volatility - Continuous delta hedge formula

When we buy a call and continuously delta hedge using some implied volatility $\sigma_i$, what is the formula for our aggregate profit given that the actual realized volatility is $\sigma_r$?

Say $S_0 = 1000, \sigma_i = 0.25, \mu = 0.10$, and the call has expiry a year from now.

How does the formula look like in terms of $\sigma_r$? What happens if $\sigma_r =0$? $> \sigma_i$? $< \sigma_i$?