How long between green days?

Question

A stock closes either green (up) or red (down) each day, and the next day depends only on today: - If green today, it is red tomorrow with probability 0.6 (and green with probability 0.4). - If red today, it is green tomorrow with probability 0.2 (and red with probability 0.8). Today the stock closed green. On average, how many days until the stock closes green again (its expected return time to a green day)?

Accepted Answer

The average time to return to a state equals 1 divided by that state's long-run share. Long-run share of green. For a two-state chain the share is \frac{b}{a+b}, with a the leaving probability and b the arriving probability. Leaving green: a = 0.6; arriving at green (from red): b = 0.2: \pi{	ext{green}} = \frac{0.2}{0.6 + 0.2} = \frac{0.2}{0.8} = \frac{1}{4}. Invert: \frac{1}{\pi{	ext{green}}} = \frac{1}{1/4} = 4 	ext{ days}. So green days recur every 4 days on average. Takeaway. Rare states have long return times: a state occupying a quarter of the days is revisited once every four days o…

How long between green days?

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