# Non-arbitrage theory and existence of a risk premium

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 risquy asset $S$ is given by $$S_t= S_0 + \int_0^t \mu_s ~ds +\int_0^t \sigma_s ~ds \quad , t \geq 0$$

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

Show that:

1. If the absence of arbitrage oportunity 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. $\nu_\sigma(O):= \int_0^T\mathbf1_{t \in B}\sigma_t ~dt$ dominates $\nu_\mu (O):= \int_0^T\mathbf1_{t \in B} \mu_t ~dt$ ( ie, $\nu_\mu \ll \nu_\sigma$) and deduce from it that there is a measurable fonction $\lambda$ such that $\mu = \sigma \lambda$.
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can you please add your definition of absence of arbitrage hypothesis? Shouldn't be the second integral a stochastic one, i.e. $\int_0^t\sigma_sdW_s$ for the $1-$d Brownian Motion $W$? –  user8 Apr 9 '13 at 6:49
@Paul : double post with this post where I made an answer : quant.stackexchange.com/questions/7697/… Best regards –  TheBridge Apr 9 '13 at 11:09