# What kind of interpolation is this?

I have Wiener process $$W_t=\int_0^t\sigma(t)dB(t)$$ where $$B(t)$$ - Brownian Motion and $$\sigma(t)$$ - piecewise constant function. I also take $$t_k where I know the values of $$W_{t_k}$$ and $$W_{t_{k+1}}$$. I found implementation of some kind of interpolation but I don't understand how it is determined. It works as follows:

1. $$D = \sigma^2(t_{k+1})\times t_{k+1} - \sigma^2(t_{k})\times t_{k}$$
2. $$N=\sigma^2(t)\times t -\sigma^2(t_k)\times t_k=\sigma^2(t_k)\times (t-t_k)$$
3. $$W_t = \sqrt{N/D}\times W_{t_{k+1}} + (1-\sqrt{N/D})\times W_{t_k}$$

And generally I would like to know what are the popular methods of interpolation for Wiener Process with stochastic\piecewise constant volatility.

I don't understand why they not just use $$\tag{1} D=\sigma^2(t_k)(t_{k+1}-t_k)$$ which leads to the theoretically correct variance of $$W_t-W_{t_k}$$.
Rewriting (3) gives for the increment over the interval $$[t_k,t]$$ $$W_t-W_{t_k}=\sqrt{N/D}\,(W_{t_{k+1}}-W_{t_k})\,.$$ This has a variance of $$\tag{2} \mathbb E\Big[(W_t-W_{t_k})^2\Big]=\frac{N}{D}\sigma^2(t_k)(t_{k+1}-t_k )=\frac{\sigma^2(t_k)(t-t_k)}{\sigma^2(t_{k+1})\,t_{k+1}-\sigma^2(t_k)\,t_k}\sigma^2(t_k)(t_{k+1}-t_k )\,.$$ From $$W_t=\int_0^t\sigma(s)\,dB_s$$ we should theoretically get $$\tag{3} \mathbb E\Big[(W_t-W_{t_k})^2\Big]=\int_{t_k}^t\sigma^2(s)\,ds=\sigma^2(t_k)(t-t_k)\,.$$ The last equals sign follows from the assumption of piecewise constancy of $$\sigma\,.$$
Obviously if (1) is used instead then (2) and (3) agree for all $$t\in[t_k,t_{k+1}]\,.$$