Model of economic cycles consistent with efficient market hypothesis I do not have background in economics, so my question may be naive or I might have taken incorrect assumptions somewhere. As far as I understand, efficient market hypothesis applies, saying that the stock prices already contain all the available information, so it is not possible to consistently beat the market without internal information on given stock.

I tried to build a simplistic toy-model of stock prices that have this property and I came with model, where each individual stock is in each time step multiplied by term $e^\varepsilon$, where $\varepsilon$ is randomly selected from uniform distribution centered around zero. 

$$
p(t+dt)=p(t)e^{\varepsilon \,dt}\approx p(t)(1+\varepsilon \,dt)
$$

Now, since every positive $\varepsilon$ has equally probable negative counter-part and since $e^\varepsilon e^{-\varepsilon}=1$, the mean value of the stock price is zero. Unless I overlooked something, it should not be possible to extract profit from stock prices following such a law, because the information about the future price is not contained in the past and it is random. We can center the distribution around positive number to add growth of the overall economy.

Now I wanted to create similar toy-model, where economic cycles are included. First thing that came to my mind is to multiply the generated prices of stocks from the previous model by a periodic function, for example:

$$
p_\mathrm{cycled}(t)=\left(1+A\sin\left(\frac{2\pi t}{T}\right)\right)p(t)
$$

But this is clearly contradicting the efficient market hypothesis, because anyone knowing the period of the function could buy the stock when the prices are low and sell when the prices are high. I thought of possible ways how to make the model consistent with the efficient market hypothesis and I came with these points:


The cycle is not a cycle, but some kind of noise. What makes the strategy of making profit buying low and selling high impossible is, that nobody knows, when the prices will go up and when down.
When buying low, some companies go bankrupt before going up to high period of the cycle. Nobody knows which one though, and it is therefore not possible to make profit on the cycle. 
Something else.


The question: Is there simple toy-model of stock prices that manifests the economic cycles and is consistent with the efficient market hypothesis? If there are more mechanisms at play, I would be very interested in seeing them all embedded as a toy model.

I am suspicious about 1., because either the noise is unbounded (as the exponential noise above), or there is memory in the noise. The memory can be used to extract the necessary information to make profit. If the noise is unbounded, then it is not really similar to how net world economy behaves. 2. seems plausible, but I didn't found any toy-model embedding such behavior yet.

-----------------------------

My takeaway:
From the answers here and after giving it some thought, I came to conclusion that if the EMH is true, there should not be any cycles with known oscillations in the prices, no autocorrelation in the noise to be exploited and the possible cycle up- and down- should not be bounded (otherwise, everyone would know when is the down-phase, even though they would not be able to predict it).

I came with example toy-model satisfying these requirements with three components:


Normal distribution random walk
Exponential growth of several % per year
Constant small probability of sudden crash with factor 0.75-1




In the model, there are "quasi-cycles" between the crashes and there is no exploitable future information. (I think.)
The EMH applies to assets, not just stocks, and its implications are more relevant for investors who own part of the market - not the entire thing.  This is important, because it's the difference between looking at a closed system versus an open one, and between populations versus samples.  People make money all the time by cycling between stocks, bonds, precious metals, real estate, etc.  Flights to "safe havens" occur all the time, rightly and wrongly.  Someone buying SNC Lavalin stock three months ago and selling it a few weeks after the Canadian election would have made some fairly predictable money.

The apparent issue you're running into here stems from the erroneous assumption that non-stock assets are "safe".  In fact, they have their own risks and cycles, which are (anti-)correlated with the stock market in various ways.  The "Risk-free rate" in the Fama-French model isn't constant, and is fundamentally no more predictable than any individual stock price.