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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.

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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 $t\in[t_k,t_{k+1}]\,.$

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