Consider the Black-Scholes model $$ dS(t) = \mu(t) S(t) dt + \sigma(t) S(t) dW^{\mathbb{P}}(t) $$ $$ dB(t) = r(t) B(t) dt$$ Steele shows now in "Stochastic Calculus & Financial Applications" (Ch. 14) that if $\nu(t)=\frac{\mu(t)-r(t)}{\sigma(t)}$ satisfies the Novikov-condition with respect to $\mathbb{P}$, then there is an equivalent measure $\mathbb{Q}$ such that $\widetilde{S}(t):=S(t)/B(t)$ becomes a $\mathbb{Q}$-local martingale. If $\sigma$ satisfies also the Novikov-condition but wrt $\mathbb{Q}$, then $\widetilde{S}(t)$ is an honest $\mathbb{Q}$-martingale. Additionally, if $\mathcal{F}_{T}=\sigma(\widetilde{S}(t)\,\vert\,t\leq T)$, then $\mathbb{Q}$ is unique. In Theorem 14.2, it is then argued that the model is complete if $$ \mathbb{P}\left[ \int_{0}^{T} \cfrac{\mu^{2}(t)}{\sigma^{2}(t)} dt <\infty \right] =1.$$ Yet, I thought that completeness of the model follows already from the previously established uniqueness of $\mathbb{Q}$ due to the Fundamental Theorem of Asset Pricing. What am I missing?


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