Derivation — Girsanov via the Doléans-Dade Exponential

1. Direct : compute the conditional moment generating function of $\tilde{W}_t - \tilde{W}_s$ under $\mathbb{Q}$ and show it matches the MGF of $\mathcal{N}(0, t-s)$.
2. Lévy characterisation (Karatzas method): show $\tilde{W}_t$ is a continuous $\mathbb{Q}$-local martingale with $[\tilde{W}, \tilde{W}]_t = t$, then apply Lévy's characterisation to conclude it is a $\mathbb{Q}$-BM.

The Lévy method is shorter and generalises more easily to the local martingale form. It also shows exactly which properties of $\tilde{W}_t$ need to be checked — quadratic variation and the local martingale property — making it clear why $\sigma$ (the diffusion coefficient) is unchanged.

1. Setup
2. Proof 1: Conditional MGF
3. Proof 2: Lévy Characterisation (Karatzas)
4. Why the Diffusion is Unchanged
5. Bayes Formula Under $\mathbb{Q}$