1 Girsanov’s Theorem
References: UChicago Note
Girsanov’s theorem is a fundamental result in probability theory giving a way of changing between different probability measures defined on the same measurable space.
Consider standard Brownian motion (BM) with respect to \(\mathbb{P}\) \(W_{t}^\mathbb{P}\) defined on filtered space \((\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})\). Let \(\theta_{t}\) be an adapted process satisfying the Novikov condition \[ \mathbb{E}^{\mathbb{P}}\left[\exp\left(\frac{1}{2}\int_{0}^T\theta_{s}^2 ds\right)\right]<\infty. \] We define a process \[ Z_{t}:=\exp\left(-\int_{0}^t\theta_{s}dW_{s}^{\mathbb{P}}-\frac{1}{2}\int_{0}^{t}{\theta_{s}^2}~d{s}\right). \] This process \((Z_{t})\) is a martingale with respect to \(\mathbb{P}\) and we can define a new probability measure \(\mathbb{Q}\) on \(\mathcal{F}_{T}\) by \[ \frac{d\mathbb{Q}}{d\mathbb{P}}=Z_{T}. \]