False proof: Every continuous-time financial market has arbitrage. We know that many continuous-time financial markets do not have arbitrage. But I have created a proof which shows a very large class of them has arbitrage, so there must be a mistake in the proof. Could you please point out where the mistake(s) are?
Assume that you have a filtered probability space $(\Omega,\mathcal{A},P,\mathcal{F}_t)$. Let $S_t$ be a risky asset on the probability space and let $B_t$ be the bank process, we assume that $B_t$ is positive. We also assume that the filtration saitisfies the usual conditions, that means the $F_0$ contains all the null-events, and the filtration is right-continuous.
Look at the discounted process $\bar{S}_t=S_t/B_t.$ We assume that this process is not constant almost surely. If it was, then $S_t=C\cdot B_t$. This does not a strong assumption as we know there are continuous time markets where the discounted proess is not constant, but they still don't have arbitrage.
Define
$$F_t=\limsup\limits_{n \rightarrow \infty}\frac{\bar{S}_{t+1/n}-\bar{S}_t}{\frac{1}{n}},$$
and also define
$$f_t=\liminf\limits_{n \rightarrow \infty}\frac{\bar{S}_{t+1/n}-\bar{S}_t}{\frac{1}{n}}.$$
We must have that $F_t$ and $f_t$ are $\mathcal{F}_t$-measurable because the filtration is right continuous.
Now we create the arbitrage process like this:
If we observe that $F_t>0$ for a $t$ we know that in the immediate future the discounted price will rise. So we loan $1$ dollar from the bank and invest it in the stock. We will have $-1/B_t$ bank units and $1/S_t$ stocks. Since $F_t$ is positive there is an $y>t$ with $\bar{S}_y>\bar{S}_t$, at this time sell the stock and pay back the bank. In this case we have $S_y/B_y>S_t/B_t\rightarrow  1/S_t \cdot S_y >1/B_t\cdot B_y$. This last equation means that the number of stocks we have times todays stock value is larger than the number of bank units we loaned times todays bank unit. Hence we have gotten free money. Something similar can be done with $f_t$.
Do you see the error in the proof?
 A: "If we observe that $F_t>0$ we know that in the immediate future the discounted price will rise."
This statement is wrong. Here is a counterexample:
Let the process be given by a Wiener process, $\bar{S}_t=W_t$. Now $F_t=\infty$ almost surely (see https://people.math.wisc.edu/~roch/grad-prob/gradprob-notes27.pdf). However, using the definition of a Wiener process $P(W_{t+h}-W_t>0| \mathcal{F}_t)=\frac{1}{2}$ for any $h>0$.
The Wiener process is well-known to be recurrent (https://en.wikipedia.org/wiki/Wiener_process), that is it reaches any point $y$ almost surely. Does this imply an arbitrage opportunity?
No. The definition of arbitrage requires that you find a self-financing strategy that cannot take negative values. If you buy the stock and sell the bond, your strategy has a positive probability of interim losses and hence is not an arbitrage strategy.
Would this still make an good investment strategy even though it is not arbitrage? Not necessarily. If you wait long enough till the process takes a positive value you have to accept that it may take an arbitrarily low value before that. This essentially requires infinitely high wealth.
