian motion with volatility scaled by $e^{-\lambda}$
DA process with jumps, because the density is exponential

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Measure change: Girsanov, Feynman–Kac and the fundamental theorems

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Under P\mathbb{P}P, WWW is standard Brownian motion. Define Q\mathbb{Q}Q by the density ZT=exp⁡(−λWT−12λ2T)Z_T = \exp(-\lambda W_T - \tfrac{1}{2}\lambda^2 T)ZT​=exp(−λWT​−21​λ2T). What is WWW under Q\mathbb{Q}Q?

More stochastic calculus questions

Under $\mathbb{P}$ , $W$ is standard Brownian motion. Define $\mathbb{Q}$ by the density $Z_T = \exp(-\lambda W_T - \tfrac{1}{2}\lambda^2 T)$ . What is $W$ under $\mathbb{Q}$ ?