With the following process: $$dS_t = r S_t dt + σ(t) St dW_t \tag1$$
and
$$ \sigma (t) = 0.1 \ \ \ if \ \ t < 0.5 \\ \sigma (t) = 0.21 \ \ \ otherwise$$
I know the general solution should be : $$S_t = S_0 e^{rt − \frac{1}{2} \int_0^t \sigma (s)^2 ds +\int_0^t \sigma (s) dW_s} \tag2 $$
and if $log(S_t) = Z_t$
$$Z_t = Z_0 + rt − \frac{1}{2} \int_0^t \sigma (s)^2 ds +\int_0^t \sigma (s) dW_s \tag3 $$
My question is how to treat the integration if $t \geq 0.5$? I think I need to split the integration but I am not sure how to compute the integration in the half open interval $t \in [0, 0.5)$. Should the upper bond of the integral for $t \in [0, 0.5)$ be treated as a number with a limit approaching 5?
For computational simulation on discrete time, I know if the volatility is a constant, I can use (4) below for simulation in discrete time. ($\delta W_s$ is just normal distribution of variance $\delta t$) $$Z_{t+\delta t} = Z_t + (rt − \frac{1}{2} \sigma )\delta t +\sigma \delta W_s \tag4 $$ However, I am not sure how to simulate especially when the time progresses from $t < 0.5$ to $t > 0.5$