I have a stochastic process defined by the intensity function $\lambda(t:F_t)$ where $t$ is time and $F_t$ is the filtration process. The stochastic process is self-exciting and models the occurrence of an event. I am interested to predict, at time $t_{i-1}$, when will the next event occur. Hence, I'm interested in finding $E[t_i|F_{t_{i-1}}]$. According to the text I'm reading, there is no closed form for $E[t_i|F_{t_{i-1}}]$. So, for starters, I have to find $t_i$ as a solution to
$$\epsilon_i=\mu(t_i-t_{i-1})-\sum^{i-1}_{k=1}\frac{\alpha}{\beta}\left(1-e^{-\beta(t_i-t_k)}\right)$$ where $t_1,..., t_{i-1} , \alpha, \beta, \mu$ are all known constants and where $\epsilon_i$ is a random variable with exponential(1) distribution. The text suggests that the solution is a non linear function of $\epsilon_1$ of the form $t_i=g(\epsilon_i;F_{t_{i-1}})$ and that is calculated by using the law of iterated expectations as $$E[t_i|F_{t_{i-1}}]=E[E[g(\epsilon_i;F_{t_{i-1}})|\epsilon_i]|F_{t_{i-1}}]$$ What I have done is simply solve the 1st equation (without any of the iterated expectations) and got that $$ \log(t_i)-\beta t_i=\log\left(\epsilon_i+\mu t_{i-1}+(i-1) \frac{\alpha}{\beta}\right)-\log\sum^{i-1}_{k=1}e^{\beta t_k} -\log\mu- \log\left(\frac{\alpha}{\beta}\right)$$ which I solve for $t_i$ by using the Lambert W function and get $$t_i = -\frac{W(-e^{x_i} \beta)}{\beta}$$ Then, I simply randomly generate my $\epsilon_i$ and plug it into my equation to get a simulated time for $t_i$ and repeat the process a large number of times only to take the average of all the generated $t_i$'s. Unfortunately, I seem to be doing something wrong as the values I get for $t_i$ are pretty different from what they should be. Should I go about this in a different way? What should I do about the iterated expectation?