No. The exchange processes between the troposphere and the stratosphere are extremely slow. In fact, usually the air in the troposphere and stratosphere don't mix.
However, the air will mix when the tops of giant thunderstorms poke through into the stratosphere. Ozone is an unstable molecule, and easily mixes with other substances, so when it's created in the troposphere, it doesn't last nearly long enough to get transported into the stratosphere. Chlorofluorocarbons (CFCs) and other ozone-depleting compounds last for hundreds of years, and therefore eventually some get into the stratosphere.
At the stratospheric level the increased radiation causes them to break down, and they mix with the ozone. That's why there was a lag from when these gases were first used until the ozone hole was seen, and why there will be another lag before we see the ozone hole disappear. The long life of CFCs and other ozone-depleting compounds also means that even if we could add ozone to the stratosphere, ... more.
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