Non-arbitrage theory and existence of a risk premium

Question

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- Brownian Motion (with $\mathcal F_T = \mathcal F$).

Also, consider a finantial market where the interest rate is nul, $r=0$, and the dynamics of the risky asset $S$ is given by $$S_t= S_0 + \int_0^t \mu_s ~ds +\int_0^t \sigma_s ~dW_s \quad , t \geq 0$$

where $t \in [0,T] \mapsto \mu_t$ and $t \in [0,T] \mapsto \sigma_t \geq 0$ are deterministic and continuous functions.

Show that:

1. If the absence of arbitrage opportunity hypothesis is verified, then $B:=\{t \in [0,T] : \sigma_t=0 \ \text{and} \ \mu_t \neq 0\}$ is a Lebesgue nul-measure set (ie, $\int_0^T\mathbf1_{t \in B} dt=0$).
2. $\nu_\sigma(O):= \int_0^T\mathbf1_{t \in O}\sigma_t ~dt$ dominates $\nu_\mu (O):= \int_0^T\mathbf1_{t \in O} \mu_t ~dt$, where $O$ is a borelian of $[0,T]$ ( ie, $\nu_\mu \ll \nu_\sigma$) and deduce from it that there is a measurable function $\lambda$ such that $\mu = \sigma \lambda$.

Answer 1

For the first one absurd reasoning allows you to construct an arbitrage (as r=0) by investing (or short selling according to the sign of $\mu$) at the time where $\sigma$ is null, or if you prefer as soon as $t$ is in $B$ (which is not a Lebesgue negligible set by hypothesis) which is absurd as no-arbitrage holds. The details that remain to be proved is that such a strategy is an admissible one.

For the second question, as soon as the first part is done, it is only the application of Lebesgue's decomposition theorem.

For the first part as any borelian set such that $\nu_\sigma(O):= \int_0^T\mathbf1_{t \in O}\sigma_t dt $ doesn't dominate $\nu_\mu (O):= \int_0^T\mathbf1_{t \in O} \mu_t dt$ is included in $B$ and as $B$ is of null probability the conclusion holds true.

Best regards