One would need access to stock price index data to make definitive statements. There are probably studies doing such analysis. However, I will make some rather safe generalised comments.
Firstly, we should expect differences in behaviour between the gold standard era and the present, as well as between developed and developing markets.
My expertise is with floating currency, developed countries. And the following statements appear safe to say.
- Other than Japan, there has been no extended period of falling CPI (“deflation”) outside of recessions/global crises in the floating currency era. Pre-2020, that deflation was mainly due to falling oil prices, core CPI remained positive.
- Stock markets tend to fall in recessions, for many reasons. Most modern recessions have been associated with financial crises, and leveraged investors needed to cover positions. This means that stock prices can be lower than fundamentals.
- The deflation in Japan was extremely mild, and I would guess that there were some periods of rising stock prices (although the Japanese stock market was in a secular bear market after the bubble pop in the early 1990s) coinciding with deflation. Someone with access to Japanese financial data would have to verify
As such, we have no real guidance for what sort of “deflation” can happen in a floating currency sovereign. However, it is very hard to see what significant difference a deflation of -1% has versus an inflation of 1% for equity valuations (with those rates being similar to historical post-1990s experience). Stocks could rise and fall based on the moods of stock investors. The difference in expected nominal growth rates can easily be counter-acted by lower discount rates.