modeling - How to compute the conditional expected value of a geometric brownian motion?

I'm working on a project, and I have to use the cumulative and conditional expected value of the variations of a stock following a Geometric Brownian Motion.

I know that the cumulative is as follows : $$ \mathbb{E}\left[ \mathbb{1}_{ \frac{S_{i+1}}{S_{i}} < z}\right] = \mathbb{P} \left[ \frac{S_{i+1}}{S_{i}} < z \right] = \Phi\left(\frac{\log(z) - (r- \frac{\sigma^2}{2})(t_{i+1}-t_i)}{\sigma \sqrt{t_{i+1}-t_i}}\right) $$

$\Phi$ being the standard normal distribution cumulative function.

But I couldn't find the expression of the conditional expected value : $$ \mathbb{E}\left[\frac{S_{i+1}}{S_i} 1_{\frac{S_{i+1}}{S_i}

Answer

Note that \begin{align*} E\bigg(\frac{S_{i+1}}{S_i}\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg) &=zE\bigg(\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg)-E\bigg(\Big(z-\frac{S_{i+1}}{S_i}\Big)\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg) \\ &=zP\bigg(\frac{S_{i+1}}{S_i}

Alternatively, note that \begin{align*} \frac{S_{i+1}}{S_i} &= e^{(r-\frac{\sigma^2}{2})(t_{i+1}-t_i) + \sigma (W_{t_{i+1}}-W_{t_i})}\\ &=e^{(r-\frac{\sigma^2}{2})(t_{i+1}-t_i) + \sigma \sqrt{t_{i+1}-t_i} \xi}, \end{align*} where $\xi$ is a standard normal random variable. Then $\frac{S_{i+1}}{S_i}