I meet some problems when calibrating the stochastic model, given in Eq(3.2) and appendix (E.2) of thesis:
https://tspace.library.utoronto.ca/bits ... thesis.pdf
I give a simplified version here.
$$d\lambda_t=\beta(\theta-\lambda)dt+\eta dM$$
where M is a poisson process.
Suppose \({t_1,t_2,...,t_n}\) is a set of observed times with \({t_n<=T}\). The hazard rate is given by \(\lambda_t=\theta+ \sum_{i=1}^{n} \eta e^{-\beta(t-t_i)} \). The log-likelihood function is
$$L=-\theta T+\sum_{i=1}^{n} \{{log(\lambda_{t_i})-\eta \frac{1-e^{-\beta(T-t_i)}}{\beta}}\}$$
The above log-likelihood function seems questionable to me, because it seems that \(L\) can be maximized by simply choosing \( \theta\rightarrow 0, \eta\rightarrow +\infty, \beta\rightarrow +\infty \). This is also what happens when I implement it with matlab fmincon. So do I miss anything? Thank you in advance.