# Incomplete market

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.

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.