$$ S_T = S_0 \exp\left(\mu T -\frac{1}{2}\sigma^2 T + \sigma W_T \right) $$

$$ S_{\Delta t - 0} = S_0 \exp\left(\mu (\Delta t - 0) -\frac{1}{2}\sigma^2 (\Delta t - 0) + \sigma (W_{\Delta t} - W_0) \right) $$

$$ S_{2\Delta t - \Delta t} = S_{\Delta t} \exp\left(\mu (2\Delta t - \Delta t) -\frac{1}{2}\sigma^2 (2\Delta t - \Delta t) + \sigma (W_{2\Delta t} - W_{\Delta t}) \right) $$

• General Form

$$ S_{t+\Delta t} = S_t \exp\left[ (\mu - 0.5\sigma^2) \Delta t + \sigma \Delta W_t \right] $$

$$ S_{t+\Delta t} = S_t \exp\left[ (\mu - 0.5\sigma^2) \Delta t + \sigma \sqrt{\Delta t} \, N(0,1) \right] $$

$$ S_b = S_a \exp\left[ (\mu - 0.5\sigma^2) (b-a) + \sigma \sqrt{b-a} \, N(0,1) \right] $$

$$ S_{c} = S_b \exp\left[ (\mu - 0.5\sigma^2) (c-b) + \sigma \sqrt{c-b} \, N(0,1) \right] $$

$$ S_{c} = S_a \exp\left[ (\mu - 0.5\sigma^2) (c-a) + \sigma \sqrt{c-a} \, N(0,1) \right] $$