The law of one price (i.e. for assets $S^{(i)}$ and $S^{(j)}$, $S^{(i)}_T = S^{(j)}_T $ almost surely implies that $S^{(i)}_t = S^{(j)}_t $ almost surely for all $ 0 \leq t \leq T$) is known to hold in discrete time when there is no arbitrage.
However, my lecturer claims that this statement might not hold in continuous time. Can anyone give me an example for that?
Let $B_t=e^{\int_0^t r_s} ds$ be the money-market account value at time $t$, where $r_t$ is the short interest rate rate. Then both $\{\frac{S_t^i}{B_t}, \, t \ge 0\}$ and $\{\frac{S_t^j}{B_t}, \, t \ge 0\}$ are martingales. Therefore, for $0 \le t \le T$, \begin{align*} \frac{S_t^i}{B_t} &= E\left(\frac{S_T^i}{B_T} \mid \mathcal{F}_t \right)\\ &= E\left(\frac{S_T^j}{B_T} \mid \mathcal{F}_t \right)\\ &= \frac{S_t^j}{B_t}. \end{align*} That is, $S_t^i = S_t^j$.
