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<t<t_{k+1} $ 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}],. $

引用自：https://quant.stackexchange.com/questions/69933