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How to prove that market with one risky asset $S_t$ and interest rate $r = 0$ is incomplete: $$dS_t = S_t (\mu dt + \sigma_t dW_t^{1}), \quad S_0 = 1,$$ $$\sigma_t = 1 + |W_t^{2}|,$$

$W_t^{1}$ and $W_t^{2}$ are independent Wiener processes.

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I edit my answer due to a probable misunderstanding. Since you reference an interest rate, I must assume you mean that you also have access to a risk-free asset $B_t$. Then we have $B_t = 1$ for all $t$, since the interest rate is zero.

The model can be shown to be free of arbitrage, using the first fundamental theorem of asset pricing, as follows. Let $\varphi^1_t = -\frac{\mu}{\sigma_t}$, $\varphi^2$ an arbitrary constant, and let $\mathbb{Q}$ be the measure given by taking the Girsanov kernel $\varphi_t = (\varphi^1_t, \varphi^2)$, i.e., $$ \left.\frac{d\mathbb{Q}}{d\mathbb{P}}\right\vert_{\mathcal{F}_t} = \exp\left\{\int_0^t \varphi_s dW_s - \frac{1}{2}\int_0^t||\varphi_t||^2ds\right\}. $$ Since $|\varphi^1_t| = \frac{|\mu|}{1 + |W^2_t|} \leq |\mu|$ and $\varphi^2$ is constant, we get that $\varphi_t$ satisfies the Novikov condition. Thus $\left.\frac{d\mathbb{Q}}{d\mathbb{P}}\right\vert_{\mathcal{F}_t}$ is a $\mathbb{P}$-martingale and the change of measure is valid.

By Girsanov's theorem, we have that $$ W^{1, \mathbb{Q}}_t := W^1_t - \int_0^t \varphi^1_s ds $$ is a Brownian motion under $\mathbb{Q}$. The stock dynamics under $\mathbb{Q}$ then become $$ \begin{split} dB_t &= 0 \\ dS_t &= (\mu + \sigma_t \varphi_t)dt + \sigma_t dW^{1, \mathbb{Q}}_t = \sigma_t dW^{1, \mathbb{Q}}_t, \end{split} $$ and the $B_t$-normalized versions of course the same. Thus $\mathbb{Q}$ is an equivalent martingale measure and by the first fundamental theorem of asset pricing the model is free of arbitrage. But $\varphi^2$ can be chosen arbitrarily in $\mathbb{R}$, yielding different equivalent martingale measures $\mathbb{Q}$. Hence, the martingale-measure is not unique. By the second fundamental theorem of asset pricing, the market in incomplete.

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